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bclo*v.
ELEMENTS OF ALGEBRA
THE MACM1LLAN COMPANY NKVV YORK
-
PAI-I.AS
BOSTON CHICAGO SAN FRANCISCO
MACMILLAN & CO, LONDON
LIMITKU HOMBAY CALCUTTA MELUCK'KNK
THE MACMILLAN
CO. OF TORONTO
CANADA,
LTD.
ELEMENTS OF ALGEBRA
BY
ARTHUR
SCJBULIi/TZE,
PH.D.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS, NKW YORK ITNIVEKSITT HEAD OF THK MATHEMATICAL DKI'A KTM EN T, HIH SCHOOL OF COMMERCE, NEW 1 ORK CUT
THE MACMILLAN COMPANY 1917 All rights reserved
COPYRIGHT,
BY
1910,
THE MACMILLAN COMPANY.
Set up and electrotyped.
Published
May,
1910.
Reprinted
February, January, 1911; July, IQJS January, 1915; May, September, 1916; August, 1917.
September, 1910
.
;
;
Berwick & Smith Co. Norwood, Mass., U.S.A.
J. 8. Cushlng Co.
1913,'
PREFACE IN
this
book the attempt
in algebra,
with
all
while
still
made
to shorten the usual course
giving to the student complete familiarity
the essentials of the subject.
similar to the author's to its peculiar aim,
"
While
in
Elementary Algebra,"
many
respects
this book,
has certain distinctive features, chief
which are the following 1.
is
owing
among
:
All unnecessary methods
and "cases" are
omitted.
These
omissions serve not only practical but distinctly pedagogic " cases " ends. Until recently the tendency was to multiply as far as possible, in order to make every example a
social
case of a memorized method.
Such a large number of methods,
however, not only taxes a student's memory unduly but in variably leads to mechanical modes of study. The entire study of algebra becomes a mechanical application of memorized rules,
while the cultivation of the student's reasoning power is neglected. Typical in this respect is the
and ingenuity
treatment of factoring in
methods which are of
many
text-books
In this book
all
and which are applied in advanced work are given, but "cases" that are taught only on account of tradition, short-cuts that solve only examples real value,
manufactured for this purpose, etc., are omitted. All parts of the theory whicJi are beyond the comprehension
specially 2.
of
the student or wliicli are logically
practical
teachers
know how few
unsound are
omitted.
All
students understand and
appreciate the more difficult parts of the theory, and conse-
PREFACE
vi
quently hardly ever emphasize the theoretical aspect of alge bra. Moreover, a great deal of the theory offered in the averis logically unsound ; e.g. all proofs for the sign text-book age
two negative numbers, all elementary proofs theorem for fractional exponents, etc.
of the product of of the binomial 3.
TJie exercises are slightly simpler than in the larger look.
The best way to introduce a beginner to a new topic is to offer Lim a large number of simple exercises. For the more ambitious student, however, there has been placed at the end of the book a collection of exercises which contains an abundance
of
more
difficult
cises in this
work.
book
With very few
differ
bra"; hence either book 4.
from those
may
exceptions
in the
all
the exer
"Elementary Alge-
be used to supplement the other.
Topics of practical importance, as quadratic equations and
graphs, are placed early in the course.
enable students
This arrangement will of time to
who can devote only a minimum
algebra to study those subjects which are of such importance for further work.
In regard
may
to
some other features of the book, the following
be quoted from the author's "Elementary Algebra":
"Particular care has been bestowed upon those chapters in the customary courses offer the greatest difficulties to
which
the beginner, especially problems and factoring. The presenwill be found to be tation of problems as given in Chapter
V
quite a departure from the customary way of treating the subject, and it is hoped that this treatment will materially diminish the difficulty of this topic for young students. " The book is designed to meet the requirements for admis-
sion to our best universities
and
colleges, in particular the
requirements of the College Entrance Examination Board. This made it necessary to introduce the theory of proportions
PREFACE
vii
and graphical methods into the first year's work, an innovation which seems to mark a distinct gain from the pedagogical point of view.
"
By studying proportions during the first year's work, the student will be able to utilize this knowledge where it is most needed,
viz. in
geometry
;
while in the usual course proportions
are studied a long time after their principal application. " Graphical methods have not only a great practical value,
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules.'
This topic has been pre-
sented in a simple, elementary way, and of the
modes of representation given
it is
hoped that some
will be considered im-
provements upon the prevailing methods. The entire work in graphical methods has been so arranged that teachers who wish a shorter course
may omit
these chapters."
Applications taken from geometry, physics, and commercial are numerous, but the true study of algebra has not been sacrificed in order to make an impressive display of sham life
applications. to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt.
McKinley
than one that gives him the number of Henry's marbles. But on the other hand very few of such applied examples are genuine applications of algebra,
nobody would find the length Etna by such a method,
of the Mississippi or the height of Mt.
and they usually involve difficult numerical calculations. Moreover, such examples, based upon statistical abstracts, are frequently arranged in sets that are algebraically uniform, and hence the student is more easily led to do the work by rote
than when the arrangement braic aspect of the problem.
is
based principally upon the alge-
PREFACE
viii
It is true that
problems relating to physics often
offer
a field
The average
pupil's knowlso small that an extensive use of
for genuine applications of algebra.
edge of physics, however, is such problems involves as a rule the teaching of physics by the teacher of algebra.
Hence the
field of
genuine applications of elementary algebra work seems to have certain limi-
suitable for secondary school tations,
give as
but within these limits the author has attempted to
many
The author
simple applied examples as possible. desires to acknowledge his indebtedness to Mr.
William P. Manguse for the careful reading of the proofs and for
many
NEW
valuable suggestions.
YORK,
April, 1910.
ARTHUR SCHULTZE.
CONTENTS CHAPTER
I
PAGB
INTRODUCTION
1
Algebraic Solution of Problems Negative Numbers
1
3
Numbers represented by Letters Factors, Powers, and Hoots
....... ...
Algebraic Expressions and Numerical Substitutions
CHAPTER
15
........ ....
Subtraction
III
...
MULTIPLICATION
Numbers
Monomials
Multiplication of a Polynomial by a
10
22
29
CHAPTER
Multiplication of
15
27
Signs of Aggregation Exercises in Algebraic Expression
Multiplication of Algebraic
7
10
II
ADDITION, SUBTRACTION, AND PARENTHESES Addition of Monomials Addition of Polynomials
6
Monomial
31 31
....
34
35
Multiplication of Polynomials
36
Special Cases in Multiplication
39
CHAPTER IV 46 46
DIVISION Division of Monomials
Division of a Polynomial by a Monomial Division of a Polynomial by a Polynomial Special Cases in Division ix
47
48 61
X
CONTENTS CHAPTER V PAGE
,63
LINEAR EQUATIONS AND PROBLEMS
.....,.
Solution of Linear Equations
Symbolical Expressions
Problems leading
to
55 67
63
Simple Equations
CHAPTER VI FACTORING
Type
76 I.
Type II. Type III. Type IV. Type V. Type VI.
Summary
Polynomials, All of whose Terms contain a mon Factor
Quadratic Trinomials of the Quadratic Trinomials of the
Com77
Form x'2 -f px -f q Form px 2 -f qx + r
The Square of a Binomial x 2 Ixy The Difference of Two Squares Grouping Terms
.
.
.... -f
/^
.
.
.
78
80 83
84 86 87
of Factoring
CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
.
.
Common Factor Lowest Common Multiple
CHAPTER
89
89
Highest
91
VIII 93
FRACTIONS Reduction of Fractions Addition and Subtraction of Fractions
93 97
102
Multiplication of Fractions Division of Fractions
104
Complex Fractions
*
,
*
.
105
CHAPTER IX FRACTIONAL AND LITERAL EQUATIONS
......
112
Literal Equations
Problems leading to Fractional and Literal Equations
108 108
Fractional Equations .
.114
CONTENTS
XI
CHAPTER X
RATIO AND PROPORTION
.........
PAGE
120
Ratio
120
Proportion
121
CHAPTER XI SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE Elimination by Addition or Subtraction Elimination by Substitution Literal Simultaneous Equations Simultaneous Equations involving More than
....
129 130 133 138
Two Unknown
....
140
....
148
Graphic Solution of Equations involving One Unknown Quantity Graphic Solution of Equations involving Two Unknown Quan-
168
Quantities
Problems leading to Simultaneous Equations
CHAPTER
143
XII
GRAPHIC REPRESENTATION OF FUNCTIONS AND EQUATIONS Representation of Functions of One Variable
.
.
164
160
tities
CHAPTER
XIII
INVOLUTION
165
Involution of Monomials
165
Involution of Binomials
166
EVOLUTION
...
CHAPTER XIV 169
Evolution of Monomials
170
.
Evolution of Polynomials and Arithmetical Numbers
.
.
171
.
1*78
CHAPTER XV QUADRATIC EQUATIONS INVOLVING ONB UNKNOWN QUANTITY Pure Quadratic Equations
178
Complete Quadratic Equations Problems involving Quadratics
181
Equations in the Quadratic Character of the Roots
Form
189 191
193
CONTENTS
xii
CHAPTER XVI PAGK 195
THE THEORT OP EXPONENTS Fractional and Negative Exponents Use of Negative and Fractional Exponents
....
195
200
CHAPTER XVII RADICALS
205
206
Transformation of Radicals Addition and Subtraction of Radicals
210
.212
Multiplication of Radicals Division of Radicals
.....
Involution and Evolution of Radicals
219
Square Roots of Quadratic Surds Radical Equations
CHAPTER
214
218 221
XVIII
THE FACTOR THEOREM
227
CHAPTER XIX SIMULTANEOUS QUADRATIC EQUATIONS I.
II.
......
Equations solved by finding x +/ and x / One Equation Linear, the Other Quadratic
III.
Homogeneous Equations
IV.
Special Devices
232
.
.
.
232
.
.
.
234
236 237
Interpretation of Negative Results
and the Forms
i
-,
.
.
241
243
Problems
CHAPTER XX PROGRESSIONS
246
.
Arithmetic Progression Geometric Progression Infinite
24(j
251
263
Geometric Progression
CHAPTER XXI BINOMIAL THEOREM
.
BEVIEW EXERCISE
.
.
.
.
.
.
..
.
.
255
268
ELEMENTS OF ALGEBRA
ELEMENTS OF ALGEBRA CHAPTER
I
INTRODUCTION 1.
Algebra
may
it
arithmetic,
be called an extension of arithmetic. Like numbers, but these numbers are fre-
treats of
quently denoted by problem.
letters,
as illustrated in
the following
ALGEBRAIC SOLUTION OF PROBLEMS 2.
Problem.
is five
The sum
two numbers is 42, and the greater Find the numbers. the smaller number. of
times the smaller. '
x
Let
5 x = the greater number, 6x the sum of the two numbers.
Then and
6x
Therefore,
= 42,
x = 7, the smaller number, 5 x = 35, the greater number.
and 3.
A problem
4.
An
is
a question proposed for solution.
equation is a statement expressing the equality of
quantities; as,
6 a?
two
= 42.
In algebra, problems are frequently solved by denoting numbers by letters and by expressing the problem in the form of an equation. 5.
6.
Unknown numbers
letters of the alphabet
are employed. B
;
are usually represented as, x, y,
1
z,
by the
last
but sometimes other letters
ELEMENTS OF ALGEBRA
2
EXERCISE
1
Solve algebraically the following problems 1.
The sum
numbers is 40, and the greater Find the numbers.
of two
times the smaller.
A man
:
is
four
and a carriage for $ 480, receiving for the horse as for the carriage. much did he receive for the carriage ? 2.
twice as
3.
A
sold a horse
How
much
and
B own
vested twice as
a house worth $ 14,100, and
much
capital as B.
How much
A
has
in-
has each
invested ? 4.
The population
of
South America
is
9 times that of
Australia, and both continents together have 50,000,000 inFind the population of each. habitants.
The
and fall of the tides in Seattle is twice that in their sum is 18 feet. Find the rise and fall and Philadelphia, 5.
rise
of the tides in Philadelphia. 6.
Divide $ 240 among A, B, and C so that A may receive much as C. and B 8 times as much as C.
6 times as
A pole 56 feet high was broken so that the part broken was 6 times the length of the part left standing. .Find the length of the two parts. 7.
off
8.
If
The sum
two
of the sides of a triangle equals 40 inches. sides of the triangle are equal, and each is twice the A remaining side, how long is each side ?
A
9.
The sum
triangle is are equal,
of the three angles of any 180. If 2 angles of a triangle and the remaining angle is 4
times their sum,
how many
degrees are
there in each ?
B
G 10. The number of negroes in Africa 10 times the number of Indians in America, and the sum of both is 165,000,000. How many are there of each ?
is
INTRODUCTION
3
Divide $280 among A, B, and C, so that much as A, and C twice as much as B.
11.
B may
receive
twice as
Divide $90 among A, B, and C, so that B may receive much as A, and C as much as A and B together.
12.
twice as
A
13.
which
is
line 20 inches long is divided into two parts, one of long are the parts ? equal to 5 times the other.
How
A
travels twice as fast as B, and the tances traveled by the two is 57 miles. 14.
sum
of the dis-
How many
A, B, C, and
15.
does
A
much
take, if
B
and
D
as B,
miles did
4
each travel ?
D buy $ 2100 worth of goods. How much buys twice as much as A, C three times as
six times as
much
NEGATIVE NUMBE EXERCISE
2
Subtract 9 from 16.
1.
2.
Can 9 be subtracted from 7 ?
3.
In arithmetic
4.
The temperature
What
is
why
cannot 9 be subtracted from 7 ? "*
\
noon is 16 ami at 4 P.M. it is 9 the temperature at 4 P.M.? State this as an at
of subtraction. 5.
less. 6.
The temperature
8.
4 P.M.
is
7, and
at 10 P.M.
it is
10
expressing the last
below zero) ? What then is 7 -10?
answer 7.
at
What is the temperature at 10 P.M. ? Do you know of any other way of (3
Can you think
of
any other
practical examples
require the subtraction of a greater
which
number from a smaller
one? 7.
Many
greater
practical examples require the subtraction of a one, and in order to express in
number from a smaller
a convenient form the results of these, and similar examples,
ELEMENTS OF ALGEBRA
4
it becomes necessary to enlarge our concept of number, so as to include numbers less than zero.
8. Negative numbers are numbers smaller than zero; they are denoted by a prefixed minus sign as 5 (read " minus 5 "). Numbers greater than zero, for the sake of distinction, are fre;
quently called positive numbers, and are written either with a prefixed plus sign, or without any prefixed sign as -f- 5 or 5. ;
The
fact that a
below zero
thermometer falling 10 from 7 indicates 3
may now
be expressed 7 -10
= -3.
Instead of saying a gain of $ 30, and a loss of $ 90 we may write
is
equal to a
loss of $ 60,
$30 The
9.
-$90 = -$60.
6,
It is convenient for
10.
number
absolute value of a
without regard to its sign. 5 is The absolute value of
is
the number taken
of -f 3 is 3.
many
discussions to represent the
numbers by a succession of equal distances laid off on from a point 0, and the negative numbers by a similar
positive
a line
series in the opposite direction. ,
I
-6
I
-5
lit -4
-2
-3
I
I
I
+\
+2
I
-1
Thus, in the annexed diagram, the line from the line from
to
4,
I
I
+4
4-5
y
I
+6
to 4- 6 represents 4- 5,
1
etc.
left.
equals 4, 5 subtracted from
EXERCISE 1.
3
The addition of 3 is repspaces toward the right, and the subtrac-
4 represents
resented by a motion of "three tion of 8 by a similar motion toward the
Thus, 5 added to
I
+
If in financial transactions
we
1 equals
6, etc.
3
indicate a man's income
by
a positive sign, what does a negative sign indicate ? 2. State in what manner the positive and negative signs may be used to indicate north and south latitude, east and west
longitude, motion upstream
and downstream.
INTRODUCTION 3.
If north latitude
is
indicated by a positive sign, by what
is
south latitude represented ?
is
north latitude represented
4.
If south latitude
5.
What
6.
A
is
5
indicated by a positive sign, by what ?
the meaning of the year 6 yards per second ? erly motion of is
20 A.D. ?
merchant gains $ 200, and loses $ 350. - 350. (b) Find 200
Of an
(a)
east-
What
is
his total gain or loss ? 7.
If the temperature at 4 A.M. is 8 and at 9 A.M. it is 7 what is the temperature at 9 A.M. ? What, therefore,
higher, is 8
- +7? 8. A vessel
sails
journey. 9.
sails
A 22
(6)
11. 12. 13.
14. 15.
16. 17.
26.
from a point in 25 north latitude, and Find the latitude at the end of the
(a)
Find 25 -38.
vessel starts from a point in 15 south latitude, and due south, (a) Find the latitude at the end of the
journey, 10.
starts
38 due south,
(b)
Subtract 22 from
From 30 subtract 40. From 4 subtract 7. From 7 subtract 9. From 19 subtract 34. From subtract 14. From 12 subtract 20. 2 subtract 5. From 1 subtract 1. From
15.
24.
To 6 2 To To 1 From 1 To - 8 To 7 From
25.
Add
18.
19. 20.
21. 22.
23.
add
1.
add
2.
subtract 2.
add
9.
add
4.
1 subtract 2.
1 and 2.
Solve examples 16-25 by using a diagram similar to 10, and considering additions and subtractions as
the one of
motions.
(a)
Which is the greater number lor -1? (b) -2 or -4?
28.
By how much
27.
12.
add
is
:
7 greater than
12 ?
ELEMENTS OF ALGEBRA 29.
Determine from the following table the range of tempera-
ture in each locality
:
NUMBERS REPRESENTED BY LETTERS 11. For many purposes of arithmetic it is advantageous to express numbers by letters. One advantage was shown in 2 others will appear in later chapters ( 30). ;
EXERCISE 1.
2. 3.
and
b
4. 5.
many
If the letter
=
What
the value of
is
the value of 17
c,
= 5?
if c
ifc
5t?
if
a=
6,
= -2?
boy has 9c? marbles and wins 4c marbles has. he ? If a
Is the last
A
marbles,
answer correct for any value of d ? m dollars and lost 11 m
merchant had 20 much has he left ?
8.
What
9.
Find the numerical value
10.
is
4?
6.
that
4
means 1000, what
What is the value of 3 6, if b = 3 ? if b = 4 ? What is the value of a + &, if a = 5, and 6 = 7?
7.
How
t
is
the
sum
of 8 &
If c represents a certain
number ?
and G
how
dollars.
b ?
of the last
answer
if b
= 15.
number, what represents 9 times
INTRODUCTION
if
11.
From 26 w
12.
What is the numerical
1
subtract 19 m.
value of the last answer
if
m = 2?
m = -2? 13.
From 22m
of the answer
if
subtract
m=
1
25m, and
14.
Add
15.
From
16.
Add -lOgand +20 q. From 22# subtract 0.
19.
find the numerical value
2.
13 p, 3p, 6p, and subtract 24 p from the sum.
10 q subtract 20
17.
q.
18.
From subtract 26 Add - 6 x and 8 x.
From
20.
x.
Wp subtract 10^).
What sign, therefore, 140. 21. If a = 20, then understood between 7 and a in the expression 7 a ? 7 a=
is
FACTORS, POWERS, AND ROOTS The
12.
signs of addition, subtraction, multiplication, division, in algebra as they have
and equality have the same meaning in arithmetic. 13.
If there is no sign between
number, a sign of multiplication 6
x a
is
generally written 6 a
Between two (either
x
or
14.
Since 24
=
Similarly,
15.
thus,
A
x n
a letter and a
is
written win.
however, a sign of multiplication has to be employed as, 4x7, or 4 7. ;
written 47, for 47
A product is
two or more
m
letters, or
understood.
figures,
)
4x7 cannot be
;
two
is
means 40
-f 7.
the result obtained by multiplying together
quantities, each of which is a factor of the product. 3 x 8, or 12 x 2, each of these numbers is a factor of 24.
7, a, 6,
is
6 aaaaaa, or a ,
c are factors of 7 abc.
is the product of two or more equal factors called the " 5th power of a," and written a5 " the 6th is power of a," or a 6th.
power
aaaaa
and
;
;
The second power is also called the square, and the third 2 power the cube; thus, 12 (read "12 square") equals 144.
ELEMENTS OF ALQEBEA
8 16.
The
base of a
is
power
number which
the
is
repeated
as a factor.
The base
of a 3
is a.
17. An exponent is the number which indicates how many times a base is to be used as a factor. It is placed a little above and to the right of the base.
The exponent
of
m
6
is
6
n
;
is
the exponent of an
EXERCISE 1.
Write and
2.
72
.
5
find the numerical value of the square of 7, the cube of 6, the fourth power of 3, and the fifth power of 2. Find the numerical values of the following powers :
If
6.
.
42
10.
.
11.
3.
2*.
7.
2*.
4.
52
.
8.
10 6
5.
83
.
9.
I 30
a=3, 6=2, c=l, and 3
10
18.
ci
.
20.
c
19.
b2
.
21.
d\
28.
If
29. 30.
.
.
.
d=^ 22.
23.
O
.
9 .
2
12.
(4|)
13.
(1.5)
.
2 .
14.
25 1
15.
.0001 2
16.
l.l 1
17.
22
.
.
.
+3
2 .
find the numerical values of:
a*. 2
(6cf)
.
3
24.
(2 c)
25.
ab.
.
26.
2
27.
(4 bdf.
at).
= 8, what is the value of a? If m = what is the value of m ? = If 4 64, what is the value of a ? a3
2
-jJg-,
In a product any factor product of the other factors. 18.
In 12 win 8/), 12 19.
8
(i)
A
is
the coefficient of
numerical coefficient
is
is
called the coefficient of the
mw 8p,
12
m is the coefficient of n*p.
a coefficient expressed entirely
in figures. In
17
When
aryx,
17
is
the numerical coefficient.
a product contains no numerical coefficient, 1 1 a, a Bb 1 a*b.
stood ; thus a
=
=
is
under-
INTRODUCTION
9
20. When several powers are multiplied, the beginner should remember that every exponent refers only to the number near which it is placed. 2
3
means 3
aa, while (3
2
)
=3ax
3 a.
= 9 abyyy. 2* xyW = 2-2.2.2. xyyyzz.
9
afty
3
1 abc*
7 abccc.
EXERCISES If
a
= 4, b = 1, c = 2, and x = ^, find the
numerical values of
:
A
21. root is one of the equal factors of a power. According to the number of equal factors, it is called a square root, a cube root, a fourth root, etc. 3
is
6
is
the square root of 9, for 32 = 9. the cube root of 125, for 6 8 = 125.
a
is
the
root of a 5 the nth root of a".
fifth
,
indicated by the symbol >/""; thus Va is the is the cube root of 27, \/a, or more simply the square root of a.
The nth
root
fifth root of a,
Va,
is
Using
(Va) 22.
n
this
= a. The
is
A/27
symbol we
index of a root
root is to be taken. sign. In v/a, 7 23.
is
The
bracket,
[ ]
may is
express the definition of root by the
number which
what
the index of the root.
signs of aggregation are ;
indicates
It is written in the opening of the radical
the brace,
j
j
;
:
the parenthesis,
and the vinculum,
.
( )
;
the
ELEMENTS OF ALGEBRA
10
They are used, as in arithmetic, to indicate that the expres* sions included are to be treated as a whole. Each 10
is
b) is
(a
1],
sometimes read "quantity a
EXERCISE
= 2, b = 3, c = 1, d
If a
+
x (4 -f 1), 10 x [4 by 4 + 1 or by 5.
of the forms 10
to be multiplied
0,
x
10 x
4"+T indicates
that
b."
7
9, find the numerical value of:
1.
Vff.
7.
Val
13.
4(a
+ &).
2.
V36".
8.
-\fi?.
14.
6(6
+ c).
3.
V2a.
9.
4V3~6c.
15.
(c-f-d)
4.
v'Ta.
10.
5Vl6c.
16.
6.
\/c.
11.
aVc^.
17.
6.
V^a6.
12.
2
[6-c]
.
3 .
AND NUMERICAL
ALGP:BRAIC EXPRESSIONS
SUBSTITUTIONS
An
24.
algebraic expression is a collection of algebraic
bols representing
A
25.
some number
monomial or term
separated by a sign (6
+ c + d}
26.
is
or
is
e.g.
;
6 a26
7
Vac
2
an expression whose parts are not as 3 cue2,
9
~* Vx,
o c ^and a monomial, since the parts are a (6 + -f-
A polynomial is an
;
c -f d).
expression containing more than one
term. y,
27.
a2
+
28.
A binomial is 62 ,
3
!^-f\/0-3
and |
-
ft,
and a 4
+ M -f c
4 -f-
d 4 are polynomials.
a polynomial of two terms.
\/a are binomials.
A trinomial
is
a polynomial of three terms.
V3
sym-
-f 9.
are trinomials.
INTRODUCTION In a polynomial each term
29.
is
11
treated as
were con-
if it
tained in a parenthesis, i.e. each term has to be computed before the different terms are added and subtracted. Otherwise operations of addition, subtraction, multiplication, and division are to be performed in the order in which they are written all
from
left to right.
E.g. 3
Ex.
4
_|_
.
5
means 3
4-
20 or 23.
28
Find the value of 4
1.
+5
32
-
*^.
= 32 + 45-27 = 50. Ex.
If a = 5, b = 3, c = 2, d = 0, - 9 aWc + f a b - 19 a 6cd
2.
2 of 6 ab
3
find the numerical value
2
6 aft 2 - 9 a& 2 c + f a 6 - 19 a 2 bcd = 6 5 32 - 9 5 32 2 + ^ 5 8 3 - 19 = 6. 5- 9-9. 6- 9- 2 + I-126- 3-0 = 270 - 810 + 150 = - 390. EXERCISE 8* 3
-
If
.
a=4, 5=3, c=l, d=Q, x=^,
2.
+ 26+3 c. 3a + 56
3.
a 2 -6.
4.
a2
5.
5a2
6.
2 a2
7.
6a2 +4a62 ~6c'
8.
27
1.
a
.
52
3
.
2
find the numerical value of: 9.
2
.
5c6 2 +-6ac3 3
8
3
17c3
-d
a
11.
3a& 2 + 3a2 6-a&c2
-f & -f c
-hl2o;.
s
10.
.
'
-5c
c
2
+-d
2
12.
.
-46c-f2^^ + 3 a& +- 4 6^9 ad. 3
- 5 ax
.
l
-+12a(i
50 a6cd.
4 .
13.
(a
14.
(a -f b)
*15. 16.
a2
+ (a + 6)c 6 (2 + a 2
c
2
-f
).
4a6-fVa-V2^.
* For additional examples see page 268,
2 .
ELEMENTS OF ALGEBRA
12 &
17
18
*
'
8
Find the numerical value of 8 a3
22.
a = 2, 6 = 1. a = 2, 6 = 2.
23.
a =3,
24.
a=3,
21.
25.
26. 27.
6=2.
28.
6 = 4. = = 5. a 3, 6
30.
29.
Express in algebraic symbols 31. Six times a plus 4 times 32. 33.
-f-
6s, if
6 a6 2
:
a = 3, 6 = 3. a = 4, 6 = 5. a =4, 6 = 6. a = 3, 6 = 6. a = 4, 6 = 7.
:
6.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square.
w
cube plus three times the quantity a minus
34.
Six
35.
The quantity a
minus
12 cr6
6
plus 6 multiplied
6,
6,
2 by the quantity a
2 .
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square. 36.
37.
Read the expressions
38.
What kind of expressions are Exs. 10-14
of Exs. 2-6 of the exercise. of this exercise?
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic, geometry, physics, and other sciences. 30.
sible to state
Ex. a, 6,
If the three sides of a triangle contain respectively c feet (or other units of length), and the area of the
and
triangle
then
is
S
square feet (or squares of other units selected),
8 = \ V(a + 6 + c) (a 4- 6 - c) (a - 6 -f c) (6
a
+ c).
INTRODUCTION
15
13
E.g. the three sides of a triangle are respectively 13, 14, 15 therefore feet, then a 13, b 14, and c
=
=
=
and
;
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14.16 1
= 84,
i.e.
the area of the triangle equals
84 square
feet.
EXERCISE
9
The
distance s passed over by a body moving with the uniform velocity v in the time t is represented by the formula 1.
Find the distance passed over by A snail in 100 seconds, if v .16 centimeters per second. A train in 4 hours, if v = 30 miles per hour. b. c. An electric car in 40 seconds, if v = 50 meters per second 5000 feet per minute. d. A carrier pigeon in 10 minutes, if v :
a.
2. A body falling from a state of rest passes in t seconds 2 over a space S (This formula does not take into ac^gt 32 feet, count the resistance of the atmosphere.) Assuming g .
=
(a)
How
far does
a body fall from a state of rest in 2
*
seconds ?
A
stone dropped from the top of a tree reached the ground in 2-J- seconds. Find the height of the tree. How far does a body fall from a state of rest in T ^7 of a (c) (b)
second ? 3.
By
using the formula
find the area of a triangle
whose
(a) 3, (b) 5, (c) 4,
sides are respectively
4, and 5 feet. 12, and 13 inches. 13, and 15 feet.
ELEMENTS OF ALGEBRA
14 4.
If
meters,
the radius of a circle etc.),
the area
square meters,
etc.).
n
If
i
i
6.
2 inches.
(b)
=p
Find by means
(b)
It
represents the simple interest of
years, then
(a)
units of length (inches,
2
square units (square inches, Find the area of a circle whose radius is
(a) 10 meters. 5.
H
is
$ = 3.14
The The
n
interest on
p
dollars at r
fo
in
*
r
or
%>
of this formula
interest
5 miles.
(c)
$800
:
for 4 years at
ty%.
on $ 500 for 2 years at 4 %.
If the diameter of a sphere equals d units of length, the
$=
2
3.14d (square units). (The number 3.14 is frequently denoted by the Greek letter TT. This number cannot be expressed exactly, and the value given above is only an surface
approximation.) Find the surface of a sphere whose diameter equals (a) 7.
8000 miles. If the
(b) 1 inch.
diameter of a sphere equals d
volume
V=
~
:
10
(c)
feet,
feet.
then the
7n
cubic feet.
6
Find the volume of a sphere whose diameter equals: (a) 10 feet.
(b)
3
feet.
(c)
8000 miles.
F
denotes the number of degrees of temperature indi8. If cated on the Fahrenheit scale, the equivalent reading C on the Centigrade scale may be found by the formula y
C
= f(F-32).
Change the following readings (a)
122 F.
(b)
to Centigrade readings:
32 F.
(c)
5
F.
CHAPTER
II
ADDITION, SUBTRACTION, AND PARENTHESES
ADDITION OF MONOMIALS 31.
While
word sum
in arithmetic the
refers only to
the
result obtained
by adding positive numbers, in algebra this word includes also the results obtained by adding negative, or positive and negative numbers. In arithmetic we add a gain of $ 6 and a gain of $ 4, but we cannot add a gain of $0 and a loss of $4. In algebra, however, we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4. Or in the symbols of algebra $4) = Similarly, the fact that a loss of loss of
$2 may be
+ $2.
$6 and a gain
of
$4
equals a
represented thus
In a corresponding manner we have for a loss of $6 and a of
$4
(- $6) + (-
$4) = (-
loss
$10).
Since similar operations with different units always produce analogous results, we define the sum of two numbers in such a way that these results become general, or that
and
(+6) + (+4) = + 16
10.
ELEMENTS OF ALGEBRA
16 32.
These considerations lead to the following principle
:
If two numbers have the same sign, add their absolute values if they have opposite signs, subtract their absolute values and ;
(always) prefix the sign of the greater. 33.
The average
of two
numbers
average of three numbers average of n numbers is the
is one half their sum, the one third their sum, and the sum of the numbers divided by n.
is
Thus, the average of 4 and 8
The average The average
of 2, 12,
(-17)
18.
15
19.
is 0.
3 J.
-
0, 10, is 2.
10
of:
Find the values 17.
is
of 2, '- 3, 4, 5,
EXERCISE Find the sum
4
of:
+ (-14).
+ (-9). + -12.
20.
l-f(-2).
21.
(_
22.
In Exs. 23-26, find the numerical values of a + b 23.
a
24.
a
= 2, = 5,
6 6
= 3, c = = 5, c =
4,
5,
d = 5. d = 0.
-f c-j-c?, if :
ADDITION, SUBTRACTION, a
25.
26. 27.
30.
31.
= -23, c=14, & = 15, c = 0, &
1?
d = l. d=
3.
What number must be added to 9 to give 12? What number must be added to 12 to give 9 ? What number must be added to 3 to give 6 ? C* What number must be added to 3 to give 6? **j Add 2 yards, 7 yards, and 3 yards. }/ Add 2 a, 7 a, and 3 a. \\ Add 2 a, 7 a, and 3 a. -'
28. 29.
= 22, = -13,
AND PARENTHESES
-
'
32. 33.
Find the average of the following 34.
3 and 25.
^
35.
5 and
- 13.
36.
12,
13,
39.
-8
'
and
37.
2, 3,
38.
- 3,
sets of
numbers:
- 7, and 4, - 4, - 5,
13. 6,
- 7,
and
1.
4
F.,
2.
Find the average of the following temperatures 27 F., and 3 F.
:
F.,
40. Find the average temperature of New York by taking the average of the following monthly averages 30, 32, 37, :
48, 60, 09, 74, 72, 66, 55, 43, 34. 41. Find the average gain per year of a merchant, if his yearly gain or loss during 6 years was $ 5000 gain, $3000 gain, $1000 loss, $7000 gain, $500 loss, and $4500 gain. :
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12, 10, -4, 1, 6, 10, 12, 10, 6, 0, - 5, -11 (Centigrade). 42.
:
34.
Similar or like terms are terms which have the same
literal factors, affected
6 ax^y and
7 ax'2 y, or
by the same exponents. 5 a2 &
and
,
or 16
Va + b
and
2Vo"+~&,
are similar terms.
Dissimilar or unlike terms are terms 4 a2 6c and o
4 a2 6c2 are dissimilar terms.
which are not
similar.
ELEMENTS OF ALGEBRA
18 35.
The sum
The sum
of 3
of
two similar terms
x 2 and
x2
is
f
another similar term.
is
x2 .
Dissimilar terms cannot be united into a single term. The indicated by connecting
sum of two such terms can only be them with the -f- sign. The sum The sum
of a of a
and a 2 and
is
a
b
is
-f-
a2
a
-f (
.
6), or
a
6.
In algebra the word sum is used in a 36. Algebraic sum. b wider sense than in arithmetic. While in arithmetic a denotes a difference only, in algebra it may be considered b. either the difference of a and b or the sum of a and The sum
of
2 a&, and 4 ac2
a,
a
is
EXERCISE
2 a&
-|-
4 ac2.
11
Add: 1.
-2 a +3a -4o
2.
ab
7
xY xY 7 #y
12
6.
7.
Find the sum of 9.
\
-f-
2 ,
-f
4 a2,
5 a2
2 wp2 - 13 rap
12
10.
dn
7 a 2 frc
:
-3a
2 a2,
2
1
13 b sx
c
,
11.
2(a-f &),
12.
5l
13.
Vm
-f- ii,
,
3(a-f-6),
5Vm + w,
,
+ 6 af
.
25 rap 2, 7 rap2. 9(a-f-6),
12Vm-f-n,
12(a-f b)
14
AND PARENTHESES
ADDITION, SUBTRACTION, Simplify
19
:
15.
-17c + 15c8 + 18c + 22c3 +c3
17.
3
xyz
3
+ xyz
12 xyz
.
+ 13 xyz + 15 xyz.
Add: 18.
ra
19.
+m """
ZL
n
n
2
21.
2
a a8
x*
**,
22.
m
20.
6
23.
c
^24.
2
^
25.
7
2
-1
i
1
-co*
l^S
26.
mn
27.
xyz
mri
Simplify the following by uniting like terms: 29. 30.
3a-76 + 5a + 2a-36-10a+116. 2a -4a-4 + 6a -7a -9a-2a + 8. 2
2
2
31.
32. 33.
"Vx + y
Vaj + y 2
2 Vi
+ + 2 Va; + / + 3 Va; + y. 2/
Add, without finding the value of each term 34.
5x173 + 6x173-3x173-7x173.
35.
4x9'
-36.
:
10x38 ADDITION OF POLYNOMIALS
Polynomials are added by uniting their like terms. It convenient to arrange the expressions so that like terms may be in the same vertical column, and to add each column. 37.
is
ELEMENTS OF ALGEBRA
20
2 Thus, to add 26 ab - 8 abc - 15 6c, - 12 a& 4- 15 abc - 20 c 5 ab 4- 10 6c 6 c 2 and 7 a&c 4- 4 6c + c 2 we proceed as ,
,
f 110WS:
,
& c~15&c
26 aft- 8
-20c2
-12a&4l5a&c
- 6a& a5c
7
+
Numerical substitution
and
a, 5,
- 3 a -f 4
sum
convenient method for
offers a
To check
the addition of
c
any convenient
assign
ft
-f 5
c, e.g.
c
-f-
= 10
3, therefore the answer
correct.
is
NOTE. While the check is almost certain an absolute test e.g. the erroneous answer ;
equal
Sum.
2
c'
a = 1, 6=2, c = 1, = - 3 + 8 + 5 = 1 0, 2 0-25- c= 2- 4-1 = -3, 4 = 7. a 4- 2 6 + 4 c = 1 +4
numerical values to
then
But 7
ca
26
6c
checking the sum of an addition. 3 a -f 4 1) 4- o c and 4- 2 a 26
the
4
4 be
9a& 38.
6ca
-f-lO&c
to
show any
a406
It is
error,
4c would
not also
7.
In various operations with polynomials containing terms with different powers of the same letter, it is convenient to arrange the terms according to ascending or descending powers 39.
of that letter. 7 4.
x
4 5 x"2 + 7 x* 4 5 -7a &+4a
6 a7
of x.
fi
5 4
is
6c
arranged according to ascending powers 4 7 a&
aW
8
cording to descending powers of
a.
EXERCISE
Add
the following polynomials
1.
2a
2.
9 q 4- 7
364-6 t
5
c, s,
3a 2?
V3. 2z2 -4?/ 2 -f2z 2
0^-9 z
7
- 3 s,
c,
and
and
-3ar -22/2 4z
,
5
46 4-
12
:
4a4-6 12 q
2 ,
4
5 2
a;
J
2c. 4-
2
s.
-3 /- 2z
2 .
* For additional examples see page 259.
2 ,
and
AND PARENTHESES
2i
14d-15e + 2/, 16e + 17/-90, -18/+6y + d,
and
ADDITION, SUBTRACTION, 4.
- 15
d.
-12a 4 15& -20c - 12 6 ~5 a - 5 c ll& -7c -6 4- a -- 1 a 4 1 0, -7ar + 3B -5,and 6. 6 # 4 5 z 4 2 7, 6 # 2
5.
and
2
2
2
2
2
,
2
2
,
,
2
2
.
2
3
2
tf
2
l
^_.Ga 4-3x45. 2
9m 48m 4- 7v/i-f- 6, 5-6 w- 7m - 8 m 2m -12, m 4 6. 8. xy3xz + yz, 2xy + 4:XZ-}-5yz,4:xy xz 6yz, and - 2 #?/ 4 5 a + 4 aft - 5 cr& + 7 6 9. 6a -5a &47a& -4& and 2
2
3
7.
9
3
,
2
and
?ft
?/z.
2
8
s
3
2
,
2
3
,
+lOa 6-ll& 10. a + 1>
10a
8
2
4
11.
a 4ar
13.
4(a
14.
a4
v 15.
.
3
3
^*
12.
-h
-f-
byb !
,
2
c^c
^2
d
a4
1 e, e
/,
+ 50 + 62 - 5a^-6 8
8
,
-f 6)
a
1,
a2
-
2
+
6(a
- (b + c)
6)
-f
a
5 (a
and
- 7^
3 ?/
a2
1,
6)
+ 3,
a,
/ 3 ?
2(6
a.
2iB 8
+ 2y + 2 8
8 ,
and
+ c) + (c -f a),
and
+ a + 1. - 9(a + &) - 12,
and
a2
and
7(a
- 12(a 4- 6) + 14(a 4 6) 4 10. 16. 7 4 5 x*y 2 y?y* 3 xf,
and a 2 4- a.
2 j
3(c -f a),
-f-
+
3 ?
+ 5)
2
2
4
4
a;
2
4 ajy 17.
a;
4
2
o^?/
4 y\
and
6
a;
4
?/.
- VS 4 2 Vc, - Va 4 2 V& 4 6 Vc, - 4 Va - 10 Vc, 4 Vc. a 4 a - a, a 4 a - 1, - a 4 a 4 1, and 1 4 a - a 3a 9 y\ 3 afy - 3 ay 6 afy + 6 ay/ 4 10 and - 3 5 Va
and 5 Vb 18. 19.
2
3
2
3
3
2
.
4
3
8
4
4
a)
?/ ,
,
4
-y -^/. v/20. w* 4 3 m n 4 3 m?i 4 2w - 2n 2
8
3
.
2
3 rz ,
in
8
3
m n 4- 3 mn 2
2
n8
,
and
ELEMENTS OF ALGEBRA
22
3
m
w + 3 m + 2 m, 5
4
21. 3
22.
a
23.
16m 7/-12my
m
24.
3
c,
4
?n
4
-2m+2m
- ra + m, 5
8 ,
and
3
-2
2-fa 3
"27.
4
3
3
1/
,
2
3
,
,
8
n + <w 2 ,4
8
2tn* Sic
2 ,
-f-3f
a
n-2<w +n ,5< 3 2
s
4^4.3^* 2n
s ,
+ n*.
-4-5a-6
-T-8a;-9aj
2 ,
2
and
,
2
+ 9 x + 12 26.
-}-
e,
3
.
l-2aj
25.
d, c
2
8
5y 3
d+e a 6. +d a, and e + 6y - 17 + 4 ?nfy - m 4m ?/-?/
6 -f c
4^ + 3t*n
and
^
6
-f-
2
and 2
6
3
7 m.
a,-
.
a3 -a 4
-a +7a, 2
SM/Z + 2
a:?/
-f
x
y
- 11 xy + 12.
3 a-f^, a s -f3o
bxyz~lx,
2,
and
5+a\
12 xyz, and
$ xy
3^2
SUBTRACTION EXERCISE
13
If from the five negative units three negative units are taken, how 1.
What
main ?
is
1,
1,
therefore the remainder
1,
1,
negative units
many
when
3
is
1,
re-
taken
-5?
from
Instead of subtracting in the preceding example, what to obtain the same result ?
2.
number may be added
The sum
3.
1,
and
total of the units -f 1, -f 1,
What
1, is 2.
ative units are taken
?
-f-
the value of the
1, -f 1,
+ 1,
sum
two neg-
if
1,
If three negative units are taken
?
away 4.
2?
away
is
What
is
When - 3
therefore the remainder is
taken from 2
when
5. What other operations produce the subtraction of a negative number?
6.
If
you diminish a person's
richer or poorer ?
2
is
taken from
?
debts, does
same
result as the
he thereby become
ADDITION, SUBTRACTION, 7.
AND PARENTHESES
23
State the other practical examples which show that the number is equal to the addition of a
subtraction of a negative positive number.
40. Subtraction is the inverse of addition. In addition, two numbers are given, and their algebraic sum is required. In subtraction, the algebraic sum and one of the two numbers is
The algebraic sum is given, the other number is required. called the minvend, the given number the subtrahend, and the required number the difference. Therefore any example in subtraction
may
be stated in a
5 take form e.g. from What 3, may be stated number added to 3 will give 5? To subtract from a the number b means to find the number which added to b gives a. Or in symbols, a-b = different
:
;
x,
x
if
Ex.
1.
From
5 subtract to
Hence,
6
2.
From
(-
Hence,
Ex.
3.
From
3 gives 5
evidently 8.
- 3. 3 gives
to
6)
is
-(-3) = 8.
5 subtract
The number which added
a.
3.
The number which added
Ex.
+b
-(-
5 subtract
3)
5
is
2.
= - 2.
+ 3.
This gives by the same method,
41.
The
results of the preceding examples could be obtained
by the following Principle.
To
subtract, change the sign
of the subtrahend and
add. NOTE. The student should perform mentally the operation of chang8 2 6 from 6 a 2 fc, ing the sign of the subtrahend thus to subtract 6 a 2 6 and 8 a 2 6 and find the sum of change mentally the sign of ;
24
ELEMENTS OF ALGEBRA
42. To subtract polynomials of the subtrahend and add.
Ex.
From _6ar3
-3z + 7 2
-6ar3 -3o2 +7 2 or3 - 3 r*-5o;-f 8
we change the
sign of each term
- 3 x* - 5 x + 8. Check, If x = l = 2
subtract 2 x
-t-
AND PARENTHESES
ADDITION, SUBTRACTION, 41.
Subtract the sum of 2
m and 7 m
from
42.
From 10 a
c
subtract
14 a
12 &
6
-f
25
10m. -f 12 b -f
5
c,
and
check the answer. 43.
From
$
7 x 2 ?/
5
a/ + ?/ subtract
ar -f
7 a 2 ?/ - 5 #?/
3 7/ ,
and check the answer.
-a
3
-7a
44.
From
45.
From 5a-(>& + 7c From 2 x2 8 a?y + 2
46.
2 aa
-f
47.
From mn -f ??/>
48.
From a3
49.
From
+ qt
-f
96
c
2 :c
-f
+
11 c-f 17
6 6
8 o#
2
.
1.
subtract 10 b
c -f d.
s
50.
From
1 -f & take 1 -f b
51.
From
6
52.
From
2 a
53.
54.
From a3 subtract 2 a3 -f- a 2 -j- a From 3 or 2 a:// + 2 subtract
55.
From 5 a 2
56.
From 6 subtract l-t-2a-f3& + 4<7. From 16 + a3 subtract 8 2 a + a2 -f a3 From a 4 - 4 a*& + 6 a & - 4 a^ 4- & 4 subtract a
57.
58.
+3x
4
a;
-f &
-f-
2
-f-
&
12 take 3
take a
&
-f-
b
4 x + 11.
-j- c.
1. #?/
2 y2 .
subtract 2 a 2
+ 2ab
3
?/
?/-'
.
3
ar -f-
2
2 ab
d.
2 ?/
mn -f wp -f- w>t.
mt subtract a
2
take 11 a
take 2
a2
+ 2 a - 7 a - 2.
2 subtract a3
8d 2 ?/
1 subtract
6a
-f-
2y
2 .
.
2
4-
6 a-&
v
59.
2
+ 4 a&
From
3
-f 6
2
4
+4
8
6
4 .
6(a-f- 6)-f- 5(6
+ c)
4(c
+ a)
subtract
7(a-f&)
REVIEW EXERCISES 1.
a
2.
tract 4 x
From
the
sum
of
a
4- b -h c
and
a
subtract
& -f c
_ 6 _ 2 c.
3.
and a 2
From x2
+
the 2 a;
From a3 + a
+ 4.
sum
of x2
4x
-f-
12 and 3 a2
3 #
3 sub-
7.
2 a2
4 a subtract the sum of a 3
-}-
a2
2a
ELEMENTS OF ALGEBRA
26
From
4.
+4x
the difference between a?
subtract
Subtract the
6.
Subtract the
7.
m +4m 9.
sum
x2 + 2 and
of
of 5 a2
sum
6 a
+
sum
from 2 ra
s
Subtract the
the
between 5 a 12.
duce 13.
duce 14.
3
sum
+4
+
and 4
a?
2
cc
7 and
2
from x3
+ a^
2a2 + 3a
4
8
2
4m*
and
+ 7 m.
sum
of #
a3
2
+ f and
+1 +a
a2
4-
3
and a2
2
and 4 a
What expression must 8a3 -2a-7? What
3
m +5 m +6m
of 6
Subtract the difference of a and a
To
+ 58+1
iE
+ 6 + c from a + b + c a +2 y from 2 2
2
2
ar*
10. To the sum of 2a + 66 + 4c and a 2 c. 10 a + 5 b sum of9ci-66 + c and 11.
2
a:
2.
Subtract the
2
8.
+
+5
+ 1.
#
-j-
5.
from 2 a2 + 2 a 5
a?
3
4 6
+a
*/
2 c add the
add the difference
a.
be added to 7 a 3
expression must be added to
+4a
2 to pro-
3a + 56
cto
pro-
~2a-6 + 2c? What
expression must be subtracted from 2 a to produce
-a+6? v
,15.
What must
be added to b
16.
a
17.
a
20.
+
= x +g
6. 6.
A is n years old. n years hence ?
2
m 21.
A
a
+b
c
z,
How
is 2 a years old. years ago ?
c
4^ + 4^ + 2
to produce
=x
find
18.
a
19.
a
2y + z, + 6 + c. 6
+
0?
:
c.
old will he be 10 years hence ?
How
old was he a
b years
ago?
ADDITION, SUBTRACTION,
AND PARENTHESES
27
SIGNS OF AGGREGATION 43.
By using the signs of aggregation, may be written as follows:
additions
and sub-
tractions
a
Hence the
it is
-f ( 4- &
+ d) = a + b
c
c
+ d.
obvious that parentheses preceded by the -f or be removed or inserted according to the fol-
may
sign
lowing principles
:
44. I. A sign of aggregation preceded by the sign -f may be removed or inserted without changing the sign of any term. II.
moved
A
may be resign of aggregation preceded by the sign inserted provided the sign of evei'y term inclosed is
w
changed.
E.g.
a+(b-c) = a +b - c.
+ c) = a =a 6 c) ( 4-= a b c a
o+(
a
45.
6
& -f-
-f- c.
& -f
c.
If there is no sign before the first term within a paren*
thesis, the sign
is
-f-
understood.
a
(b
c)
=a
6 4-
c.
46. If we wish to remove several signs of aggregation, one occurring within the other, we may begin either at the innermost or outermost. The beginner will find it most convenient at every step to
remove only those parentheses which contain
no others. Ex. Simplify 4 a -
f
(7 a
+ 5&)-[-6& +(-25- a^6)]
4a-{(7a + 6&)-[-6&-f(-2&- a~^~6)]} = 4 a -{7 a 6 b -[- 6 b -f (- 2 b - a -
-f-
= 4a sss
4a
7a 12
06 6.
66
2&-a + 6
Answer.
}
.
ELEMENTS OF ALGEBRA
28
EXERCISE 15* Simplify the following expressions 1.
x + (2y-z).
2.
a-(3b
3.
4. 5.
2c).
f
+ (2a -6 + c ). a -(a + 26 -c ). 2a -(4a -26 +c ). a3
3
2
2
3
3
2
2
2
2
11.
2a
2
2
:
a
7.
a -f (a -
8. 9.
2
(-a + 6).
6.
10.
- (a + 6).
6)
a-(- + 6)-f (a-2 b). 2m a
4a-f-
2a;-y
(60;-
?
+ 5a-(7-f 2a )-f (5-5a). 2
13. 14.
(m
-f- 7i
n p) ___
(m
-h jp)
m~n-\-p.
.
15.
m+n
16.
a2
+ [#
(m
?*,)
+ M> + w
(
m -f
ft)-
5
(r
a;)].
17.
22.
18.
a
(6
19.
a
[36+
21.
7
(a
By removing
c)
+ [3 a {3c
6)+
{a
a)}
6a].
-2c].
[a-
parentheses, find the numerical value of
1422 - [271 47.
26 (c
Signs of aggregation
{
271 + (814 - 1422)
may
J
]
:
.
be inserted according to
43.
In the following expression inclose the second and third, the fourth and fifth terms respectively in parentheses,: Ex.
1.
Ex.
2.
last three
Inclose in a parenthesis preceded by the sign terms of
See page 260.
the
AND PARENTHESES
ADDITION, SUBTRACTION, EXERCISE
29
16
In each of the following expressions inclose the last three terms in a parenthesis :
c
+
d.
1.
a-\-l>
2.
2m-n + 2q-3t.
3.
5 a2
4.
4 xy
>
7 x* 4-9 x + 2. - 2 tf - 4 y* - 1.
In each of the following expressions inclose the last three in a parenthesis preceded by the minus sign
terms
:
5.
m
6.
x
7.
p + q + r-s.
2
-27i2 -3^ 2 + 4r/. y
-f-
z
8
_ r)X - 7-fa.
+ d.
EXERCISES
ALGEBRAIC EXPRESSION
IN"
EXERCISE Write the following expressions I.
5^2
.
The sum^)f
m
and
n.
17 :
The
2.
NOTE. The minuend is always the of the two numbers mentioned.
difference of a
3.
The sum
The
difference of the cubes of
5.
The
difference of the cubes of n and m.
6.
The sum
7.
The product The product
8. 9.
10.
II. 12.
of tKe squares of a
and
m
b.
and
of the fourth powers of a
m
and
n.
and
6.
n.
of the cubes of
m and
n.
Three times the product of the squares of The cube of the product of m and n.
m and n.
The square of the difference of a and b. The product of the sum and the difference
Nine times the square of the sum of a and by the product of a and b. 13.
6.
and the subtrahend the second,
4.
of
and
'
first,
of
m and n.
6 diminished
ELEMENTS OF ALGEBRA
30 14.
The sum
square root of
of the squares of
a and
b increased
by the
x.
15.
x cube minus quantity 2 x2 minus 6 x plus
16.
The sum
ference of a and
of the cubes of a,
b,
and
c
6.
divided by the
dif-
d.
Write algebraically the following statements:
V 17.
The sum
of a
and
18.
The
.
difference of the cubes of a
difference of
a and
uct of a and
6,
s
by the difference of a and a 2 and b 2
b multiplied
b is equal to the difference of
and
6 is equal to the square of
plus the square of
b divided
by the
a plus the prod-
b,
-19. The difference of the squares of two numbers divided by the difference of the numbers is equal to the sum of the two numbers. (Let a and b represent the numbers.)
CHAPTER
III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS EXERCISE
18
In the annexed diagram of a balance, let us consider the and JB, and forces produced at by 3 Ib. weights, applied at let us indicate a downward pull at by a positive sign.
A
A
A
3.
By what sign is an upward pull at A represented ? What is the sign of a 3 Ib. weight at A ? What is the sign of a 3 Ib. weight at B ?
4.
If the
1. 2.
two loads balance, what force is produced by the Ib. weights at A ? Express this as a multi-
addition of five 3 plication example. 5.
If the two loads
balance,
taking away 5 weights from
A?
what What,
force is produced therefore, is
5
by
X 3?
6. If the two loads balance, what force is produced by the addition of 5 weights at B ? What, therefore, is 5 x ( 3) ?
7.
If the
two loads balance, what force
ing away 5 weights from
is
produced by tak-
B ? What therefore is 31
(
5)
x(
3) ?
ELEMENTS OF ALGEBRA
32 If
8.
method of the preceding what would be the values of
the signs obtained by the
examples were generally
true,
5x4, 5x(-4), (-5)X4, 9 9.
x
11,
(- 9) x
11, 9 x
(-
(
11),
(-
9)
x (-
11) ?
State a rule by which the sign of the product of
two
fac-
tors can be obtained.
48. Multiplication by a positive integer is a repeated addition; 4 multi44-44-4 12, thus, 4 multiplied by 3, or plied by 3, or
4x3 =
(_4) X The preceding
=
3=(-4)+(-4)+(-4)=-12.
definition,
however, becomes meaningless
the multiplier is a negative number. To take a number 7 times. times is just as meaningless as to fire a gun
if
7
Consequently we have to define the meaning of a multiplicaif the multiplier is negative, and we may choose any definition that does not lead to contradictions. Practical examples^
tion
however, such as given in the preceding exercise, make venient to accept the following definition
it
con-
:
49.
by a negative
Multiplication
integer is a repeated
sub-
traction. 4
Thus,
x(-8) = ~(4)-(4)-(4)=:-12,
(- 4) x
(
~ 3> = -(- 4)-(- 4)-(-4) = +
12.
NOTE. This definition has the additional advantage of leading to algenumbers which are identical with those for positive numbers, a result that would not be obtained by other assumptions.
braic laws for negative
In multiplying integers we have therefore four cases trated
by the following examples
4x3 = 4-12.
:
4x(-3)=-12.
illus-
MULTIPLICATION
We
50.
33
assume that the law illustrated for positive is true for all numbers, and obtain thus
shall
and negative integers the
Law
TJie
of Signs:
product of two numbers with
positive; the product of two
numbers with unlike
(-a)(+6) = -a&; (- a)(-
Thus,
EXERCISE
&)
=+
a&;
6
1.
3.
X(-5). (-7) X (-12).
4.
(-2)X
6.
NOTE. tors
is
If
5.
6.
no misunderstanding
etc.
19
Find the values of the following products
2.
like signs in signs is negative.
:
6x-7. (-2)x9. (-4)X(-15).
possible, the parenthesis
is
about
fac-
frequently omitted.
7. 8. 9.
10. 11.
-5x-3. 4 - 7.
13.
.
_3.(-4J).
_2. -. 3. _2^ -3.
2
14.
(-2) 8 (- 3)
15.
(-4)'.
16.
(-1)
17.
(-10)
18.
-1- -2- -3- -4. +5.
.
.
7 .
4 .
12.
6.-2--f
19.
Formulate a law of signs for a product containing an
even number of negative factors. 20. Formulate a law of signs for a product containing an odd number of negative factors.
If a cal
=
2, b
= 3, c =
1,
x=
0,
and y
= 4,
find the numeri-
values of:
21.
22. 23. 24.
4a6c.
3 a2?/2
2a
2
25. .
6c.
Ua b z
s
x.
aW.
3
8 4
26.
2a6
27.
11 aWcx.
28.
(c#)
c
8 .
.
29.
- (a&c)
2 .
31.
4a -f-26 2 2a + 3&2 -6c*
32.
4 a2 - 2
30.
2
.
f+x
2 .
ELEMENTS OF ALGEBRA
34
Find the numerical value
a8
of 8
a=2,
= -3. 4. 6 =
34.
a=
51.
By
&
33.
1,
- 1 2 a 6 -f 6 aW - 6 if 35. a = 2, 5 = 2. 36. a = 3, 6 = 1. 2
3
,
MULTIPLICATION OF MONOMIALS 3
Hence 2 x 2
Xa
= (a =aa
n
=2
m and
general, if
am
a
a
=2
23
definition,
2 2
x2
n are two positive a
to
(m
52.
known
is
(a
factors) -f n) factors.
X
fl
w
=
2
fl
=2
23 + 5
8 *.
,
.
2
2.
Or
in
-
integers,
m
to
-
fl*"
This
-
= 2- 2 -2
25
and
2 2, 2 2 2 .
/w
+w
a
a
n
to
factors)
.
as
The Exponent Law
of Multiplication
of the factors. 6 aWc x - 7 &*# =(6
The exponent of
:
product of several powers of the same base
is
equal to the
the
sum
oj
the exponents
Ex.
1.
-
2
(a
7)
a8 )
8
.
.
(ft
c
&*)
d*.
53. In multiplying a product of several factors by a number, only one of the factors is multiplied by the number. Ex. 2. 2 x (2* 5 7 2 )= 26 5 7 2 25)
.
.
.
Ex. 3. 4 x (2
=8
.
25, or 2
.
100,
i.e.,
EXERCISE
200.
20
Express each of the following products as a power 1. 2.
m*.m a
3
-
a
4
B.
(^
12
2
.
6
3
3
4
9.
-(a?
+
4
2/)
-(aj
U U .(-12) .12
+
3a-7abc.
13.
4- (2- 25-
7).
78
-
5
2
3
(-a)
5
.
:
.
127 9
6" 127 U
.
7 .
.(-7).7.
.
Perform the operation indicated 12.
10.
2/).
a
5
3
127
6.
.
.
8 2/)
5
5.
.
9
(a6) -(a5)
11.
3
4.
.
7.
+
2 -2
3.
.
2
2
14
3
:
14.
2(7.3-5),
16.
5(7-11.2).
IB.
50(11-2.3).
17.
2(14.50-3),
M UL TIPLICA TION 18.
7(6- f-
19.
20.
11(3- 6- A). 4 aft -5 aft 2
21.
19
27.
2).
28. 29.
.
22. '
23.
mV
lla
25.
(
2
5 2
3 ft
.
3 tfy 2z*.
(- 2
aft
4
).
2
3
)
(-
6
e
- 7 w'W (-8 n^W). 2
5
2
9 afy
32.
(- 4
33. 34.
/).
35.
8
3
aft
31.
C a 2ftc).
(-
4
-7p*q r*.-7pqt. _4aft.-4a#.
30.
c-(-4a ftc ).
aft
6 e/ a
2 ran4
-
- 5 xy 19 aW
24.
26.
5
35
ftc
2ac).
(
2 a3 ?/ ). 2
(
a 2 ft 3) 2
.
3
ax /) 2 4 1 (- 3 win ) .
(2
.
(-
2 a2 ) 3
.
MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL we had to multiply 2 yards and 3 inches by 3, the would obviously be 6 yards and 9 inches. Similarly the for quadruple of a 4 2 b would be 4 a -f 8 If
54.
results
ft,
= (a + 26)+(a + 2
ft)
-f (a 4-
2
ft)
+ (a + 2
ft)
55. This principle, called the distributive law, is evidently correct for any positive integral multiplier, but we shall assume it for any number.
Thus we have
in general
a(b 56. tet^m
-f c)
= ab
+ac.
To multiply a polynomial by a monomial, multiply each by the monomial.
- 3 a2 6(6 a*bc + 2 be -
1)
=-
18 a 4 6 2 c
EXERCISE
- 6 a2 62c -f 8 a2 6.
21
Find the numerical values of the following expressions, by first multiplying, and then adding :
1.
2(5-fl5-f25).
3.
2.
6(104-20430).
4. 7.
3(124342). 2(645410).
5.
12(| + 1
4 i).
17(10041042). 23(10004100420). 6.
ELEMENTS OF ALGEBRA
36
Express as a sum of several powers 8.
9.
10.
5(5 2
6 (6
+
+
52
5 7 ).
+6 +6 2
7 3 (7 3 -f-7
2
3
+7
:
4 13 (4 9
11.
-4
5
-4).
12.
).
10
).
Perform the multiplications indicated: 13.
2
14.
~2mn(m +n -p ).
m(m-hn -\-p). 2
2
aW(
2
2
16.
4 %Pq\
17.
-5 x\-
aW + 3 a
5
2
pqr + 5 pr 5 x2 - 5 x-
- 6 a6).
19.
5
20.
c(- ^ c + 2 - 2 mn(9 mV - 5 w*V -f 7 wn).
21.
3
7 a 6
2
2
2
?/
6 c
7).
5).
- 3 aftc).
22. 23.
24.
By what
-: expression must
be multiplied to give
2
26.
4o; -f7a;asa product. Express 3a^ Find the factors of 3x + 3 y + 3z.
27.
Find the factors of 6
or*
28.
Find the factors of 2
ofy
25.
2
-f
3
3 a4
x* -f
4
arty
+8
- 60 a&
2
.
4 a; .
29.
Find the factors of 5 a
30.
Find the factors of 6 ary - 3 x2y 2 + 3
6
10
aft.
xy.
MULTIPLICATION OF POLYNOMIALS 57.
x
polynomial may be written as a monomial by inb by within a parenthesis. Thus to multiply a the distributive and write + z,we y z) apply (a b) (x
Any
closing
+y
law.
(a
it
- 6) (x -f y
z)
= x(a = (ax
b)
+ y(a
z(a
b)
bx) -f (ay
by)
by
az
b)
(az
+ bz.
bz)
M UL TIP LICA TION
37
58. To multiply two polynomials, multiply each term of one by each term of the other and add the partial products thus formed.
The most convenient way of adding the partial products is to place similar terms in columns, as illustrated in the following example
:
Ex.1. Multiply 2 a - 3
b
by a
5
b.
2a-3b a-66 2 a - 3 ab 2
2 a2
-
10 ab
-
13 ab
+ 15 6 2 + 15 6 2
Product.
59. If the polynomials to be multiplied contain several powers of the same letter, the work becomes simpler and more symmetrical by arranging these expressions according to either ascending or descending powers.
Ex.
2.
Multiply 2
+ a -a- 3 a 3
2
by 2 a
Arranging according to ascending powers
a2 + l.
:
Check. 2
a
- 3 a 2 + a8
a 4- 4.a
- 3 a 2 + a8 - 2 a2 6 a8
-
60.
Examples
2"
a2
-7
a2
a = =-
I 1
=2
a 2
If
-f
2 a*
+ a8 + 3 - a6 4 a 8 + 5 a* - a6 *
in multiplication can be checked
=2 by numerical
substitution, 1 being the most convenient value to be substituted for all letters. Since all powers of 1 are 1, this method tests only the values of the coefficients
and not the values of
the exponents. Since errors, however, are far more likely to occur in the coefficients than anywhere else, the student should
apply this test to every example.
ELEMENTS OF ALGEBRA
38
EXERCISE 22* Perform the following multiplications and check the results
+ 2y).
17.
(2 x*
36).
18.
(4ra
3.
(5c-2d)(2c-3d).
19.
4.
(2w
1.
(2s
3y)(3a?
2.
(4a-f 76)(2tt
3n)(7m
+ 8n). 3<7).
5c)(2a-6c).
22.
(6p
6.
(2
7.
(9m-2n)(4m + 7tt).
8.
(aj-f6y)(aj
9.
(6i-7n)(llJ-n).
-f-
-f-6<7)(5^)
a
23.
a-l)(2a?-fl).
24.
7y).
25. 26.
-2) (3 A: -1).
(13
(6xy + 2z)(2xy
12.
(8r-7*)(6r-39.
28.
(2m
13.
(llr + l)(12r
1).
29.
(a
12)(a?^2-|-l).
30.
(4 a
15. 16.
(rcya
(a&c 2
4 2).
+ 7)(2a&c-3).
(a -|-2a
6
2
33.
(4a
35.
(m?n?p
36.
(x
37.
(a
OQ OO.
//)4 lA/
QQ O7.
I
27.
31.
+ 2)(a-3). 2
l)(ra-f 2).
2
10.
14.
-f-ra
(a
11.
A;
2
20.
21.
5.
- 4) (x + 1).
x
2
- 1 - 2m)(l -m). 2
36)
.
2 .
I)
(6a~7)
32.
2 .
(6a&c-5) -3a6-f-2)(2a6~l).
2
-^
2
2 .
mnp -f- 4) (mnp 4- 2).
+ & + 1-f a^faj -1). 2
-
'
//j.2 ^/
40.
(m-fn)(m-4- n)(m
41.
(a-^-26)
n)(m
n).
8 .
* For additional examples see page 261.
;
MUL TIPLICA TION
39
SPECIAL CASES IN MULTIPLICATION 61.
The product
two binomials which have a common term.
of
The product of two binomials which have a common term equal to the square of the common term, plus the sum of the two unequal terms multiplied by the common term, plus the product 62.
in
of the two unequal terms. 6 ft) (5 a 9 ft) is equal to the square of the common term, 25 a 2 , (5 a plus the sum of the unequal terms multiplied by the common terms, i.e. 16 ft) (5 a) 75 ab, plus the product of the two unequal terms, i.e. ( 2 Hence the product equals 25 a'2 54 ft 2 . 75 ab -f 54 ft .
=
+
EXERCISE Multiply by inspection 1.
2. 3.
23
:
+ 2) (a -f 3). (a-3)(a + 2).
15.
(a
16.
(a
_3)(a _4).
17.
(ra- n)(w-f w). 2 5 b z) (a2 -f 4 (a
(a;
(a
-9) (a + 9).
+ 3) (a -7).
6.
(-!)(* -5).
19.
(a5
6.
(p-12)(p + ll).
20.
(a
7.
(wi
21.
(100
8.
(6
9.
10.
+ 9)(m+9).
-7)
3
(a
-8).
+2) (100 + 3). 1) (10 + 2). + 5) (1000 + 4).
(10+
(2,-25)(y+4).
(1000
+ 60)(f-2).
24.
(100-1) (100
(J
(*- !!)(
12.
(a
13.
2^*-12)(ajy
3
22.
11.
+ 21).
- 2 6) (a -f 6). -2 (a 6) (a -3 6).
+
2 6) (a
(1000
26.
102 x 103.
27.
1005x1004.
28.
99
X
+ 2).
-2) (1000 + 3).
25.
102.
14.
(a
29.
Find two binomials whose product equals
6).
2 ).
2
23.
-12) (6 -f- 13).
ft
- 4). (ofy* -f 3) (tfy*
18.
4.
2 a?
3x
+ 2.
ELEMENTS OF ALGEBRA
40
Find two binomial factors sions
of each, of the following expres-
:
30.
ar'-Sz +
31.
33.
+ 6 a + 8. 7 a + 10. w 2 ro - 15.
63.
Some
32.
a
G.
35.
w + 2 w - 15. m2_ 3m _ 4
36.
n2
2
34.
2
a2
2
37.
p
2
10ii+16.
-p- 30.
special cases of the preceding type of examples
deserve special mention
:
II.
III.
(a-
Expressed
in general language
:
sum of two numbers
is equal to tlie square I. 77ie square of the of the first, plus twice the product of the first and the second, plus
of the second.
the square
square of the difference of two numbers is equal to the square of the Jirst, minus twice the product of the first and the II.
71ie
second, plus the square of the second.
is
III.
The product of the sum and
equal
to the difference
the difference
of two numbers
of their squares.
The student should note that the second type
(II) is only a
special case of the first (I). Ex.
(4
x3
+
2
7
i/
the product of the
second,
i.e.
49
2 is )'
equal to the square of the
.
y*.
Multiply by inspection
2. 3.
+ 6) (a + 2)
2
(a
(a
-a)
.
2
2
16 y* t plus twice
,
EXERCISE
1.
first, i.e.
and the second, i.e. oft x 3 y'2 plus the square of the Hence the required square equals 16 xP -f- 66 s; 8j/ 2 + 49 y4 first
.
24
:
4.
(a-2)
5.
+ 3)
6.
(p
(*-5)
a
J
.
7.
<J>-7)
.
8.
(a-26)
.
9.
(x+3i/)
2
2
.
2 .
2 .
MULTIPLICATION 10.
2
12.
(a
19.
(6afy
2
20.
(m -f
n)(ra
21.
(2m + 3)(2m-3).
-3) 2 2
2
22.
(
tt
-5)
26
(^
2 .
2 .
2
34.
104 2
35.
(1000
+ 2) (100 -2).
29.
(a
-I)
2
x*+2xy+y\ a 2 -2a6 + &
43.
2
44.
.
Pind two binomial sions
-5c )
Z
-
)
2 (5 r -f 2
2
)
5
2 .
.
2
37.
991
38.
(20
39.
2
99x101.
m
46.
n 2 -f4n+4.
49.
a 2 + 10 ab -f 25 b\
-2m-hl.
5
).
2
22
42.
45.
).
.
+ 5)(5+a). (m -27i )(m + 2n 2
2
J
2
-f-
.
1)
.
998x1002. :
47.
n*-6n+9.
48.
a 2 -8a6+166 2
.
factors of each of the following expres-
:
51.
50.
y?-f.
54.
9a -496
55.
16aW-25.
64.
The product
2
a2 -9.
52.
2 .
of
m
2
16.
56.
25 a
57.
9 a2
4
53.
62
-25n
2
-9.
- 30 ab + 25 6
2 .
two binomials whose corresponding terms
are similar.
By
.
.
Extract the square roots of the following expressions 2
2
2 2
2
2
2/
r*-2t )
2
41.
(4 a 6
2
(5
.
2 2
2
28.
2
+ 3z)
2
-11 # )
.
36. ,998
.
(2a#
+ 11 -2 (5 r
30.
).
(2a6-c)
17.
2
-
27.
+ 5).
2
(100
2
w )-
2
16.
18.
.
(a;
+ 1) (100 + 2)
103
2 2
25.
31.
33.
.
.
(^-.ll^X^+lly
40.
2
-7& )
2
24.
32.
(6a
2
-f 7). 2
(100
(3p -9)
2
-5)(c d
(c-d
2
15.
.
-7)(a 2
23.
G> +5g)*.
14.
.
2
2
13.
(2x-3yy. 2 (4a-36)
11.
41
actual multiplication,
3x 5x
we have
+
2y 4y
2xy-Sy*
ELEMENTS OF ALGEBRA
42
The middle term
of the result is obtained
by adding the These products are frequently called the cross products, and are represented as
product of 5 x
2 y and
4y
3 x.
follows:
or
Wxy-12xy Hence in general, the product of two binomials whose corresponding terms are similar is equal to the product of the first two
sum of the
terms, plus the
cross products, plus the product of the
last terms.
EXERCISE Multiply by inspection
(2a-3)(a + 2). (3m + 2)(m-l).
1.
2. 3.
4. 5.
6.
8. 9.
2
(2x y
10.
(6
(5a-4)(4a-l). (4s + y)(3-2y).
11.
(-
12.
(5a?
13.
(10
2
i-
(x
65.
2 2 2 2 (2a 6 -7)(a & +
(2m-3)(3m + 2).
(5a6-4)(5a&-3).
7.
25
:
5
x 2 -3 6 s). ) (2
2 ft
The square
(a 4- &
5).
+ z )(ary + 2z ). 2
2
+ 2) (10 4-3). (100 + 3)(100 + 4).
of a polynomial.
+ c) = a + tf + c 2
14.
2
2
2
,-f
2 a&
-f
2 ac
+ 2 &c.
7%e square of a polynomial is equal to the sum of the squares of each term increased by twice the product of each term with each that follows it.
The student should note
that the square of each term is the while always positive, product of the terms may have plus or
minus
signs.
M UL TIPLICA TION EXERCISE Find by inspection
n+p)
(m-f
2.
(x-y+z)*.
3.
(a
+ 6-5)
6.
.
.
(,i-2&-c)
5.
(u-4& +
2 .
3c'.
Find the square root 11.
s?
12.
m
2
7.
2
4.
26
:
2
1.
43
(.r
_
8.
(2a-36 + 5c)
9.
4y
(3
2 .
s-f n)
2 .
10.
of
:
+ y + z + 2xy + 2yz + 2 xz. 2
2 -+-
n2
z
2 "-f-
jp
-f 2
mn
2 ?wp
2 np.
13.
66. In simplifying a polynomial the student should remem. ber that a parenthesis is understood about each term. Hence, after multiplying the factors of a term, the beginner should inclose the product in a parenthesis.
Ex.
Simplify (x + 6) (a
- 4) - (x - 3) (x - 5). x
=
-4) =
-
Check.
+ 6)( - 4) - (>-.3)(z- 5) = (7 - 3) - (= [ Xa + 2 - 24] - [a? - 8 x + 15] - X2 + 2 x - 24 - y? + 8 - 1 5 = 10 - 39. = 10 x - 39. -
(
2
If
1,
20
a;
a;
EXERCISE
= - 29.
27
Simplify the following expressions, and check the answers !.
2.
6(a
-2)-6. 6~2(a + 7). 5.
2(m 3
n)
3.
4(* + 2)-5(-3).
4.
4(aj-2)-h3(-7).
3(m + n)H- (m
n).
+ 6 )-2(6 + &)~(&4-& ). 2
2
8
6.
3(6
7.
(m-f n)(m+2)-3m(n + m).
8.
(a-2)(a-3)~(a-l)(a-4).
:
ELEMENTS OF ALGEBRA
44 9.
10. 11. 12.
4(m + 2)
+ 5(w
5)(oj-2)
(a?
3)
(a;-
- 2) + (n - 7) (n + 4) - 2 (n* - 2)
13.
(n -f 5) (w
14.
6(p+2)-7(p-9)-2(i> + l)(p-l).
15.
16.
x- 2 y)(3 x -f 2 y) - (4 - y) (a3 (a -f 6) - 4 (a + &) (a -f 2 6) + (a (5
2
17.
18.
19.
20. 21.
22.
2
(a
-fa-f
1)
(
a - 1)
- (a + 1) (a - 1). 8
CHAPTER
IV
DIVISION is the process of finding one of two factors and the other factor are given. The dividend is the product of the two factors, the divisor the given factor, and the quotient is the required factor.
Division
67.
if
their product
Thus by
-f
12
to divide
3 gives
by
But
12.
Since
68.
+
3,
this
-f
we must find
number
a
-
it
the ;
number which
hence
_
12 r +3
multiplied
=4.
= -f ab = ab b = ab b = ab, b
-f-
a
and
4
-f b
-fa
_a
is
is
-f-
=+b
follows that
4-a
ab
a ab
a
Hence the law
69.
multiplication
Law
70.
a8 -5- a5
Or
=a
of
3 ,
:
is
the same in division as in
It follows from the definition that Exponents. X a5 a8
=
for a 3
in general, if
greater than
of signs
Like signs produce plus, unlike signs minus.
m n, a
-f-
.
m
and n are positive integers, and m ~ n an = a m a" = a'"-", for a <
45
m
is
ELEMENTS OF ALGEBRA
46
71. TJie exponent of a quotient of two powers with equal bases equals the exponent of the dividend diminished by the exponent
of the divisor.
DIVISION OF MONOMIALS
2x y
7 3 72. To divide 10x y z by number which multiplied by number is evidently
Therefore,
*
the quotient
Hence,
coefficient is the quotient
and whose
sign,
found
literal
,
6
we have
2 ,
2x*y
z
of their part
the
find
This
= - 5 a*yz.
of two monomials
in accordance with the
to
gives 10 x^ifz.
is
a monomial whose
preceded by the proper
coefficients,
quotient of their law of exponents.
is the
literal
parts
73. In dividing a product of several factors by a number, only one of these factors is divided by that number. Thus (8 12 20)-?-4 equals 2 12 20, or 8 3 20 or 8 12 5. -
-
.
-
EXERCISE Perform the divisions indicated
28
:
'
2
.
3.
-39-*- 3. 15
2
5.
3 19 -j-3
10.
12
5
38 '
35
13
-j-2
(3 -
7
7'
-4* 4
.
2
-2 4 )^(3 4 .2 2). 56
'
2V
14
11.
y-ffl-g
15
3
(2
68
.3*.5 7 )-f-( 2
36 a '
-5.25
''
5 11
9
'
12
4.
'
3"
76-H-15.
'
-12 a
2abc
-56aW
-42^ '
.
-
.
'
UafiV
DIVISION lg
-^1^. 16 w 7
7i
20>
_Z^L4L.
01
-240m
9
14 132 a V* 1
10 iy.
*
40
3J)
23.
2 (15- 25. a ) -=- 5.
24.
2
)
22.
fl
6c
/5i.
120m-
(7- 26 a
47
-f-
c
25.
13.
(18
26.
.
5
.
2a )-f-9a. 2
(
DIVISION OF POLYNOMIALS BY MONOMIALS
To divide ax-}- fr.e-f ex by x we must find an expression which multiplied by x gives the product ax + bx -J- ex. 74.
But
+ b e) ax + bx + ex. + bx -f ex = a 4- b +
x(a
TT
aa?
Hence
-\-
,
.
.
c.
a?
To divide a polynomial by a monomial, cfc'wde each term of the dividend by the monomial and add the partial quotients thus formed.
3 xyz
EXERCISE
29
Perform the operations indicated 1.
2. 5.
fl
o.
_5* + 52)
(5*
-5.
52
.
(G^-G^-G^-i-G (11- 2
+
11 -3
+ 11
18 aft- 27 oc
9a
-14gV+21gy Itf
97 .
:
4
-25 -2 )^-2
3.
(2
4.
(8- 3
-5)-*- 11.
Q y.
2
.
+8- 5 + 8-
5a5 +4as -2a
7) -*-8.
2
-a
15 a*b
-
12
aW + 9 a
3a
2
2
ELEMENTS OF ALGEBRA
48
22
,
4,
m n - 33 m n s
4
2
-f
55
mV
- 39 afyV + 26 arVz 3
- 49 aW + 28 a -W - 14 g 6 c 4
,
15.
2 (115 afy -f 161 afy
16.
(52
17.
(85 tf
afyV - 39
- 69 4
oryz
a;
4
3 ?/
- 23 ofy
4
-5-
)
- 65 zyz - 26 tf#z) 3
- 68 x + 51 afy - 34 xy* -f 1 7 4
4
23 x2y.
-5-
13 xyz.
2
a;/)
?/
-f-
- 17
as.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL 75.
Let
2 a 2 -f 3 a, divide
it
4
- 12 -f 6 a - 20 a 3
be required to divide 25 a
6a3 -20a
2
-f
25a-12
2
by
descending powers of
a, or, arranging according to
2 by 2a -
The term containing the highest power of a in the dividend (i.e. a 8 ) is evidently the product of the terms containing respectively the highest power of a in the divisor and in the quotient. Hence the term containing the highest power
If
the product of 3 a and 2
2
4 a
+
3, i.e.
of a in the quotient is
6 a3
12 a 2
-f
9 a, be sub-
8 a 2 -f 16 a tracted from the dividend, the remainder is 12. This remainder obviously must be the product of the divisor and the rest of the quotient. To obtain the other terms of the quotient we have
therefore to divide the remainder,
8 a2
-f-
16 a
12,
2 by 2 a
4 a
+
3.
We
consequently repeat the process. By dividing the highest term in the new dividend 8 a 2 by the highest term in the divisor 2 a 2 we obtain ,
the next highest term in the quotient. 4 by the divisor 2 a2 4 a Multiplying
4,
8 a2
-I-
16 a
no remainder. Hence 3 a
12,
4
is
+ 3, we
obtain the product
which subtracted from the preceding dividend leaves the required quotient.
DIVISION The work
usually arranged as follows
is
- 20 * 2 + 3 - 12 0aa2 + a3
-
25 a {)
8 a? 4- 16
-
12
_
a
a-
I
I
49 :
2 a2
-
8 a
+3
4 a 4
12
76. The method which was applied in the preceding example may be stated as follows 1. Arrange dividend and divisor according to ascending or :
descending powers of a common letter. 2. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient. 3.
Multiply this term of the quotient by the whole divisor, and
subtract the result 4.
Arrange
the same order as the given new dividend, and proceed as before.
remainder in
expression, consider 5.
the dividend.
from
the
as a
it
Continue the process until a remainder zero is obtained, or of the letter according to which the dividend
until the highest poiver
was arranged
is less
than the highest poiver of the same
letter in
the divisor.
77.
Checks.
Numerical substitution constitutes a very con-
venient, but not absolutely reliable check. An absolute check consists in multiplying quotient and divisor. The result must equal the dividend if the division
was
exact, or the dividend diminished by the remainder division was not exact.
Ex.
1.
Divide 8 a3
-f
8 a
- 4 + 6 a - 11 a 4
2
by 3 a
Arranging according to descending powers, 6 a4
+ 8 a8 -
6 a4
+
11
a2
-11
a2
-
a'
-f
8a
4
4 a3 12 a 8
3 a2
3
2
+ +
8 a 2 a
-4 + 6a - 4
I
3 a
-2
2 a8
-f
4 a2
,
,
if
- 2. ^ _ _ ,
= a _+ 2 .
the
,
7-r-l,
=
7
ELEMENTS OF ALGEBRA
50 Ex.
2.
Divide a4
- 46 -6a6 4
3
2
9
-f-
6
2 l by 26 -3a& + a
2
.
Arranging according to descending powers of
a
6 a36
- 3 a8
fr
-f
9 a2 6 2
-f
2 a2 6 2 2
-46*
I
|
we have
a,
a2
-
8 ab
a*
-
3 ab
+ 2 6^ - 2 62
-46*
- 3 a^ + 9a 2 6 - 6 ab 8 2
+ 6 a& a - 4 6 4 - 2 a^a + 6 aft - 4
ft*
Check.
The numerical
example since larger
number
it
substitution a
=
1,
=
&
1,
cannot be used in this
Hence we have
renders the divisor zero.
either to use
for a, or multiply. 2 - 8 ab + 2 & 2 ) ( a _ 3 ab - 2 6 2 ) (a = [(a2 - 3 aft) + 2 62 ] [(a2 - 3 a&) - 2 62 ] = (a 2 -3 aft) 2 -4 6* = a2 - 6 8 6 + 9 a2 6 2 - 4 5*.
EXERCISE
30 *
Perform the operations indicated and check the answers
2.
(jf_2y-15)-i-
3.
2 (15 a
4.
- 46 a# -f 16 ) _ 26 mn 4- 5 n ) (5 m 2
i/
2
2
-5-
-*-
(5
a-
(m
5 w).
5. 6.
7.
(6^-53^ + 40)^(6^-5).
8.
(56
2 a; -f-
19 x
-15) --(8
-3).
9.
10.
11. 12.
13.
2 (25 a
- 36
2 ft
)
-j-
(5
a
-f-
* See page 263.
6
6)
:
a
DIVISION
51
+ 23a& + 20)-*-(2a& + 6). (8xy + lo-22x' y)-+(2x y-3). - 11 a + 9 a - 2) (3 a - 2). (3 a 13 m + 47 m + 35 w (1 (5 m -f
14.
(6a
2
&
2
2
2
15.
3
16.
2
-f-
2
v/17.
3
-f-
18.
(a? s
19.
(aj
)
-8)
-*-(
(81
1) .
2).
-3aj-2)-^(oj-2).
m + 1 - 18 m 4
20.
-5-
2
)
-f-
G
(1
m -f 9 m
2
).
SPECIAL CASES IN DIVISION 78.
Division of the difference of two squares.
Since
(a -f b) (a
,
a I.e.
b
V)
=a
2
b
2 ,
.
a
-f b
the difference of the squares of two numbers is divisible of the two numbers.
by the difference or by the sum Ex.l.
EXERCISE
31
Write by inspection the quotient
of
--
2
x c
3
2
c
6
'
1
v7
-^.
+ 3*
169 a<6 2
f
ISVft-Qc 8 64
'
- 81 c8
'
a2 -166 2 '
:
'
10
a?
-1 '
ELEMENTS OF ALGEBRA
52
Find exact binomial divisors of each expressions 9.
10.
,
of
the following
:
w a
4
4
-!.
-b.
11.
aW
12.
a;
12
13.
-r/
14.
36 a4 ?/ 4
121a
- 49.
100
-9&
2 .
f
1. 16 .
16
15.
a
16.
1,000,000-1.
-100ry.
CHAPTER V LINEAR EQUATIONS AND PROBLEMS 79.
The
first
member
or left side of an equation
is
that part
The secof the equation which precedes the sign of equality. ond member or right side is that part which follows the sign of equality. Thus, in the equation 2 x 0. second member is x
+
x
4
9,
80. An identity is an equation of the letters involved.
the
first
which
is
member
is
2 x
+
4,
the
true for all values
a2 6 2 no matter what values we assign to a Thus, (a + ft) (a b) and b. The sign of identity sometimes used is = thus we may write ,
;
(rt+6)(a-ft)
=
2
-
2
b'
.
81. An equation of condition is an equation which is true only for certain values of the letters involved. An equation of condition is usually called an equation. .r
-f9
= 20
is
true only
when
a;
hence
=11;
it
is
an equation
of
condition.
A set of numbers which when substituted for the letters an equation produce equal values of the two members, is said to satisfy an equation. 82.
in
Thus x
12 satisfies the equation x
the equation x
=
+
1
13.
x
20,
y
=
7 satisfy
13.
An
equation is employed to discover an unknown num(frequently denoted by x, y y or z) from its relation to
83.
ber
y
known numbers. 63
ELEMENTS OF ALGEBRA
54
an equation contains only one unknown quantity, an^ unknown quantity which satisfies the equation is
If
84.
value of the
a root of the equation. 9
85.
To
86.
A numerical
tities are
=
- 2. 87. A
solve
an equation
is
20.
to find its roots.
equation is one in
which
all
expressed in arithmetical numbers
the
known quan
as (7
;
x) (x -f 4)
2
a;
literal
equation
known
is
quantities as x -f a letters
= bx
;
A
88.
first
one in which at least one of the
is
expressed by a letter or a combination of c.
linear equation or
which when reduced
A
+2=
a root of the equation 2 y
is
an equation of the first degree is one form contains only the
to its simplest
power of the unknown quantity;
9ie
as
2
= 6#-f7.
linear equation is also called a simple equation.
The process
89.
of solving equations depends upon the
lowing principles, called axioms
fol-
:
sums are
1.
If equals be added
2.
If equals be subtracted from equals, the remainders are
to equals, the
equal.
equal. 3.
If equals be multiplied by equals, the products are equal.
4.
If equals be divided by equals, the quotients are equal.
5.
Like powers or
NOTE.
= 0x5, 90.
Axiom
4
is
of equals are equal.
like roots
not true
but 4 does not equal
if
the divisor equals zero.
Transposition of terms.
A
its sign.
x + a=.b. Consider the equation b Subtracting a from both members, x
right
0x4
term may be transposed from
one member to another by changing
I.e.
E.g.
5.
a.
(Axiom
2)
the term a has been transposed from the left to thQ
member by changing
its
sign.
LINEAR EQUATIONS AND PROBLEMS x
Similarly, if
Adding a
to both
The result is first member to
a
b
+ a.
Consider the equation Multiplying each member by
Ex.
1,
x-\-
a=
x
a
When x =
changed
6-fc. b
(Axiom
3)
6# = 4x + l + 6. 4x 1 + 6. 2 x = 6. x = 3. (Axiom
4)
c.
Qx
3.
The first member, The second member, Hence the answer, x =
To
be
a?
Subtracting 4 x from each term, Uniting similar terms, Dividing both members by 2,
93.
=
may
SOLUTION OF LINEAR EQUATIONS 1. Solve the equation Qx 5 = 4 -f 1.
Adding 5 to each term,
Check.
1)
if
91. The sign of every term of an equation without destroying the equality.
92.
(Axiom
we had transposed a from the member and changed its sign.
the same as the right
= b.
x
members,
55
6a-5 = 18-5 = 13. 4-fl = 12-fl = 13 3, is correct.
solve a simple equation, transpose the
unknown terms
member, and the known terms to the second. Unite similar terms, and divide both members by the coefficient of the
to the first
unknown
quantity.
Ex.2. Solve the equation (4 Simplifying,
Transposing, Uniting,
- 8, -f If y
Dividing by Check.
y) (5
y)
= 2 (11
3 y)
+ #*.
20 - 9 y + y2 = 22 - 6 y -f y\ - 9 y + 6 y = 20 -f 22. 3 y - 2 y= f .
The first member, (4-y)(6- y) = C4 + })(5-f The second member, 2(11 - 3 y) + y 2 = 2(11 +
1 4 = 26 i + | = 26 -f f = 26$
i)^ V= 2)
JI
ELEMENTS OF ALGEBRA
56 Ex.
Solve the equation | (x
3.
= \ (x + 3).
Simplifying,
\x
2-^x-fl.
Transposing,
x
=2 = 3. x = 18.
Dividing by If
Cfcecfc.
The The
x
-f-
1
.
x
Uniting,
it
4)
J,
x
= 18.
member right member left
{(x (x
- 4) = + 3) =
\
x 14 x 21
= 7. = 7.
NOTE. Instead of dividing by \ botli members of the equation \ x would be simpler to multiply both members by 0.
= 3,
BXEECISB 32* Solve the following equations by using the axioms only 1.
5# = 15+2a;.
2.
7a?
3.
3
a;
4.
7
a;
a?.
a?.
Solve the
following
check the answers 9.
10.
11.
6.
Xx 7 = 14. 4a + 5 = 29.
7.
17
8.
7
5.
= 5a?+18. = 60 -7 = 16 + 5
equations
a?
a;
:
+ 16 = 16 + 17. 3 = 17 3 a?
a?.
by transposing,
etc.,
and
:
- 17 + 4y = 36. = 2 ?/- 7. 12. 9 9a? = 7 13. 13 y -99 = 7 y- 69. 13a? 3a?. 14. 3-2 = 26-4. 24-7y = 68-lly. 15. 17 + 5a;-7a: = 39-4a; + 22. -50. 16. 17 -9 x + 41 = 12 -8 17. 14y = 59-(24y + 21).
4y
11
?/
a?
18.
19.
20.
87-
21.
9(5 x -3)
22.
6(3
a?
= 63.
-3)= 9(3
+ 24) = 6 (10 x + 13). + 7(3 + 1) =63.
v23. 7 (6 x a;
-16).
24.
7
* See page 264.
a;
aj
LINEAR EQUATIONS AND PROBLEMS 25.
73-4* = 13*~2(5*-12).
26.
6(6a;-5)-5(7a>-8)=4(12-3a5) + l.
27.
7(7 x
28.
y
+ 1) -8(7-5
a?)
57
+24 = 12 (4 - 5) + 199. a?
-
29. 30. 31.
5)
(as
- 1) (a
(a?
(a;
+ 3) = - 5) =
(a;
(a;
-7) (a; + 4).
+ 7)
(.7;
- 3) + 14.
.32.
33.
34.
(aj-
35. 36.
37. 38. .
39. 40.
41.
- 12) (2 + 5) - (2 + 6) (4 - 1 0) = 0. - 7) (7 x + 4) - (14 x + 1) + 7) = 285 + 21 a* (z + 2) -(a-5) :=2. - 3) + - 4) (x + I) + (x + 2) = (x 2(* + l) -(2J-3)( + 2) = 12. - 2) (M - 3) - 5(2 u - 1) (u - 4) + 4 w - 14 = 0. (6 u =5 44. | +6= |aj (4
t
t
1
t
(5 x
(a?
2
2
2
2
2
2
.
(a?
2
2
*
42.
a?
-Jaj.
43.
SYMBOLICAL EXPRESSIONS 94.
Suppose one part of 70 to be
a?,
and
let it
be required to
If the student finds it difficult to answer find the other part. this question, he should first attack a similar problem stated in arithmetical
numbers
the other part.
Hence
if
one part
only,
e.g.
:
One part
Evidently 45, or 70 is
a?,
the other part
of 70 is 25
;
find
25, is the other part. is
70
x.
WJienever the student is unable to express a statement in algebraic symbols, he should formulate a similar question stated in arithmetical numbers only, and apply the method thus found to the algebraic problem.
ELEMENTS OF ALGEBRA
58 Ex.
What must
1.
be added to a to produce a sum b ?
Consider the arithmetical question duce the sum of 12 ?
The answer is 5, or 12 7. Hence 6 a must be added
Ex. If 7
x
2.
-f-
:
What must
to a to give
y yards cost $ 100
;
find the cost of one yard.
if
x
-f
y yards cost $ 100, one yard will cost
EXERCISE
2. 3.
4. 5. 6.
7. 6. 9.
10.
5.
--
$> 100 yards cost one hundred dollars, one yard will cost -
Hence
1.
be added to 7 to pro-
100
dollars.
33
By how much does a exceed 10 ? By how much does 9 exceed x ? What number exceeds a by 4 ? What number exceeds m by n ? What is the 5th part of n ? What is the nth part of x ? By how much does 10 exceed the third part of a? By how much does the fourth part of x exceed b ? By how much does the double of b exceed one half Two numbers differ by 7, and the smaller one
Find the greater one. 11. Divide 100 into two
parts, so that
one part equals
of c ? is
p.
a.
13.
Divide a into two parts, so that one part Divide a into two parts, so that one part
is b.
14.
The
two numbers
is
d,
and the
Find the greater one. 15. The difference between two numbers Find the smaller one. greater one is g.
is
c?,
and the
12.
smaller one
16. 17. is
a?
2
?
difference between
is 10.
is s.
What number divided by 3 will give the quotient a? ? What is the dividend if the divisor is 7 and the quotient 2
LINEAR EQUATIONS AND PROBLEMS 18.
What must
19.
The
59
be subtracted from 2 b to give a?
smallest of three consecutive numbers
is a.
Find
is x.
Find
the other two. 20. The greatest of three consecutive the other two.
21. is
A
A
is
# years
A
is
and
B
is
How many years
y years old.
B?
older than
22.
old,
numbers
y years
old.
How
old was he 5 years ago ?
How
old will he be 10 years hence ? If A's age is x years, and B's age is y years, find the of their ages 6 years hence. Find the sum of their ages
23.
sum
5 years ago. 24.
A
has ra dollars, and B has n dollars. amount each will then have.
If
B
gave
A
6
dollars, find the
25.
How many
26.
A has
a
cents are in d dollars ? in x dimes ?
dollars, b dimes,
and
How many
c cents.
cents
has he ? 27.
A man
had a
dollars,
and spent
How many
5 cents.
cents had he left ? 28. A room is x feet long and y feet wide. square feet are there in the area of the floor ?
29.
and 3 30.
and 4
Find the area of the
floor of
feet wider than the one
a room that
Find the area of the feet
floor of a room that is 3 feet shorter wider than the one mentioned in Ex. 28.
A
33. 34.
35.
2 feet longer
mentioned in Ex. 28.
31. rectangular field is x feet long and the length of a fence surrounding the field. 32.
is
How many
What What What What
?/
feet wide.
is
the cost of 10 apples at x cents each ?
is
the cost of 1 apple
is is
Find
x apples cost 20 cents ? the price of 12 apples if x apples cost 20 cents ? the price of 3 apples if x apples cost n cents ? if
ELEMENTS OF ALGEBRA
60
36. If a man walks 3 miles per hour, how many miles he walk in n hours ?
man walks
wil\
r miles per hour,
how many
miles will
38. If a man walks n miles in 4 hours, he walk each hour ?
how many
miles does
37.
If a
he walk in n hours
?
If a man walks r miles per hour, in how many hours he walk n miles ?
39.
will
40.
How many
miles does a train
move
in
hours at the
t
rate of x miles per hour ? 41.
x years ago
A
was 20 years
A
cistern
A
cistern can be filled
How
old.
old
is
he
now ?
by a pipe in x minutes. What fraction of the cistern will be filled by one pipe in one minute ? 42.
43.
alone
in
fills it
is
filled
by two pipes. The first pipe x minutes, and the second pipe alone fills it in
y minutes. What fraction of the cistern will be second by the two pipes together ? 44.
Find 5
45.
Find 6
48.
Find
49.
The numerator
by
If
3.
-.50.
a;
% % %
of 100 of
47.
Find x
% %
per
of 1000. of 4.
of m.
m is the
The two
-46. Find a
a.
x.
filled
of a fraction exceeds the denominator
denominator, find the fraction.
digits of a
number
are x and
y.
Find the
number.
To express in algebraic symbols the sentence: " a exceeds much as b exceeds 9," we have to consider that in this by statement "exceeds" means minus ( ), and "by as much as" Hence we have means equals (=) 95.
b
as
a exceeds
b
a
b
by as much
=
as c exceeds 9. c
-
9.
LINEAR EQUATIONS AND PROBLEMS Similarly, the difference of the squares of a
a2
-
by 80 equals the excess of a over i<5
=
80
a3
Or,
-
-}-
80.
80.
2
(a
b increased
'
2
b'
-b ) + 80 = a
2
and
61
8
-80.
cases it is possible to translate a sentence word by in algebraic symbols in other cases the sentence has to be changed to obtain the symbols.
In
many
word
;
There are usually several different ways of expressing a symbolical statement in words, thus:
a
b
= c may
be expressed as follows
The
difference between a
a exceeds b by c. a is greater than b by b is smaller than a by
The
excess of a over b
EXERCISE
:
and
b is
c. c.
is c, etc.
34
Express the following sentences as equations
2.
The The double
3.
The sum
1.
double of a
is
of a and 10 equals 2
One
third of x equals
5.
The
difference of x
6.
The double
7.
8.
c.
x.
c.
and y increased by 7 equals
a.
of a increased
by one third of b equals 100. Four times the difference of a and b exceeds c by as as
d exceeds
The product
diminished by 90 b divided by 7. 9.
:
10.
of x increased by 10 equals
4.
much
c.
9.
of the is
sum and the difference of a and b sum of the squares of a and
equal to the
Twenty subtracted from 2 a
subtracted from
a.
gives the
same
result as 7
ELEMENTS OF ALGEBRA
62
Nine
10. 11.
12.
as
much below a
#is5%of450.
100
x
14.
50
is
6
%
of m.
20, express in algebraic
a;
as 17
13.
is
is
If A's age is 2 x, B's age
16.
4
is
A is twice as old as B. A is 4 years older than
(a) (b)
is
x%
x % of is
above
a.
3x
symbols
a.
of 700. 15.
m is x %
of n.
and C's age
10,
is
:
B.
Five years ago A was x years old. (d) In 10 years A will be n years old. (e) In 3 years A will be as old as B is now. (c)
->.,*(/)
(g) (Ji)
Three years ago the sum of A's and B's ages was 50. In 3 years A will be twice as old as B. In 10 years the sum of A's, B's, and C's ages will be 100.
If A, B, and C have respectively 2 a, 3 1200 dollars, express in algebraic symbols
17.
x
4-
a;
-700, and
:
(a)
A
(6)
If
has $ 5 more than B.
A
gains
$20 and B
loses
$40, they have equal
amounts. If each
(c)
money (d)
A
and
(e)
If
B
18.
5x
man
gains $500, the
sum
of A's, B's,
and C's
will be $ 12,000.
B together have $ 200 less than C. pays to C $100, they have equal amounts.
A sum of money consists of x dollars, a second sum. of 30 dollars, a third sum of 2 x + 1 dollars. Express as
equations of the (a) 5 :
(b) (c)
(d)
(e)
third
00
% a%
x c/ of a % of 4
%
sum
sum equals $ 90. sum equals $20. the first sum equals 6 % of the third sura. the first sum exceeds b % of the second sum by first
of the second
of the first plus 5
%
of the second plus 6
%
equals $8000.
x % of the
first
equals one tenth of the third sum.
of the
LINEAR EQUATIONS AND PROBLEMS
63
PROBLEMS LEADING TO SIMPLE EQUATIONS The simplest kind of problems contain only one unknown number. In order to solve them, denote the unknown 96.
number by x (or another letter) and express the yiven sentence as an equation. The solution of the equation (jives the value of the unknown number. The equation can frequently be written by translating the sentence word by word into algebraic symbols in fact, the ;
equation is the sentence written in alyebraic shorthand.
Ex.
much
Three times a certain number exceeds 40 by as Find the number.
1.
as 40 exceeds the number.
Let x = the number. Write the sentence in algebraic symbols. Three times a certain no. exceeds 40 by as much as 40 exceeds the no. = x x
3x
-40
40-
3z-40:r:40-z.
Or,
3x
Transposing, Uniting, 3 x or 60 exceeds 40
Check.
+ x = 40 + 40. 4 x = 80. x = 20, the required
by 20
;
number.
40 exceeds 20 by 20.
Ex. 2. In 15 years A will be three times as old as he was 5 years ago. Find A's present age.
= A's present age. verbal statement (1)
Let x
The
(1) In 15 years
+
x
(2)
A
will
16
may be expressed in symbols (2). be three times as old as he was 5 years ago.
=
3
(x
Simplifying,
Transposing,
x
Dividing,
In 15 years
Check.
=3
x
3x
16
-
p)
15.
-23 =-30. x= 15.
Uniting,
30
x
x+16 = 3(3-5). 15. x + 15 = 3 x
Or,
A
will
be 30
;
6 years ago he was 10
;
but
10.
NOTE. The student should note that x stands for the number of and similarly in other examples for number of dollars, number of
years,
yards, etc.
ELEMENTS OF ALGEBRA
64 Ex.
W
56
3.
is
what per cent
= number
Let x
of 120 ?
of per cent, then the
Uldbe
=
66
-*-. 300
| x x
or,
Dividing,
Hence
120,
56.
= 46f. % of
40
5(5 is
120.
EXERCISE 1.
- 2. 3.
4.
problem expressed in symbols
What number added
35
to twice itself gives a
sum
of
39?
Find the number whose double increased by 14 equals Find the number whose double exceeds 40 by 10.
44.
Find the number whose double exceeds 30 by as much
as 24 exceeds the number. 5.
A
original 6.
number added number.
42 gives a
to
sum
equal to 7 times the
Find the number.
47 diminished by three times a certain number equals 2. Find the number.
twice the number plus 7.
Find 8.
A will
Forty years hence
be three times as old as to-da3r
.
his present age.
Six years hence a
12 years ago.
How
old
is
man will be he now ?
twice as old as he was
9. Four times the length of the Suez Canal exceeds 180 miles by twice the length of the canal. How long is the Suez
Canal?
what per cent of 500 ? % of what number?
10.
14
is
11.
50
is
12.
What number
4
is
7
%
of
350?
Ten times the width of the Brooklyn Bridge exceeds 800 ft. by as much as 135 ft. exceeds the width of the bridge. 13.
Find the width of the Brooklyn Bridge. 14. A train moving at uniform rate runs in 5 hours 90 miles more than in 2 hours. How many miles per hour does it run ?
LINEAR EQUATIONS AND PROBLEMS
A
15.
and
$200, and as
How
15.
16.
B
A
have equal amounts of money. If A gains A have three times as much
B
will loses $100, then ? dollars each has many
and
B
have equal amounts of money.
$200, B will have lars has A now? 17.
A has A
give to
to
65
five
$40, and
times as
B
much
has $00.
make A's money
If
A
gives
How many
as A.
How many dollars
equal to 4 times B's
must
money
B
dol-
B
?
A man
wishes to purchase a farm containing a certain He found one farm which contained 30 acres too many, and another which lacked 25 acres of the required number. If the first farm contained twice as many acres as 18.
number
of acres.
the second one,
how many
acres did he wish to
buy
?
19. In 1800 the population of Maine equaled that of Vermont. During the following 90 years, Maine's population increased by 510,000, Vermont's population increased by 180,000, and Maine had then twice as many inhabitants as Vermont. Find
the population of Maine in 1800.
97. If a problem contains two unknown quantities, two verbal statements must be given. Ill the simpler examples these two
statements are given directly, while in the more complex probWe denote one of the unknown
lems they are only implied.
numbers (usually the smaller one) by
x,
and use one of the
given verbal statements to express the other unknown number in terms of x. The other verbal statement, written in algebraic
symbols,
Ex. 14.
1.
is
the equation, which gives the value of
One number exceeds another by
8,
x.
and their sum
Find the numbers.
The problem consists of two statements I. One number exceeds the other one by II. The sum of the two numbers is 14. :
F
8.
is
ELEMENTS OF ALGEBRA
66
Either statement may be used to express one unknown number in terms of the other, although in general the simpler one should be selected. If
we
Let x
and
select the first one,
= the
smaller number, 8 the greater number.
Then x -+- = The second statement written the equation ^
/
,
#4a;
Simplifying,
,
o\
(o?-f 8)
+
Transposing,
in algebraic
symbols produces
= 14. = 14. -i
<
a-
-f
8
x
-f
x =14
8.
= 6. = 3, the smaller number. 8 = 11, the greater number.
2x
Uniting, Dividing,
a?
x
-j-
Another method for solving this problem is to express one unknown quantity in terms of the other by means of statement II viz. the sum of the two numbers is 14. ;
Let
the smaller number.
x
Statement
=
x
14
Then,
the larger number.
in
I
expressed symbols is (14 x) course to the same answer as the first method.
x
= 8,
Ex. 2. A has three times as many marbles as B. 25 marbles to B, B will have twice as many as A. The two statements I.
A
II.
If
are
Use the simpler statement, x 3x
Then,
To
If
ot
A gives
:
has three times as many marbles as B. A gives B 25 marbles, B will have twice as
terms of the other. Let
which leads
viz. I, to
express one
many as A. unknown quantity
in
= B's number of marbles. = A's number of marbles.
express statement II in algebraic symbols, consider that by the
exchange Hence,
A will lose, x 3x
4-
and
B
will gain.
= B's number of marbles after the exchange. 26 = A's number of marbles after the exchange.
26
LINEAR EQUATIONS AND PROBLEMS x
-f
25
x x
+
25
Therefore, Simplifying,
Transposing,
= 2(3 x = 6x
Qx
67
(Statement II)
25). 50.
25
60.
- 5 x - - 75. x = 15, B's number of marbles. Dividing, 3 x = 45, A's number of marbles. Check. 45 - 25 = 20, 15 + 25 = 40, but 40 = 2 x 20.
Uniting,
'
*
98.
.
*
The numbers which appear
in the equation should
be expressed in the same denomination. of dollars to the number of cents, the
always
Never add the number number of yards to their
price, etc.
Ex. 3. Eleven coins, consisting of half dollars and dimes, have a value of $3.10. How many are there of each ? The two statements are I. The number of coins II. The value of the half :
is 11.
dollars
and dimes
is
$3.10.
= the number of dimes, then, x = the number of half dollars.
Let
x 11
from
I,
Selecting the cent as the denomination (in order to avoid fractions),
we
express the statement II in algebraic symbols.
-)+ 10 x = 310.
Simplifying,
50(11 660 50 x
Uniting,
Dividing,
Check.
= 310. - - 550 -f 310. 40 x - - 240. x = 6, the number of dimes. 11 x = 5, the number of half dollars. cents, 6 half dollars = 260 cents, their sum
50 x
Transposing,
6 dimes
= 60
+ +
10 x
10 x
is
.$3.10.
EXERCISE
v
1.
Two numbers
the smaller. v,
2.
differ
by 44, Find the numbers.
Two numbers
differ
by
36
and the greater
60,
and their sum
is five
is 70.
times
Find
the numbers.
w'3.
The sum of two numbers is 42, and the Find the numbers.
6 times the smaller.
greater
is
ELEMENTS OF ALGEBRA
68 4.
One number
is
six
times another number, and the
greater increased by five times the smaller equals 22. the number.
Find
5.
Find two consecutive numbers whose sum equals 157.
6.
Two numbers
differ
by
and twice the greater exceeds Find the numbers.
39,
tnree times the smaller by 65. 7.
The number
of volcanoes in
Mexico exceeds the number
of volcanoes in the United States by 2, and four times the former equals five times the latter. How many volcanoes are in the 8.
United
A
States,
and in Mexico
?
cubic foot of iron weighs three times as much as a If 4 cubic feet of aluminum and
cubic foot of aluminum.
2 cubic feet of iron weigh 1600 foot of each substance. 9.
find the
Ibs.,
weight of a cubic
Divide 20 into two parts, one of which increased by
3 shall be equal to the other increased by
9.
A's age is four times B's, and in 5 years A's age will be three times B's. Find their ages. 10.
11. Mount Everest is 9000 feet higher than Mt. McKinley, and twice the altitude of Mt. McKinley exceeds the altitude of
Mt. Everest by 11,000
mountain 12.
Two
as the larger one.
A
is
the altitude of each
vessels contain together 9 pints.
one contained 11 pints more,
much
What
feet.
?
it
If the smaller
would contain three times as
How many
pints does each contain ?
14 years older than B, and B's age is as below 30 as A's age is above 40. What are their ages ? 13.
is
much
A
line 60 inches long is divided into two parts. Twice 14. the larger part exceeds five times the smaller part by 15 inches. How many inches are in each part ? 15.
On December
longer than the day.
21, the night in
How many
Copenhagen
lasts 10 hours
hours does the day last ?
LINEAR EQUATIONS AND PROBLEMS
problem contains three unknown quantities, three One of the unknown num-
If a
99.
69
verbal statements must be given. bers is denoted by x, and the other of x
two are expressed in terms by means of two of the verbal statements. The third
verbal statement produces the equation. Tf it should be difficult to express the selected verbal state-
ment
directly in algebraical symbols, try to obtain
it
by a
series of successive steps.
Ex.
A, B, and C together have $80, and B has three as A. If A and B each gave $5 to C, then
1.
times as
much
sum of A's and B's money would exceed much as A had originally.
three times the as
money by
The three statements
are
C's
:
A, B, and C together have $80. II. B has three times as much as A. III. If A and B each gave $5 to C, then three times the sum of A's and B's money would exceed C's money by as much as A had originally. I.
x
the
number
of dollars
A
3 x
the
number
of dollars
B
has,
the
number
of dollars
C
has.
Let
According to
and according
II,
to
I,
80
4
x
=
has.
To
express statement III by algebraical symbols, let us consider the words ** if A and B each gave $ 5 to C."
x 8x
90
first
= number of dollars A had after giving $5. = number of dollars B had after giving $5. 4 x = number of dollars C had after receiving $10. 5
5
Expressing in symbols Three times the sum of A's and B's money exceeds C's money by A's 3 x ( x _5 + 3z-5) (90-4z) = x. original amount. x = 8, number of dollars A had. The solution gives :
3x
80 Check.
If
respectively.
A
4x
= 24, = 48,
number
of dollars
B
had.
number
of dollars
C
had.
to C, they would have 3, 19, and 68, or 66 exceeds 58 by 8.
and B each gave $ 5
8(8
+ 19)
ELEMENTS OF ALGEBRA
70
A
man spent $1185 in buying horses, cows, and Ex. 2. sheep, each horse costing $ 90, each cow $ 35, and each sheep $ 15. The number of cows exceeded the number of horses by and the number of sheep was twice as large as the number How many animals of each kind did he buy ?
4,
of horses and cows together.
The I.
IT.
III.
three statements are
:
The total cost equals $1185. The number of cows exceeds the number of horses by 4. The number of sheep is equal to twice tho number of horses and
cows together. Let x
then, according to II,
and, according to III, 2 (2 x -f 4) or 4 x
x 4
+
8
90 x
Therefore,
85 (x 15 (4 x
and,
Hence statement 90 x Simplifying, 90
Transposing, Uniting, Dividing,
Check.
-j-
I
may
+ 4) +
8)
the
= the
number of horses, number of cows,
= the number of sheep. = the number of dollars spent for horses, = the number of dollars spent for cows, = the number of dollars spent for sheep
be written,
+ 35 (x +-4) -f 15(4z-f 8) = 1185. + 35 x 4- 140 + (50 x x 120 = 185. 90 x -f 35 x + GO x = 140 20 + 1185. 185 a = 925. x = 5, number of horses. x -f 4 = 9, number of cows. 4 x -f 8 = 28, number of sheep.
x
1
1
5 horses, 9 cows, and 28 sheep would cost 6 x 90 -f 9 + 316 420 = 1185; 9 -5 = 4 ; 28 2 (9 5).
28 x 15 or 450
=
+
EXERCISE
x 35
-f
+
37
Find three numbers such that the second is twice the first, the third five times the first, and the difference between the third and the second is 15 1.
2. first,
first
Find three numbers such that the second is twice the 2, and the sum of the
the third exceeds the second by and third is 20.
LINEAR EQUATIONS AND PROBLEMS
71
Find three numbers such that the second is 4 less than the third is three times the second, and the sum of the first and third is 36. 3.
the
-
first,
4.
"Find three
the third
is 4,
second by 5.
numbers such that the sum of the first two times the first, and the third exceeds the
is five
2.
Divide 25 into three parts such that the second part first, and the third part exceeds the second by 10.
is
twice the 6. v -
Find three consecutive numbers whose sum equals
63.
The sum
of the three sides of a triangle is 28 inches, the second one is one inch longer than the first. If twice
7.
and
the third side, increased by three times the second side, equals 49 inches, what is the length of each?
New York
has 3,000,000 more inhabitants than Philaand Berlin has 1,000,000 more than Philadelphia (Census 1905). If the population of New York is twice that of Berlin, what is the population of each city ? 8.
delphia,
9.
180.
The
three angles of any triangle are together equal to
first,
If the second angle of a triangle is 20 larger than the and the third is 20 more than the sum of the second and
first,
what are the three angles ?
10. In a room there were three times as many children as If the number of women, and 2 more men than women. men, women, and children together was 37, how many children
were present ? x
11.
A
is
A
twice as old as B, and is 5 years younger than sum of B's and C's ages was 25 years.
Five years ago the What are their ages ? C.
v .
12.
the
Find three consecutive numbers such that the sum of and twice the last equals 22.
first
13. The gold, the copper, and the pig iron produced in one year (1906) in the United States represented together a value
ELEMENTS OF ALGEBRA
72 of
The copper had twice
$ 750,000,000.
arid the value of the iron
of
Find the value of each.
the copper. 14.
the value of the gold,
was $300,000,000 more than that
California has twice as
many
electoral votes as Colorado,
and Massachusetts has one more than California and Colorado If the three states together have 31 electoral votes, together.
how many
has each state
?
If the example contains Arrangement of Problems. 3 or 4 of different such as length, width, and kinds, quantities
100.
area, or time, speed, and distance, it is frequently advantageous to arrange the quantities in a systematic manner.
A and B
start at the same hour from two towns 27 miles walks at the rate of 4 miles per hour, but stops 2 hours on the way, and A walks at the rate of 3 miles per hour without stopping. After how many hours will they meet and how
E.g.
apart,
B
many
miles does
A
walk
?
Explanation. First fill in all the numbers given directly, i.e. 3 and 4. Let x = number of hours A walks, then x 2 = number of hours B walks. Since in uniform motion the distance is always the product of
and time, we obtain 3 a; and 4 (x But the 2) for the last column. statement "A and B walk from two towns 27 miles apart until they meet " means the sum of the distances walked by A and B equals 27 miles.
rate
Hence Simplifying,
Uniting, Dividing,
3x
+
4 (x
2)
=
27.
3z + 4a:-8 = 27.
= 35. = 5, number of hours. 8 x = 15, number of miles A
7
x
x
walks.
LINEAR EQUATIONS AND PROBLEMS Ex.
73
The length
of a rectangular field is twiee its width. were increased by 30 yards, and the width decreased by 10 yards, the area would be 100 square yards less. Find the dimensions of the field. l.
If the length
" The area would be decreased by 100 square yards," gives (2.x
+ 00) 2 x2
(a -10) = 2s -100. + 10 x 300 = 2 z2 100. 2
-
-
Simplify, Cancel 2 # 2
and transpose,
The
40 yards long and 20 yards wide.
field is
Check.
70x10 Ex.
The
or 700.
2.
terest as a
A
= 200. z = 20. 2 a = 40.
10 x
an area 40 x 20 =800, the second
original field has
But 700
= 800
sum invested
certain
sum $200
larger at
x
at 5
4%.
= =
%
What
brings the same is the capital?
+ 200). + 8.
Therefore
.05
Simplify,
.053;
Transposing and uniting,
.01
Multiplying, Check.
= 800; $ 800 = required sum. x .06 = $ 40; $ 1000 x .04 = $ 40.
x
x
$ 800
fid
1
100.
.M(x .04
x
8.
in-
ELEMENTS OF ALGEBRA
74
EXERCISE
38
A
rectangular field is 10 yards and another 12 yards wide. The second is 5 yards longer than the first, and the sum Find the length of their areas is equal to 390 square yards. 1.
of each. 2.
A
rectangular field is 2 yards longer than it is wide. were increased by 3 yards, and its width decreased
If its length
Find the dimen-
by 2 yards, the area would remain the same. sions of the field. 3.
as a 4.
A
certain
sum $ 50
A sum
sum
invested at 5
larger invested at 4
% %.
brings the same interest Find the first sum.
invested at 5 %, and a second sum, twice as large, What are the
invested at 4 %, together bring $ 78 interest.
two sums
?
Six persons bought an automobile, but as two of them were unable to pay their share, each of the others had to pay 5.
$ 100 more.
Find the share of each, and the cost
of the auto-
mobile.
Ten yards
and 30 yards of cloth cost together much per yard as the cloth, how much did each cost per yard ? 6.
$
of silk
If the silk cost three times as
42.
For a part he 7. A man bought 6 Ibs. of coffee for $ 1.55. paid 24 ^ per pound and for the rest he paid 35 ^ per pound. How many pounds of each kind did he buy ? 8.
Twenty men subscribed equal amounts
to raise a certain
money, but four men failed to pay their shares, and in order to raise the required sum each of the remaining men had to pay one dollar more. How much did each man subscribe ?
sum
of
A
walking at the rate of 3 miles per hour, and follows on horseback traveling at the rate of 5 miles per hour. After how many hours will B overtake A, and how far will each then have traveled ? 9.
sets out
two hours
later
B
LINEAR EQUATIONS AND PROBLEMS v
10.
A
and
direction, but
B
set out
A has
75
walking at the same time in the same If A walks at the rate
a start of 2 miles.
miles per hour, and B at the rate of 3 miles per hour, how must B walk before he overtakes A ?
of 2 far
A
walking at the rate of 3 miles per hour, and from the same point, traveling by coach in the opposite direction at the rate of 6 miles per hour. After how many hours- will they be 36 miles apart ? 11.
sets out
two hours
later
B
starts
New York to Albany is 142 miles. Albany and travels toward New York at the rate of 30 miles per hour without stopping, and another train starts at the same time from New York traveling at the rate of 41 miles an hour, how many miles from New York will they meet? X
12.
The
distance from
If a train starts at
CHAPTER
VI
FACTORING
An
101.
expression is rational with respect to a letter, if, it contains no indicated root of this letter
after simplifying,
;
irrational, if it does contain
a2
\-
V&
is
a
to 6.
some indicated root of
rational with respect to
,
and
this letter.
irrational with respect
102. An expression is integral with respect to a letter, if this letter does not occur in any denominator. -f-
6
db
+ 62
is
integral with respect to a, but fractional with respect
to b.
103.
An
expression
and rational, if it is integral to all letters contained in it; as,
is integral
and rational with respect
a-
104.
The
factors of
+
+ 4 c2
2 ab
.
an algebraic expression are the quantities
which multiplied together
will give the expression.
In the present chapter only integral and rational expressions are considered factors. J Although Va'
b~
X
V
stage of the work, consider 105.
A
<2
a2
Ir
vV
2
b'
2 ?>
,
we
shall not, at this
a factor of a 2
6
factor is said to be prime, if it contains
factors (except itself
The prime
and unity)
;
factors of 10 a*b are 2, 5,
76
otherwise ,
a, a, 6.
it is
2 .
no other
composite.
FACTORING 106.
the process of separating an expression expression is factored if written in the
is
Factoring
77
An
into its factors.
form of a product. 2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product.
The factors
107.
The prime
of a
factors of 12
4)
form
+3,
It (a; 8) (s-1). for this result is a sum,
monomial can be obtained by inspection
&V
2
Since factoring
108.
in the
is
are 3, 2, 2,
01,
x, x,
y.
?/,
the inverse of multiplication,
it fol-
lows that every method of multiplication will produce a method of factoring. E.g. since (a + 6) (a 2 IP factored, or that a
=
= a - 62 + &)(a 2
6)
(a
,
it
that a 2
follows
- 62
can be
&).
Factoring examples may be checked by multiplication by numerical substitution.
109.
or
TYPE
I.
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
mx + my+ mz~m(x+y + z). Ex.
110.
The
1.
Factor G ofy 2
greatest factor
common
6
and the quotient But, dividend
- 9 x if + 12 xy\
to all terms
a% - 9 x2 y 8 + 12
2 x2
is
2
3 xy
-f
55.)
4
flcy*
8
by
2
xy'
3
.
Divide
xy\
2 1/
.
= divisor x quotient. - 9 x2^ + 12 sy* = 3 Z2/2 (2 #2 - 3 sy + 4 y8).
6 aty 2
Hence
Ex.
is
(
2
2.
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
- 3 6a + 1).
ELEMENTS OF ALGEBRA
78
EXERCISE Resolve into prime factors
- 12 cdx.
1.
6 abx
2.
3x*-6x*.
3.
15
2
4
&-{-20a
4.
14a
5.
Ilro8
6
2
&3
39
:
6.
4 tfy -f- 5 x*y 2
7.
17 a? - 51 x4
8.
.
.
s
.
s
.
2
2
2
.
9. -7a & 10. + llm -llm. 11. 32 a *?/ - 16 a'V -f 48 ctfa^ 2
2
4
6 xy
+ 34 X 8 a*b -f 8 6V - 8 c a 15 ofyV - 45 afy - 30 aty. a -a '-J-a 4
3
2
:
4
.
8 .
12.
13.
34
14.
a^c 8 - 51
aW + 68
a6c.
15.
16. 17. 18.
19.
q*-q*-q
2
+ q.
a(m-f-7i) + & ( m + 3 (a + 6) -3 /(a + 6). 7i
)-
2
a;
+ 13 -8.
21.
13- 5
22
2.3.4.5 + 2.3.4.6.
-
23.
2
3
5-f 2
.
3
5
6.
20.
TYPE
IT.
QUADRATIC TRINOMIALS OF THE FORM
111. In multiplying two binomials containing a common 3 and 5 to obterm, e.g. (as 3) and (cc-f-5), we had to add tain the coefficient of x, and to multiply 3 and 5 to obtain the term which does not contain x or (x 3)(x -f 5) 15. x2 -f-2 x
=
In factoring x2
15 we have, obviously, to find two numbers whose product is 15 and whose sum is -f- 2. 2 Or, in general, in factoring a trinomial of the form x -f-/>#-f q,
we have
to find
whose product
2x
-f
two numbers m and n whose sum is p and and if such numbers can be found, the y
is g;
factored expression
is
(x -}-m)(x
+ n).
FACTORING Ex.
Factor a2
l.
-4 x - 11.
We may consider or
77
1,
79
77 as the product of 1 77, or 7 11, or 11 and 7 have a sum equal to 4. .
11
7,
but of these only
Hence
a:
2
- 4 x - 77 =
(a;- 11) (a
+
7).
Since a number can be represented in an infinite number of ways as the sum of two numbers, but only in a limited number of ways as a product of two numbers, it is advisable to consider the factors of q first. If q is positive, the two numbers have both the same sign as p. If q is negative, the two numbers
have opposite
signs,
and the greater one has the same sign
as p. of this type, however, can be factored.
Not every trinomial Ex.
2.
Factor a2
- 11 a + 30.
The two numbers whose product and -6. a2
Therefore Check.
Ex.
If
3.
tf
30 and whose
sum
11 are
5
a 4- 30 = (a - 5) (a 6). + 30 = 20, and (a - 5) (a - G) = - 4 - 6 = 20. .
+ 10 ax - 11 a
2 .
11 a2 and whose sum The numbers whose product is and a. 2 11 a?=(x + 11 a) (a- a). Hence fc -f 10 ax
is
10 a are 11 a
12 /. Factor x? - 1 afy 8 The two numbers whose product is equal to 12 yp and whose sum equals 3 8 7 y are -4 y* and -3 y*. Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ).
Ex.
-
is
11
a = 1, a 2 - 1 1 a
Factor
is
112.
+
4.
In solving any factoring example, the student should first all terms contain a common monomial factor.
determine whether
EXERCISE
40
Besolve into prime factors : 4.
tf-
5.
3.
m -5m + 6. 2
6.
a2 -
ELEMENTS OF ALGEBEA
80
x*-2x-8. + 2x-S.
22.
8.
x2
9.
y_ 6y
24.
7.
2
10.
?/
2
11.
?/
12.
?/
2
23.
16.
+6y
16.
-15?/
+
25.
44.
26.
-5?/-14.
27.
+ 4?/-21. + 30. or - 17 + 30. 2
13.
28.
?/
14.
15.
a 2 +11 a
29.
a?
30.
^
16.
2
2
a2
21.
a4
TYPE 113.
.
6
8
8
4
2
a;
x*y ra
-9a&-226 + 8 a -20.
.
2 .
2
ITT.
3
4
32.
2
.
2
4xy
21y. 21 a 2
4 wia 2
2
a' 2
.
- 70 x y - 180 2
34.
10 x y 2
35.
200 x2
36.
4 a 2 - 48
+
+ + 446
400 x aft
a;
2 .
200. 2 .
QUADRATIC TRINOMIALS OF THE FORM
According to 66, - 2) = 20 x2 + 7 x - 6. (4 x + 3) (5 x 20 x2 is the product of 4 a; and 5 x. 6 is the product of + 3 and 2. .
+7 Hence
.
2
4
33. 2
.
2
+ 7ax 18. -17a& + 7(U
a 2^ 2
20.
2
?/
2
31.
18.
a2
2
-7p-8. + 5
17.
19.
ay -11 ay +24. ra + 25ra + 100. 3?/-4 + a' -2a&-24& n + 60+177> a + 7 a -30. a -7 a -30. a? + 5 + 6 a. 100 xr - 500 x + 600. 6 a -18 a + 12 a
a?
is
the
in factoring 6
x2
sum of the 13 x
+ 5,
cross products.
we have
to find
two
bino-
mials whose corresponding terms are similar, such that
The The
first
last
two terms are factors of 6 x 2 two terms are factors of 5,
and the sum
By
give the correct
of the cross products equals
we find which of the sum of cross products.
actual trial
.
13
x.
factors of 6 x 2
and 5
FACTORING If
we consider that the
factors of -f 5
and that they must be negative, sible
81
as
must have
13 x
is
combinations are contained in the following
6x-l
6.e-5
x-5 - 31 x
x-1
3xl \/ /\
V A
V A
Evidently the
combination
last
G
2
a;
3
2x- 5 - 17 x
11 x is
like signs,
negative, all pos:
5
V A
a;
2o?-l
-
13 a
the correct one, or
- 13 x + 5 = (3 x - 5) (2 x - 1).
In actual work
it is not always necessary to write down and after a little practice the student combinations, possible should be able to find the proper factors of simple trinomials
114.
all
The work may be shortened by the
at the first trial.
ing considerations
follow-
:
If p and r are positive, the second terms of the factors have same sign as q. 2. If p is poxiliw, and r is negative, then the second terms of 1.
the
the factors
have opposite signs.
a combination should give a sum of cross products, which has the same absolute value as the term qx, but the opposite sign, exchange the If
signs of the second terms of the factors. 3. If py? -\-qx-\-r does not contain any monomial factor, none of the binomial factors can contain a monomial factor.
Ex.
Factor 3 x 2
- 83 x
-f-
54.
The and
factors of the first term consist of one pair only, viz. 3 x and x, the signs of the second terms are minus. 64 may be considered the
product of the following combinations of numbers 1 x 54, 2 x 27, X x 18, 6 x 9, 9 x 6, 18 x 3, 27 x 2, 54 x 1. Since the first term of the first fac:
tor (3 x) contains a 3, we have to reject every combination of factors of 54 whose first factor contains a 3. Hence only 1 x 54 and 2 x 27 need
be considered.
ELEMENTS OF ALGEBRA
82
3s-2
3x-l
X
X
x-54
x
-27
- 83 x - 163 x 2 = 83 x + 64 Therefore 3 z 2) (x 27). (3 a;
The type
115.
2
pa; -f go; -h r is
the most important of the
trinomial types, since all others (II, IV) are special cases of In all examples of this type, the expressions should be it.
arranged according to the ascending or the descending powers of some letter, and the monomial factors should be removed.
EXERCISE Kesolve into prime factors 1.
2. 3.
4. 5. 6.
7. 8.
2x* + 9x-5. 4a2 -9tt + 2. 3x*-Sx + 4.
5m -26m -f 5. 6n + 5?i-4. 3a + 13a; + 4. Sar' + Sa-G. 2
2
13. 14.
10a?2
10. 11. 12.
15.
16. 17. 18.
2
*
2
2
2
2
-9a;-7. 12^-17^-1-6. 6n 2 -f 13w + 2. 2
2.y
+ 172/-9. 2
14 a -fa -4.
15
21.
10a2
40*.
- 77 xy + 10 y -23afc + 126
2
aj*
24.
- 13 xy + 6 y2 12 x -7 ay- 10 4a? + 14oj + 12.
25.
2
G a2
2
.
2 .
.
2
i/ .
ar*
+ 11
2
or
+ 12 a.
26.
12y -2/-6. 2
SoJ + llay
23.
2
2i/
19.
20.
22.
2
+ 2/-3. - 17-9. 10 a - 19 a -f 6. 9 y + 32^-16. 2m -t-7w + 3.
9.
41
:
x
27.
28.
100^-200^ + 100^.
29.
5 a6
30.
-9 a - 2 a 90 x*y - 260 xy - 30 y 6
.
2
31.
90 a
32.
8
33. 34. 35.
4
2
2
- 300 ab
250
-f-
2 .
2 fc
.
f-3y -4y 40a -90aV + 20aV. 144 x - 290 xy -f 144 y* 4x 8 ofy + 3 y 2
4
.
2
2
4
2
-f-
4
.
FACTORING
TYPE
83
THE SQUARE OF A BINOMIAL
IV.
2
Jr
2 xy
+/.
form are special cases of the preceding type, and may be factored according to the method used In most cases, however, it is more convenient for that type. Expressions of this
116.
them according
to factor
a2
A
-
to
2 xy
65.
+ if = (x
2 ?/)
.
trinomial belongs to this type, i.e. it is a perfect square, of its terms are perfect squares, and the remaining
when two term
equal to twice the product of the square roots of these
is
terms.
The student should note that a term, must have a positive sign.
in order to be a perfect
square,
24 xy
16 y?
Evidently 10
+ 9 y'
2
&
is
24 xy
2VWx
a perfect square, for + 9 y2 = (4 x - 3 y) 2
2
x V0y2" = 24
To factor a trinomial which
is
a perfect square, connect the
square roots of the terms which are squares by the sign of the
maining term, and
EXERCISE
42
:
2.
m + 2mn + n c -2cd-d 2
2
8.
.
2
2
9.
.
2
-10g-f25.
10.
9
+6a6 2
4
wi -f
4.
x* - 10 x -f 16.
11.
2/
5.
a 2_4 a &
12.
9
13.
25
-10a6-25.
6.
a
7.
m -14ww + 49n 2
2 .
14.
-f
a4
.
a -flOa&4-6 4
9
+ 462.
2
2
3.
2
re-
indicate the square of the resulting binomial.
Determine whether or not the following expressions are feet squares, and factor whenever possible 1.
xy.
.
2
6 m*ti
-f
.
9
n*.
x>
16 a
2
- 20 xy -f 4 y\ - 26 ab + 9 6 2
.
per-
ELEMENTS OF ALGEBRA
84 15.
16a 2 -24a&4- 9& 2
16.
+ GO + 25. 225 ofy - 60 a# + 4.
3<>
4
a;
4
19.
a;
2
17.
- 20 ab + 10 b a - 2 ofy + ofy m - 6 m* + 9 m. 10 a 2
18.
.
2
2 .
2
.
3
20.
Make the following expressions perfect squares by supplying the missing terms :
-6& +
21.
u2
22.
x*-Sx + (
(
).
24.
+ 6a + ( 9a -( ) +
25.
144 a 2
23. a2
-
2
TYPE
+(
).
16&*. 2
)-f816
.
4
27.
64 a 4
28.
4m
2
29.
m 4a + 12a + (
30.
!Gar
2
20
-f-
(
9
-(
According
to
2
-/.
65,
^//c
Ex.
1.
Ex.
2.
Ex.
3
8
10
8
10
4
5
3
4
)
a4 a2
2 -f 6 is
4
2
-f b
2
2
)
2
2.
a -9.
3.
36
-6
2 .
5
2
(a
prime.
EXERCISE
tf-y\
4
-b) = (a* + b*)(a + b)(a-b).
- 6 = (a
Resolve into prime factors 1.
^
to the
- 3 * ). aV - 9 z* = (2 ary + 3 z ) (2 1G a - 64 6 = 16(a - 4 6 = lG(tt +2Z> )(a -26 ). 4
3.
NOTE,
).
)+25.
difference of the squares of two numbers is equal the sum and the difference of the two numbers. of product i.e.
).
2
THE DIFFERENCE OF TWO SQUARES
V.
JT
117.
100w +( )+49. -48 a +( ).
26.
).
43
:
4.
4a2 -l.
5.
1-49 a
6.
81
-*
2 .
2 .
7.
100a2 -68
8.
a2 & 2 -121.
9.
9a2
.
FACTORING
One or both terms are squares
118.
Ex.
Factor a
1.
2
- (c 4- d)
a2 - (c
Ex.
85
of polynomials.
2 .
+ d) 2 = (a + c + cZ) (a - c - (I)
.
Resolve into prime factors and simplify
2.
EXERCISE 44 Resolve into prime factors 1. 2.
3.
2
(m-7?)
(m -f #
5.
16
T.
8.
-y.
9.
_p
2
2
16p
n)
.
2
2
4.
6.
.
(m-f-n) 2
:
2
(?/
4-
.
2:)
2
2
(y -f
cc
2
.
a:)
25a -(&-c) (m-h2n)
2
2
-
.
2
36|>
.
(m
3n)
2 ( 2
(2a-5&) -(5c-9ef) 2
10.
(a
11.
x?
(x
12.
(x -f
3
13.
(2a
14.
(2s
6
6)
-f-
2 .
y)*. 2
9 2/ 2
.
?/)
+ 5) -(3a-4) 2
2 .
2 .
ELEMENTS OF ALGEBRA
86
TYPE
GROUPING TERMS
VI.
By the introduction of parentheses, polynomials can frequently be transformed into bi- and trinomials, which may be factored according to types I- VI. 119.
A. After grouping tain a
Ex.
Ex.
the terms, ive find that the
1.
2.
Factor ax ax
+
Factor
or
x8
-f-
bx
5
bx
+
-f
ay
+ by
5 x2
- 6z2
ay
con-
x x
+
-f by.
= x(a + &) +
Resolve into prime factors
= z2
ax + bx
2.
ma
3.
2an-3&n + 2ag-3&?.
4.
4:cx
5.
10ax-5ay-6bx + 3by.
6.
a?
+ ay+by.
+ m&
+ 4cy--5dx
+ x + 2x + 2.
By
2
+
6)
(.r,
5)
- (x -
5)
45
:
1.
?*a
y(a
-f 5.
5
EXERCISE
B.
new terms
common factor.
7.
nb.
5dy.
8.
raV + nV
9.
3 a 2ic
10.
a3
11.
c
3
12.
a5
m ?/ 2
n
- 4 6 x -f 3 a y 2
2
+ ab 6 - 7 c + 2c - 14. - a a - ab + bx. a 26
2
3
.
2
4
grouping, the expression becomes the difference of two
squares.
Ex.1. Factor 9 x*-y*-4:Z 2 -f 4 yz.
= (3 x + y - 2
)
(3 x
- y + 2 2).
FACTORING Ex.
Factor 4 a2
2.
-
6
2
87
+ 9 tf - 4 f - 12 aaj
-f-
4
6y.
Arranging the terms, 4 a2
- 62 + 9
_ 4 _ 12 ax + 4 6y 2 = 4 a 2 - 12 ax + 9 a2 + 4 &t/ 4 y2 = (4 a 2 - 12 z + 9 x2)_ (&2 _ 4 ty + 4 ^2) a;*
*/2
ft
EXERCISE Kesolve into prime factors 1.
2. 3.
+ 2xy + y*-q*. l~a -2a5-6
x*
2
2
.
a
4.
36
5.
9
-4a6 + 46 -25.
2
2
7. 8.
a 2 -10a6 4
46
:
m - 6 ww + n 2
2
<
6.
+ 256 2
x -ar -2a;-l. 2
SUMMARY OF FACTORING First find
I.
monomial factors common
to all terms.
Binomials are factored by means of the formula
II.
a 2 -6 2
= (a + 6)(a-6).
Trinomials are factored by the method of cross products, although frequently the particular cases II and IV are more conIII.
venient.
IV.
Polynomials are reduced to the preceding cases by grouping
terms.
EXERCISE
47
MISCELLANEOUS EXAMPLES* Resolve into prime factors !.
2. 3.
m
2
16.
+ 16. w -m 2.
8ra 2
2
:
6
6a4 + 37a2 + 6. 6a4 -12a2 + 6.
6.
2a3/
4.
+c+
2
2
2/
.
* See page 266.
7.
a8 - 9 a2
8.
4 v*
$-
m -Gw + 9-n 2
.
- 10 xy + 4 y\ 2 .
ELEMENTS OF ALGEBRA
88 10.
x*-xif.
11.
10 a 2
12.
4a
13.
49 a 4
14.
-32 aft + 6
4
4ft
2 ft .
4
3
2
.
4
2
17.
1
18.
?v
7#2
3
or
_w 8
2
ft
ft
16. 2
4 a;
+ 14.
19. 5a' 20.
a6
22.
3 a2
156.
23.
24.
a;
5
3 25. a
2
ft
2
a.
+ a + a + l. 2
2
z
.
6
2
:J
2
ft
2
+ 2 ?<s
>r
_|_ ft)2
256
35. 2
38.
a5
39.
a
42 s 2
(a;
2
2 ?/)
.
4
V 2
51 xyz
+ 50.
(
1
40. 3 41.
.
__ G4.
a -128.
36. any
48. + 6 aft + 3 80 a 310 x 40. 4
20
34.
-50^ + 45. a3
32.
29.
8
4
tt
33. (^
.
n Qy 2 . 2
30.
27. 28.
.
-42 a + 9 a 20a -90a -50. 4
-2a + a*-l. 42 x - 85 xy + 42 y 10 w 43 w 9. 25 a + 25 aft - 24. 13 c - 13 c - 156.
26.
3#4 -3a2 -36.
CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR 120.
The
common
highest
factor (IT. C. F.) of
two or more
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II. C. F. of a 7 and a e b 7 is
.
;
Two
expressions which have no are prime to one another. 121.
The H.
common
factor except unity
two or more monomials whose factors
C. F. of
are prime can be found by inspection. The H. C. F. of a 4 and a 2 b is a2 .
The H. The H.
C. F. of
aW, aW,
C. F. of (a
+
8 ft)
and
and (a
+
cfiW is 2
fc)
a 2 /) 2
(a
.
4 ft)
is
+ 6)
(a
2 .
122. If the expressions have numerical coefficients, find by arithmetic the greatest common factor of the coefficients, and prefix it as a coefficient to H. C. F. of the algebraic expressions. Thus the H. C. F. of 6 sfyz, 12 tfifz, and GO aty 8 is 6 aty.
The student should note H. C.
F. is the lowest
that the power of each factor in the power in which that factor occurs in any
of the given expressions.
EXERCISE Find the H. C. F. of
4.
2.
3.
15
aW,
13 aty
8 ,
25
48
:
W.
39 afyV.
5. 6.
89
33
3
-
,
5
2 7
22 3 2 2
5
3
s ,
7
24
23 3 ,
3 ,
2
2
2
5,
.
5 7
s
.
34 2s ,
.
II
2 .
54
-
32
.
ELEMENTS OF ALGEBEA
90
6 rarcV, 12 w*nw 8, 30
7.
mu\
39 afyV, 52 oryz4, 65 zfyV. 38 #y, 95 2/V, 57 a>V.
8. 9.
aWd,
10.
225
11.
9
4a
12.
4(m -f ?i)
10
8 a
,
a&X -15 bed
75
16 a
,
3 ,
11
24 a
,
5(w + w)
2 ,
8
6(m+l) (m+2),
14.
6
-
3 a;
(a7
5 ?/)
,
9
7(m + n}\m 2
8(?/i-f-l)
aj*(a?
.
6.
3
13.
2
- y)\
O+
12
0^(0;
3),
ri).
4(m+l)
- y)
2 .
3 .
123. To find the H. C. F. of polynomials, resolve each polynomial into prime factors, and apply the method of the preceding article. Ex. 1. Find the H. C. F. of + 4 if, x2
^-4^
and
tf
-7 xy + 10 f. - 3 xy + 2 y* = (x - 2 ?/) (x - y) - 7 xy + 10 2 = (x - 2 y) (a; - 5 y). = x 2 y.
x*
x2
Hence the H. C. F.
.
7/
EXERCISE Find theH. 1.
4 a3 6 4 8 a663 - 12 as 66
2.
15 x-y^ 2 10 arV - 5 x3?/ 2
3.
25 m27i, 15
4.
.
,
,
4
3ao;
49
C. F. of:
3
7/i
-3^
4
n2
10
mV.
6 mx - 6 4
,
.
4 ?io; .
5.
6 a2
6.
y?
7.
a2
8.
ar*
- 6 a&,
10.
12. 13. 14.
15. 16.
2
2
2
a
-
2
2
2
a;?/
,
2
2
,
a;
^-707 + 12, 0^-80:4-16, ^a + 5^ + 6,^-9, ^-f a;-6. a2 - 8 a + 16, a3 -16 a, a -3a-4. a2 + 2a-3, a2 + 7a-f!2, a3 -9a. y + 3y-64,y + y-42, 2a -f5a-f 2, 4a -f 4a2
-5^
- # 4 afy -f 4 - 6 a' + 2 a& + 6 - 5 + 6, ^-
9.
11.
5 a6
.
2 .
3 .
LOWEST COMMON MULTIPLE
91
LOWEST COMMON MULTIPLE
A
multiple of two or more expressions is an be divided by each of them without a which can expression 124.
common
remainder.
Common
2 multiples of 3 x
The
125.
lowest
and 6 y are 30 xz y, 60
common
2
x^y'
,
300 z 2 y,
etc.
two or more
multiple (L. C. M.) of
expressions is the common multiple of lowest degree; thus, ory is the L. C. M. of tfy and xy*. 126. If the expressions have a numerical coefficient, find by arithmetic their least common multiple and prefix it as a coefficient to the L. C.
The The
M of the algebraic expressions.
L. C.
M.
of 3
L. C.
M.
of 12(a
aW,
+
2
a^c8 3
ft)
,
6
c6 is
and (a
C a*b*c*.
+ &)*( -
&)
2
is
12(a
+ &)( - 6)2.
127. Obviously the power of each factor in the L. C. M. is equal to the highest power in which it occurs in any of the
given expressions. 128. To find the L. C. M. of several expressions which are not completely factored, resolve each expression into prime factors and apply the method for monomials.
Ex.
1.
Find the L.
C.
M.
of 4 a 2 6 2 and 4 a 4
4 a 2 &2
2.
.
_
Find the L.C.M. of as -&2 a2 + 2a&-f b\ and 6-a. ,
= (a -f
Hence the L.C.M. NOTE.
2
=4 a2 62 (a2 - 6 3 ).
Hence, L. C. M.
Ex.
-4 a 68
The
L. C.
M. of the
last
2
&)'
(a
-
6) .
- (a + &) 2 (a
In example ft). two lowest common multiples, which is
also
general, each set of expressions has
have the same absolute value, but opposite
signs.
ELEMENTS OF ALGEBRA
92
EXERCISE Find the L. 1. 2.
a,
a 2 a3 ,
C. .
xy\
afy, 3
y*. 2
4a
3.
2
7.
4 a 5 6cd, 20
8.
9
ic
a,
10.
,
,
50
M. of:
8 a.
3
5.
6
6.
afc'cd
3 ab, 3(a
4.
2
40 abJ, 8 d 5
,
+ b).
9.
6
-f
2
6 y, 5
5
a?
a2
2 a?-b\ a + 2ab + b' 2a-2b.
a~b,
b
a?
14.
.
2
7i)
,
3(m
18.
19.
x2
3
a,-
a;
-f2, x
2
a
5
4,
a;
-f-
2
3#
+ 2,
5
21.
a 2 -fa6, a&
5
#,
x2
3 a
20.
or -f-
+ 5 a + 6,
2
+&
22.
a -!, a^-1,
23.
ic
24.
ax -{-ay ~
7ic+10, bx
15
3, 2
a2
,
4
2
2 .
a 2 -f 4 a +4.
2
-f
x2 + 4 a
-f 4,
#.
~ab
6b 2
.
1. a?
2
8
lOaj-f-lfi, by,
3 a
a.
+ n) 4 m
-4)(a-2)
,
2
30
a, a.
2
+ 6. + 2, a -f 3, a 1. 2 a - 1, 4 a - 1, 4 a -f 2.
17.
G
,
,
?/.
2
15. 16.
2
2
ic
y,
a
b,
+
(a
,
13.
-{-
3 Z>
,
3 a
,
x.
.
,
a;
3
2(m
,
12.
a?b,
5 a 2 ^ 2 15
T a
(a-2)(a-3) ( a -3)(a-4) 2 2a?b-'2ab 2 a, 2 a
3
8 afy, 24
.
2
11.
xif,
3
b,
x*
~5a;-f 6.
2 x -\-2 y.
(For additional examples see page 268. )
2
CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
A
129.
with a
-f-
fraction is
an indicated quotient; thus -
is identical
The dividend a is called the numerator and the The numerator and the denominator
b.
divisor b the denominator.
are the terms of the fraction. All operations with fractions in algebra are identical
130.
with the corresponding operations in arithmetic. Thus, the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number; the product of two fractions is the product of their numerators divided by the product of their denominators, etc. In arithmetic, however, only positive integral numerators shall assume that the
and denominators are considered, but we
arithmetic principles are generally true for
all
algebraic numbers.
131. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered. rni
Thus
A
132.
and
Reduce
1.
Remove tor, as 8,
TT
= ma
b
mb
is
i
,
Hence
~-
successively all 2
a?,
j/' ,
and z 8
and
mx = -x my y
in its lowest
denominator have no
its
Ex.
fraction
a -
terms
common
when
its
numerator
factors.
to its lowest terms.
common
divisors of
(or divide the terms
6
2
.ry ^
24
2 z = --
3x
by
numerator and denomina-
their H. C. F.
ELEMENTS OF ALGEBRA
94
133. To reduce a fraction to its lowest terms, resolve numerator and denominator into their factors, and cancel all factors that are
common
Never cancel terms of the numerator or the
to both.
denominator; cancel factors only.
Ex.
2.
a*
Keduce
~
6 a'
6a qs
_. 6
tf
Ex.
3.
Keduce
a*
*8a
to its lowest terms.
24 a2
4
n2 + 8 a 24 a*
-
_ ap 2 - 6 a + 8) 6 d\a* - 4)
~ 2 62
--
to its lowest terms.
a2
62
_Q
2 6
EXERCISE 51* Reduce i
to lowest terms
9-5
:
3
o 3
*'
*
32
78
2.33
-7 a
2
'
12a4"
T^
3
3T5"**
'
36 arV 18 x2^'
* See page 268.
K 6
'
39 a2 6 8c4
FRACTIONS 7-
^-.
9.
10.
11
'
22 a 2 bc 1
8.
4-
m-
n h
g
95
2
m
3
11
21.
J-
~__ 9n _ 22
^+3*. LJZJ^JL.
9x +
23.
.
^ Mtr
"a"
04
.
!l
9
'-M
12.
3
3
??i
2
6
or
it*?/
2fi 25.
.
_
7i
-9 - 10 a + 3 2
rt<
3a
15
^
^
x1
4 xy
"-^
^
'
+
//(/
' ft<
4
'
_.*..
m
2
5
?tt
^"
" 16.
.7
,
2
27.
?/
-*-7 * ,
OQ
12 15
2
m m
3 a3
2
7 w,n
2
8
_6a
4 18.
19.
a;
^'
+
?
wn + n 2 ?i
T>
2
a/i
2
+6 ^.
29.
nx 17.
2
f
26.
14.
+y
30.
5^-10 y
^
rt
"-""-;'
-
31.
32.
ny
ELEMENTS OF 'ALGEBRA
96
Reduction of fractions to equal fractions of lowest common Since the terms of a fraction may be multiplied
134.
denominator.
by any quantity without altering the value of the fraction, we may use the same process as in arithmetic for reducing fractions to the lowest
Ex.
mon
common
denominator.
^
Reduce -^-, and 6rar 3 a? kalr
1.
,
to their lowest
com-
,
T denominator.
The
L. C.
M.
To reduce
of
//-*
2 ,
3 a\ and 4
aW
is
to a fraction with the
-
12 afo 2 x2 .
denominator 12 a3 6 2 x2 numerator ,
^lA^L O r 2 a 3
and denominator must be multiplied by
22
'
Similarly, multiplying the terms of
***- by ^ 3
A
2 , '
we have
by 4
-
-M^-1^22 , '
.
,
2>
2
6' .r
2 ,
and the terms of
and
Tb reduce fractions to their lowest common denominator, C.M. of the denominators for the common denominator. Divide the L.C.M. by the denominator of each fraction, and 135.
take the L.
multiply each quotient by the corresponding numerator.
Ex
-
Reduce
to their lowest
common
TheL.C.D. =(z
denominator.
+ 3)(z- 3)O -
1).
Dividing this by each denominator, (x
+
3),
and
(a-
we have
the quotients (x
1),
8).
Multiplying these quotients by the corresponding numerators and writing the results over the common denominator, we have
(a
+ 3) (a -8) (-!)' NOTE.
Since a
=
(z
,
-6 + 3)(s-3)O-l)'
we may extend this method
6a;~16 (a
+ 3) (x- 3) (-!)'
to integral expressions,
FRACTIONS EXERCISE
97
52
common denominator
.Reduce the following to their lowest 1.
5?, JL. 22 a2
^*
8
5a
n">
.T
2,^1.
3.
a?
^'
* .
.,
'
S?
.,
m^
?y2"
/
.
o o
a.
2aj ~
>
7^
2
i, i.
26
5c
5
1
m
i
m
**.
S*
3
7i
5 o>
",oj
o atf
2 ab*
2a-l _ n.
5a -
.
6.
.
o*
77"
or
3
9a ~l' 3a-l 2
a
8
3
2 a8
zl
a
^
9
3.T
4a
+ 6 a-
jj
,
j
y
'
8'
'
5
6 *
'
3y
Ga-1 9
'
ay
bxby
IB.
?--,
ax
a g
'
a
2a ;
-f-5
a2
2
25
!
.
- a+2
18.
5
'
a 2 -3a-f 2
ADDITION AND SUBTRACTION OF FRACTIONS 136.
Since --{-c
c
= 5L^ c
(Art. 74), fractions having a
common
denominator are added or subtracted by dividing the sum or the difference of the numerators by the common denominator. 137. If the given fractions have different denominators, they must be reduced to equal fractions which have the lowest common denominator before they can be added (01 subtracted).
ELEMENTS OF ALGEBRA
98
Ex
-
Sim
'
The L.
C. D.
^ is
4(2 a
- 3 ft)(2 a
-f
Ga-6
+
2^JT)
:
3
ft).
Multiplying the terms of the first fraction by 2(2 a the second by (2 a - 3 ft), and adding, we obtain 2 a
2(2 a
+3 -3
6 a
-
ft
4(2 a
-f
3
ft
ft)
_ 2(2 a +
3
ft)
(2 a
4(2 a
ft)
_ 8 a 2 -f 24 aft
~~
ft
+
2
12 a 2
The
138.
4
-f
aft -f
21
ft)
- 20 aft
4(2a-3ft)(2a-f 20 a 2
the terms of
+ 3 ft) -f (2 a - 3 - 3 ft)(2 a + 3 ft)
18
-f
+ 3 ft),
(6 a -ft)
-f
3
2
2
ft
and subtraction should be
results of addition
ft
3ft)
re-
duced to their lowest terms. T?
Ex.
2.
cr
_T__
+
-r-
-\-t
Simplify
_
,
*
a2
^
ab
a-3b
^
a(a
~
ft).
_ 3 ab + 2 = ( a _ ft)( _ 2 a 2 -2 aft :=(- 2 = a(a - (a 2 6). L. C. D. 3 a 2a + "~ a2
ft2
ft).
ft).
Hence the a a2
-f
2 6
ft)
ft
ft
- aft
a2
-
3
aft
+
2
a2
ft2
_(a +
-
2
aft
2ft)(a-2ft) +a (2q + a(a - ft) (a
2 =a -4
ft
2
+
a(a
=a
2
^.
4 6
Ca2
2 (2 a 4- aft)
-
ft)~.
(a-8ft)(a~-ft)
2ft)
- 4 aft +
8
ft
2
)
2ft)
ft)(a
+ 2qg+6~ag-f4a&-8
ft
2
a(a-ft)(a -2ft) ...
NOTE. (a
3
ft)
a(a
+ 5 aft - 7 - ft)(a - 2
a(a
-2
2 a2
ft
2
ft)
_. (2
~
a
a(a
+
-
7 ft)fa ft)(a
-
-
ft)
2
ft)
'
ft)
In simplifying a term preceded by the minus sign, e.g. (a ft), the student should remember that parentheses are
understood about terms ( 66) hence he should, in the beginning, write 2 the product in a parenthesis, as 4 aft -f- 3 ft 2). (a ;
FRACTIONS
99
EXERCISE 53* Simplify
:
2a-4
2.
9m + 7n
3.
2x + 3y
3x
5a-76
8.
9.
6
3a
46
2
106
6a-116
15a-26
13 a
116
a
2a
4a
4a
10.
'
7.
++. 6
c
36
36
12.
3u
v
2v
12 uv
5 wv
8v
30 u
18 v
13.
-+-
19.
_H_ + _*_. a+6 a 6
14.
? + i-
20.
2
15.
A+2_3.
21.
j>0
16.
i>
-1*
+
18-
t-3
q *
M.
m-f 3
23.
m-2*
-^4-f25.
+
1 -f
1
1
1 17.
5n
x + 2y 45
y
15
e
6m
3
5
24.
a "" 2 6 '
,
m m
2L + 2a
1
a _2 6 a 4- 5
1 1 -f
w
.
a-2 a + 3* 2 a -7 a-f-1
+
2) * See page 270.
'
ELEMENTS OF ALGEBRA
LOO
26
-
27
x*3x + 2 x-2
,9.
3a
4
a
2x
x2
+. a
9
+ 3y
x
Gx
5x
x-3y
~.
'
30.
2
a-f-1
31.
32.
_m
1
2
i
+m
36
6
&
34.
a2+ a ^_2&2
x-2
35.
Qfi ou
Q 3 *
_
/j.
1
I
'
TTo
<1
-
1
i
n.
1
'>
-.9
a2
i
- 679 2
/ IIlNT:
+ 1.
i. _ '
a
+b
Let a
+a=
42.
a
-
?^
(
-_ + a?
L
\_
1
!
-
ic
41.
9
O :_
"I
L "I
IT-i ~T~ 7
a-f 1-f
39.
40.
/Yl
7
37
38.
_w
i
a;
+ y.
43.
?/
^-2-^+6m 3 45
'
x2
-7x+12~x
44.
a 2
' }
-l7x + 4:~
^>
2
FRACTIONS To reduce a
139.
101
fraction to an integral or *
mixed expression.
= + ceo Hence
2
5a2 -15a-7 = 5 a2 5a
oa
v Ex.
,
1.
*-
3
4
,
T,
Reduce
or
-
4 x3
-
2a; 2
+ 4tf
2x
3
(S74) v '
15a
7
oa
5a
17
2 x2
;
+ 4x 4 x2 - 6 x + 10x-
+
3
5a
,
.
,
to a
.7
=a
mixed expression.
2 g 4- 6
4 x2
x
Therefore
y
3g
- 17
17
(2^ + 2x
5-3
-f
(2x-,'3) 2
EXERCISE
54
Keduce each of the following fractions expression
to a
:
a
a
9a2 -6a + 2 3a
m
2
*-
m
5
m -f 6
7
4
n 2 + 7n + 14
fi
+1
mixed or
integral
ELEMENTS OF ALGEBRA
102
MULTIPLICATION OF FRACTIONS 140. Fractions are multiplied by taking the product of tht numerators for the numerator, and the product of the denominators for the
or,
expressed in symbols:
a
c
denominator;
_ac b'd~bd'
Since -
141.
= a,
we may extend any
fractions to integral numbers,
To multiply a fraction by an
-x
e.g.
c
b
principle proved for
= b
integer, multiply the
numerator by
that integer.
142. Common factors in the numerators and the denominators should be canceled before performing the multiplication. (In
order to cancel
common
factors, each
nator has to be factored.)
Ex.
!.
Simplify 1 J
The
Ex.
2.
expreeaion
F J Simplify
=8
6
.
2
a
numerator and denomi-
FRACTIONS EXERCISE Find the following products 8
'
"
^
'
2
5
48
..
4
'
53 *38
14 b*
10 a 8
5c
36C2
"
'
76
10 (a
56
5#
3a 2 6
c&
/
GoA
V
ai>
9m _ JO.
2
-25n
3m +&n
a2
12 2
ar "
'
"
1
3m
,. 14.
5n
15.
_G x
7
2 ab
+
2
+ 1"
"
a
fc
12 d6 4- 20
b*
~
'
a 2 -5a-h4 <
2
(x
I)
a2 -5a-6
a;
5
aj
1
18. a?
2
-f
5
a;
o,
2
o?-f
a2 4-3a-4 a2 3 a 4
x2 + x
17.
'
14m4
q~. 4
7a-216
ot
"
2
34 ab 2
4a-f-86 "
56 2c
V
17 ab '
'
'
6)
"
21m*
36^
as b*'
4- 6
'
4 ac2
'
'
8.
55
:
a2
2!v! 2 4
103
50
-
ELEMENTS OF ALGEBRA
104
DIVISION OF FRACTIONS 143. To divide an expression by a fraction, invert the divisor and multiply it by the dividend. Integral or mixed divisors should be expressed in fractional form before dividing.
The
144.
dividing 1
number
reciprocal of a
is
the quotient obtained by
-f-
|
by that number.
The
reciprocal of a
is
The
reciprocal of J
is
1
+*
The reciprocal of ? Hence the
a |.
+ + * = _*_. x a + b
1
is
x
reciprocal of a fraction
obtained by inverting
is
the fraction, and the principle of division follows
may
be expressed as
:
145. To divide an expression by a fraction, expression by the reciprocal of the fraction.
the
multiply
8
Ex.
1.
Divide X-n?/
.
by
x** -f xy 2
s^jf\ = 2
x'
y
-f
3
x*y
x*
- y3 +
x3
2/
+y
3
x*y~ -f y
8
2
x'
xy*
EXERCISE 56* Simplify the following expressions x*
2
'""*'-*-
ft2 '
om
'
13 a& 2 5
:
+a
. :
2 a2 6 2
r -
3
i_L#_-i-17
J
ar
u2
a 4-1
* See page 272.
a-b
FRACTIONS
-.-
-
^-5^+4
*
'
105
t '
'
a^-3^-4 ?
4*
. '
'
>
m 12
2
a a2 6
2
a
a?-~ab
2
2a
2
2
4- 5
4-g-20
80 50
??i
.
2
^y
?/
.
4-
4
a:
+3
mm
5
ga2
25
15 #4- 10 ?/ _._ ~#
a*?/
in^o
2
a2
8
4
.T ?/
45
a
4-g-2 2
w
w + 56
5
-f- 1
4a
a-
5 a 4-
y
'
"xy
14
15
a2
+
(Jf
a
fr
b
a
.
a2
4- 6
+ 064- 6
s
COMPLEX FRACTIONS 146. A complex fraction is a fraction whose numerator or denominator, or both, are fractional.
Ex.
l.
Simplify
c
a a2 c
-L
4.
&
c
a
_6
6
c
4-
ab 2 4-
4-
a6 2 4- &c 2
&c*
a ,
c
ac
a
a2c
~
2
4- afr 4- ^c
2
ELEMENTS OF ALGEBRA
16
147.
In
many examples
the easiest
mode
of simplification
ia
multiply both the numerator and the denominator of the mplex fraction by the L. C. M. of their denominators. If the numerator and denominator of the preceding examples multiplied by a&c, the answer is directly obtained.
B
x -}- ?/ x y _x^_l X ~V x+y .
Ex.
2.
Simplify
xy
x
.
+y
Multiplying the terms of the complex fraction by (x y), the expression becomes (x
EXERCISE Simplify
57
:
x 2.
y X
4* 2 y
3.
JL.
4.
+6.
&
6.
7i+~ .
,a
.
c
^c
-n a
7.
a
9.
_^
,y
.
m ""
-.
32
a
8.
i. c
x*
10.
FRACTIONS
m 11.
,
o
1 15.
:
~
*
a "~ 2
12.
1
i
~T"
!^-5n
107
y 1
-
-
&
a
a
+2
1
i
2
4
'
5
,
-i
19
20
-
a4-6 13.
m^n* n
17.
i
1
1
a
+
"
"
f
(
1 /*-_i_i
4-
14.
L
s-y
18.
1
+
1+ 1 ti
(For additional examples see page 273.)
flg-f-l
a?l
a;-~l
ic+1
CHAPTER IX FRACTIONAL AND LITERAL EQUATIONS
FRACTIONAL EQUATIONS If an equation contains fracbe removed by multiplying each term by the may L. C. M. of the denominator. 148.
Clearing of fractions.
tions, these
^-2^ 63 =
Ex.1. Solve
-* + *-*.
12
2
Multiplying each term by 6 (Axiom 2 x
-
Removing parentheses, 2x
2(x 2
-
a;
3)
+
6
9x
Uniting,
x
Ex.
If
2.
x
= =
89),
72
-3
72
(a; 4-
Bx
4)
-
12
x.
Qx.
2z-2a;-f3# + C:E=-6-f72-12.
Transposing,
Check.
3,
= 6,
each
member
-I
reduced to
1.
14
5
Solve
is
= 64. = 6.
x
+1
x
+3
-!)(&+ 1) (x + 3), + 1) (a + 3) - 14 (a; - l)(z + 3) = - 9(se + !)( 14 x 2 - 28 x + 42 = - 9 x2 + 9. 5 x2 + 20 x + 15 15 - 42 + 9. 14 z 2 + z 2 + 20 x - 28 a = 5 x2 - 8 x = - 48, = 6.
Multiplying by (x
5(3
Simplifying,
Transposing, Uniting,
85
Check.
If
x
6,
each
member
is
reduced to
108
1.
I).
FRACTIONAL AND LITERAL EQUATIONS EXERCISE Solve the following equations
^
3
109
58
:
_ +7 a?
4
-
32
'2
--. 3
a?
4
"T"" 4
"
2
4- 1
_7-7
o
""~TiT"
a?
a;
'
3
3 10.
^-1 = 9. a/
-
-
12.
4
5
= 12.
+4 14.
1+5
= 19
-^^0
= 5.
&
a/
'
1
11.
- = 2.
a:
16.
= xx a?
a?
1
*>
7
a;
13.
= 2.
a?
18.
hi-
15.
x
+^ + 3 = 11. ^'
2,
4
a;
20
on
y
334
+2
y-2
+1
y-3 ==
2 ^
16
ELEMENTS Of ALGEBRA
110
24.
?_=_.
29.
y+3~2
25.
l-~.
26 26.
4a4-l4* +
27
.
2^12 = 2
28
.
= 34.
35.
36.
31 31.
.
32
.
3
_J_ = _J3 ._ _
-
20
x+3 x-3
33.
.
2
6 .
-
2
o^-
-
13
J_.
.
3x
3
3x-2
3x-2
3x*-2x 23 x
51
+
2a?-3 A*
1
1
22
26
4^-9
2^4-3 37.
-
38
=
7
^^
'
39
'
x- 11_4 x-
40.
149. If two or more denominators are monomials, and" the remaining one a polynomial, it is advisable first to remove the monomial denominators only, and after simplifying the resulting equation to clear of all denominators.
FRACTIONAL AND LITERAL EQUATIONS Ex.1.
Solve
5#
10 Multiplying each term by tors,
16 x
Transposing and
5
1
L. C. M. of the monomial denomina-
10, the
2( +3-~
~ &Q n =:
x
a;
26
1,
a;
-
5
x
:
a;
=
-
20 g
~ Jff
each
9,
member
Solve the following equations 41
is
2.
.
1
a:
= 20 x -
5x
Dividing, If
16 x
5
Transposing and uniting,
Check.
=
5
uniting.
Multiply ing by 6
60.
45.
=
9.
reduced to
^.
:
5a;-2
42 9
,,
43.
24
a;
-f
8#-f 2__ 2x
13
15
44.
5
7
~~7-16*
10 x -f 6 __ 4a;-r-7 5
6a?
6a-fll~~
+l
3
6x-flO '
5
2a?~25
15
28
64-14
17a?~9 14
~
-
.2
18
3
==
9
111
7a;-29 507-12'
ELEMENTS OF ALGEBRA
112
LITERAL EQUATIONS 150. Literal equations ( 88) are solved by the same method as numerical equations. When the terms containing the unknown quantity cannot be
actually added, they are united
factoring.
= (a -f 6) mnx = (1 4- m
ax
Thus,
by
-f-
x -f- m 2*
bx
jr.
2
mn) x.
Ex.1.
a
a
b
ax-
Clearing of fractions,
a
z
+ bx ax
Transposing, Uniting,
6)z
l
= !=?_=^6?
a;
=a
-f
-f
(a
= 3 & 2 ab. = 2 -f b 2 - 3 6 2 = a' - & - 2 62
IP
bx
Dividing,
Reducing
151.
It
a to
lowest terms,
frequently occurs that the
expressed by
Ex.
2.
or
x, y,
L=
If
3a-c
Multiplying by 3 (a
a
=
Transposing
all
, ?
Uniting the Dividing, 5>
a,
fr
unknown
not
letter is
find
a
in
terms of b and
c)
ac
+6
= (2a + &)(3a-c). = 6 a2 - 2 ac + 3 aft - be.
6c
terms containing a 6 ab
Simplifying,
-
-f 6.
3(a-c)
- c) (3 a -
6 a&
2
.
z.
6(rt-fc)(a-c) 6 a2
ab.
2
6 ac
to
one member,
+ 2 ac
3 ab
9 a&
4 ac
and multiplying by
1,
= =
a(9 b
= -l^ 9 b
4-
4 c
6 6c
~
5c.
7 6c. -f
4 c)
= 7 &c.
c.
FRACTIONAL AND LITERAL EQUATIONS EXERCISE Solve the following equations
59
:
*,
2. 3.
4.
6.
+ 3a; = 8 4 #. a + 26+3aj=2o + 6 + 2a?. mx = n.
11.
ax
13. 14.
-4-- = c
21.
a
.
Z>
3(*-
12.
10.
_
iw
= 3 (6 a). = 2(3a = aaj-ffta? + 7^ = 0*+^
8.
9.
113
4 (a
1
x)
3(2a +
aj)
a).
25
1
a;
?+l a/'~~
= 2L
-f- a;
l
1
.
;i
c.
^
^o;
-f-
+ &o; = 6 (m -f n) = 2 a + (m-?i)a?.
a^
co?.
a?
n) x
15.
(wi
16.
- = H.
18.
-
=px + q.
- = n.
17.
+ = xx
1.
*
*
-
26.
m
= 5.
a?
x!7
-
x
a ITo
n
IIL
x
b
,
T
= vt, = rt, s =
_ ~
2 8.
If s
solve for v.
29.
If s
solve for
30.
If
31.
If
V-t
2
If
Solve the same equation for^).
34.
The formula
for simple interest
-, solve for a. c
-=-+!,
f P
33.
solve for y a.
^^a = 1
32.
,
(
t.
solve for/.
q
30, Ex. 5) is
t
=^,
denoting the interest, p the principal, r the number of $>, and n the number of years. Find the formula for: i
() The (6) (c)
The The i
principal. rate.
time, in terms of other quantities.
ELEMENTS OF ALGEBRA
114
(a) Find a formula expressing degrees of Fahrenheit terms of degrees of centigrade (<7) by solving the equation
35.
(F)
in
(ft)
Express in degrees Fahrenheit 40
C
If
36.
is
100
C.,
- 20
the circumference of a circle whose radius
= 2 TT#.
then
C.,
Find
R in terms of C and
C. is
R,
TT.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS 152. Ex. 1. When between 3 and 4 o'clock are the hands of a clock together
At
?
3 o'clock the hour hand
is
15 minute spaces ahead of the minute
hand, hence the question would be formulated After how many minutes has the minute hand moved 15 spaces more than the hour hand ? :
x x
Let then
= the required number of minutes after 3 o'clock, = the number of minute spaces the minute hand moves over,
= the number
and 12
Therefore x
of minute spaces the
hour hand moves
over.
~ = the number of minute spaces the minute hand moves more than the hour hand. x
Or
~^ = 15
'
!i^=15.
Uniting,
Multiplying by
- 180. = 16^- minutes after x=
11 x
12,
^
Dividing,
3 o'clock.
Ex. 2. A can do a piece of work in 3 days and B in 2 days, In how many days can both do it working together ? If
by
1,
we denote then
respectively ff
/-
the required
A would do
number
each day ^ and
of
B
days by x and the piece of work while in x days they would do
j,
~ and and hence the sentence written in algebraic symbols ^, 2 3
FRACTIONAL AND LITERAL EQUATIONS
115
A
more symmetrical but very similar equation is obtained by writing ** The work done by A in one day plus the work done by B in one day equals the work done by both in one day." in symbols the following sentence
:
Let
x
= the
required
Then
-
= the
part of the
Therefore, Solving,
32
x
x
= |,
number
of days.
work both do
or 1J, the required
in
number
one day.
of days.
Ex. 3. The speed of an express train is $ of the speed of an If the accommodation train needs 4 accommodation train. hours more than the express train to travel 180 miles, what is the rate of the express train
?
xx*
180
Therefore,
=
152
+4
(1)
= 100 + 4 x. 4x = 80.
180
Clearing,
Transposing,
Hence fx
Explanation
:
If
x
is
=
36
= rate
of express train.
the rate of the accommodation train, then
the rate of the express train.
Ox j
5
But
in
uniform motion Time
=
a
Distance
Rate Hence the rates can be expressed, and the statement, u The accommodation train needs 4 hours more than the express train," gives the equation /I).
ELEMENTS OF ALGEBRA
116
EXERCISE 1.
60
Find a number whose third and fourth parts added
together
make
21.
Find the number whose fourth part exceeds part by 3. 2.
3.
Two numbers
ceeds the smaller by 4.
What 5.
2.
The sum of two numbers numbers ?
is oO,
is
Find two consecutive numbers such that
Two numbers
^ of the other.
J-
of the greater
9.
by 3, and J of the greater Find the numbers.
differ
Twenty years ago A's age was |
Find A's 8.
and one
ex-
are the
l of the smaller. to s
7.
fifth
by 6, and one half the greater Find the numbers.
differ
increased by ^ of the smaller equals 6.
its
is
equal
of his present age.
age.
The sum
of the ages of a father and his son is 50, and of the father's age. -|
10 years hence the son's age will be
Find their present ages. 9 its
A
post
is
a fifth of
length in water,
and 9
its
length in the ground, one half of What is the length
feet above water.
of the post ?
A man left ^ of his property to his wife, to his daughand the remainder, which was $4000, to his son. How
10 ter,
much money
did the
man
leave ?
11. A man lost f of his fortune and $500, and found that he had \ of his original fortune left. How much money had he at first?
12 left
After spending ^ of his
^ of his money and $15.
at first?
money and $10, a man had How much money had he
FRACTIONAL AND LITERAL EQUATIONS 13.
The speed
of an accommodation train
is
117
f of the speed
of an express train. If the accommodation train needs 1 hour more than the express train to travel 120 miles, what is the rate of the express train? 152, Ex. 3.) (
An express train starts from a certain station two hours an accommodation train, and after traveling 150 miles overtakes the accommodation train. If the rate of the express train is -f of the rate of the accommodation train, what is the 14.
after
rate of the latter ? 15.
At what time between 4 and
a clock together? 16.
(
5 o'clock are the hands of
152, Ex. 1.)
At what time between 7 and 8
o'clock are the
hands of
a clock together ? 17.
At what time between 7 and
8 o'clock are the hands of
a clock in a straight line and opposite 18.
A man
has invested
the remainder at
6%.
J-
of his
19.
money
4%, ^ at 5%, and has he invested if
at
How much money
his animal interest therefrom is
more
?
$500?
A has invested capital at
4%.
investments.
at 4J % and P> has invested $ 5000 They both derive the same income from their How much money has each invested ?
20. An ounce of gold when weighed in water loses -fa of an How many ounce, and an ounce of silver -fa of an ounce. ounces of gold and silver are there in a mixed mass weighing
20 ounces in 21.
A can
22.
A
air,
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days, and B in 4 days. In how many days can both do it working together ? ( 152, Ex. 2.) can do a piece of work in 2 days, and
how many days can both do
it
B in 6 days.
working together
In
?
23. A can do a piece of work in 4 clays, and B In how many days can both do it working together
in ?
12 days.
ELEMENTS OF ALGEBRA
118
The
and their solutions differ only two given numbers. Hence, by taking for these numerical values two general algebraic numbers, e.g. m and n, it is possible to solve all examples of this type by one example. Answers to numerical questions of this kind may then be found by numerical substitution. The problem to be solved, therefore, is A can do a piece of work in m days and B in n days. In how 153.
last three questions
in the numerical values of the
:
many days we
If
let
method of
can both do
x
= the
it working together ? required number of days, and apply the
we
170, Ex. 2,
Solving, 3;=
m
-f-
obtain the equation
n
Therefore both working together can do
To
-- = -. n x
m
it
in
mn
m
-f-
n
days.
A can do this work in 6 days Q = 2. and n = 3. Then
find the numerical answer, if
ft
and
B
in 3 days,
they can both do
make it
m
6
i.e.
6
in 2 days.
Solve the following problems 24.
In
piece of
how many days
work
if
can
(a) (6) (c)
(d)
3
:
A
and
each alone can do
ofdavs:
I
A in 5, A in 6, A in 4, A in 6,
it
B
working together do a
in the following
number
B in 5. B in 30. B in 16. B in 12.
25.
Find three consecutive numbers whose sum
is 42.
26.
Find three consecutive numbers whose sum
is 57.
The
last
two examples are
special cases of the following
problem 27. Find three consecutive numbers whose sum equals m. Find the numbers if m = 24 30,009 918,414. :
;
;
FRACTIONAL AND LITERAL EQUATIONS
119
Find two consecutive numbers the difference of whose
28.
squares
is 11.
Find two consecutive numbers -the difference of whose
29.
squares
is 21.
30. If each side of a square were increased by 1 foot, the area would be increased by 19 square feet. Find the side of the square.
The
last three
examples are special cases of the following
one:
The
31.
difference of the squares of
two consecutive numbers
find the smaller number.
By using the result of this problem, solve the following ones Find two consecutive numbers the difference of whose squares is ?n
;
:
is (a)
51, (b) 149, (c) 16,721, (d) 1,000,001.
Two men
same hour from two towns, 88 one traveling 3 miles per hour, and the second 5 miles per hour. After how many hours do they meet, and how many miles does each travel ? 32.
miles
start at the
apart, the
first
Two men start at the same time from two towns, d miles the first traveling at the rate of m, the second at the apart, After how many hours do they rate of n miles per hour. 33.
meet, and how many miles does each travel ? Solve the problem if the distance, the rate of the
first,
and
the rate of the second are, respectively (a) 60 miles, 3 miles per hour, 2 miles per hour. 2 miles per hour, 5 miles per hour. (b) 35 miles, :
(c)
64 miles, 3J miles per hour,
4J-
miles per hour.
by two pipes in m and n minutes how In many minutes can it be filled by the respectively. two pipes together ? Find the numerical answer, if m and n are, respectively, (a) 20 and 5 minutes, (b) 8 and 56 minutes, 34.
(c)
A cistern can
6 and 3 hours.
be
filled
CHAPTER X RATIO AND PROPORTION 11ATTO
The
154.
Thus the
two numbers number by the
ratio of
dividing the
first
ratio of a
and
b
is
is
the quotient obtained by
second.
- or a *
b.
The
ratio is also frequently
b
(In most European countries this symbol is employed as the usual sign of division.) The ratio of 12 3 equals 4, 6 12 = .5, etc.
written a
:
the symbol
b,
:
A
155.
:
being a sign of division.
:
ratio
is
used to compare the magnitude of two
numbers. " a Thus, instead of writing
a
:
b
is
6 times as large as
?>,"
we may
write
= 6.
156.
The
first
term of a ratio
is
the antecedent, the second
term the consequent. In the ratio a
:
ft,
a
the
is
numerator of any fraction
is
The
antecedent, b is the consequent. the antecedent, the denominator
the
consequent.
157.
The
ratio -
is
the inverse of the ratio -.
a 158.
Since a ratio
fractions if its
may
b is
a
fraction, all principles
be af)plied to ratios.
E.g. a ratio
is
relating
terms are multiplied or divided by the same number,
Ex.
1.
Simplify the ratio 21 3|.
A somewhat shorter way
:
would be to multiply each term by 120
to
not changed
6.
etc.
AND PROPORTION
RATIO Ex.
2.
Transform the
ratio 5
1.
equal
5
*~5
3J so that the
:
33 :
~
72:18.
3.
J:l.
4.
$24: $8.
4|-:5f
6.
5 f hours
3:4.
9.
8.
3:1}.
10.
8^-
hours.
4
.
27 06: 18 a6.
11.
16 x*y
12.
64 x*y
:
:
24
xif.
48
a-y
3 .
ratios so that the antecedents equal
:
16:64.
15.
159.
|
:
:
7|:4 T T
Transform the following
two
:
5.
7.
unity
61
ratios
62:16.
Simplify the following ratios
term will
'4*
EXERCISE
1.
first
3
5
Find the value of the following
2.
121
A
16.
17.
7f:6J,
is
proportion
:
1.
18.
16a2 :24a&.
a statement expressing the equality of
ratios.
= |or:6=c:(Z are
160.
The
first
proportions.
and fourth terms of a proportion are the and third terms are the means. The last
extremes, the second
term
is
the fourth proportional to the
first three.
In the proportion a b = c c?, a and d are the extremes, b and c the means. The last term d is the fourth proportional to a, b, and c. :
:
If the means of a proportion are equal, either mean the mean proportional between the first and the last terms, and the last term the third proportional to the first and second
161.
is
terms. In the proportion a b :
and
c,
and
c
is
=
b
:
c,
b is the
mean
the third proportional to a and
b.
proportional between a
ELEMENTS OF ALGEBRA
122 162.
Quantities of one kind are said to be directly proper
tional to quantities of another kind, if the ratio of any two of the first kind, is equal to the ratio of the corresponding two
of the other kind. ccm. of iron weigh ,30 grams, then G ccm. of iron weigh 45 grams, 6 ccm. = 30 grams 45 grams. Hence the weight of a mass of iron is proportional to its volume. " we " NOTE. Instead of u If 4
or 4 ccm.
:
:
may say,
directly proportional
pro-
briefly,
portional.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind, if the ratio of any two of the first kind is equal \o the inverse ratio of the corresponding two of
the other kind. If 6 men can do a piece of work in 4 days, then 8 men can do it in 3 days, or 8 equals the inverse ratio of 4 3, i.e. 3 4. Hence the number of men required to do some work, and the time necessary to do it, are :
:
:
inversely proportional.
163. In any proportion product of the extremes.
t/ie
a
Let
:
product of the means b
=c
:
is
equal
to the
d,
!-; Clearing of fractions, 164.
ad =
The mean proportional
the square root
be.
bettveen two
numbers
is
equal to
of their product.
Let the proportion be
Then Hence
=b = ac.__(163.) b = Vac.
a b :
6
:
c.
2
165. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means, and the
other pair the extremes, of a proportion. If
mn = pq, and we
divide both
?^~ E. q~~ n
(Converse of
members by
nq,
163.)
we have
AND PROPORTION
PATIO Ex.
Find
1.
x, if
6
:
x = 12
Ex.
rn
a?
8:6 =
4$ = 35,
If
a
6
a
166. I.
III.
7
=c
d,
6
:
=d
:
:
c.
= 35
;
7
(Called Alternation.)
:
:
of these propositions
To prove
may be proved by
is
true
example
ad
if
-
Division.)
a method which
:
d
b
= be = be. ad = be. bd
bd.
ad
if
But
163.)
(
^ =^'
Hence
d
o
These transformations are used to simplify proportions. Change the proportion 4 5 = x 6 so that x becomes the
167.
:
last term.
By
true.
(Frequently called Inversion.)
illustrated by the following
I.
is
then
:
V.
Or
true
4|.
hence the proportion
+ b:b = c + d:d. (Composition.) a + b:a = c + d:c. d d. (Division.) a b b=c = b d. a+b a c-)-d:c (Composition and
IV.
This
:
is
a
Or
Any
:
and 5 x
a:c=b:d.
II.
is
Determine whether the following proportion
2.
t:
8 x
7.
= 42. (163.) = f f = 3 J.
12x
Hence
:
123
inversion 5
:
4
=6
:
x.
:
ELEMENTS OF ALGEBRA
124 IT. its
Alternation shows that a proportion is not altered when its consequents are multiplied or divided by
antecedents or
the same number. E.g. to simplify
consequents by
3:3
Or
1:1
To
III.
divide the antecedents by 16, the
48:21=32:7x,
7,
= 2:3. = 2:x,
i.e.
5:6
simplify the proportion
Apply composition,
11
:
6
=4
:
5
division,
Divide the antecedents by
V. To simplify
?
5, 1
m 3n
Apply composition and
=4
x
:
x.
x.
IV. To simplify the proportion 8 Apply
= 2.
x
:
:
3 3
= =
5
:
1
:
= 5 -f x
3
:
:
x.
jr.
x.
= + *.
mx
= ^-
division,
2x
tin
.!=!*.
Or
3n
=-
Dividing the antecedents by m, JJ
n
A parenthesis is understood
NOTE.
x x
about each term of a proportion.
EXERCISE
62
Determine whether the following proportions are true 1.
2. 3.
5^:8 = 2:3. = 7:2f 3J.:J
:
= 12 5ft. 8ajy:17 = i^:l-^. 11
4. 5.
:
5
:
15:22=101:15.
Simplify the following proportions, and determine whether they are true or not :
6.
7.
10.
= 20:7. 8. = 9. 72:50 180:125. m n (m n) = (m + rif m 2
2
2
:
:
= 24:25. 13 = 5f llf
18:19
120:42
6 2
:
n 2.
:
RATIO AND PROPORTION Determine the value of x 11. 12.
:
40:28 = 15:0;. 112:42 = 10:a.
13.
03:a?=135:20.
14.
a?:15
17.
1, 3, 5.
21.
3, 3t, f.
20.
2, 4, 6.
22.
ra, w,j>.
25.
16 and
31. 35.
to
37. 38. 39.
40.
,
rap, rag.
1
and
a.
27.
29.
a and
1.
34.
ra
to
:
8 a 2 and 2 b 2
Find the
2
28.
2 a and 18
If ab
ra
a 2 and ab.
33.
x 10
23.
:
32.
Form two
:
14 and 21.
and 2/.
equation 6 36.
:
4 and 16. |-
96.
26.
Find the mean proportional 30.
:
to:
19.
28.
4z = 72
= 35:*. 4 a*:15ab = 2a:x. 16 n* x = 28 w 70 ra.
18.
Find the third proportional
:
2.8:1.6
16.
Find the fourth proportional
9 and 12.
21
15.
= l^:18.
24.
125
a.
+ landra
proportions commencing with 5 from the
= 5 x 12.
= xy,
form two proportions commencing with
ratio of
x
:
y, if
6x = 7y. 9 x = 2 y. 6 x = y. mx = ny.
41. 42. 43. 44.
+ fyx = cy. x:5 = y:2. x m = y n. 2 3 = y #. (a
:
:
:
45.
7iy = 2:x.
46.
y
:
47.
y
:
=x 1 =x
b
2:3 = 4- x:
49.
6:5
50.
a
x.
= 15-o;:ff. 2= 5 x x. :
:
a.
:
a2
.
:
Transform the following proportions so that only one 48.
b.
:
contains x:
:
1.
.
= 2 + x: x. = 3 43 + x. 5=
51.
22: 3
52.
19
53.
2
:
:
18
a?
a;
:
:
a?.
terra
ELEMENTS OF ALGEBEA
126
State the following propositions as proportions : T (7 and T) of equal altitudes are to each, othei
54.
(a) Triangles
as their basis (b
and
b').
(6) The circumferences (C and C ) of two other as their radii (R and A"). (c) The volume of a body of gas (V) is 1
circles are to
each
inversely propor-
tional to the pressure (P).
The
(d)
(A and
areas
A') of two circles are to each other as
(R and R'). The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work. the squares of their radii (e)
55. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk, and the :
total cost.
The time a
(b)
train needs to travel 10 miles,
and the speed
of the train.
The length
(c)
of a rectangle of constant width, and the area
of the rectangle.
The sum
(d)
of
money producing $60
interest at
5%, and
the time necessary for it. (e) The distance traveled by a train moving at a uniform rate, and the time.
A
56.
line 11 inches long
A
22 miles.
The
57.
their radii.
4
:
on a certain
areas of circles are proportional to the squares of If the radii of two circles are to each other as
and the area of the smaller
7,
what
is
58.
map corresponds to how many miles ?
line 7^- inches long represents
circle is
8 square inches,
the area of the larger?
The temperature remaining
the same, the volume of a
A
body of gas inversely proportional to the pressure. under a pressure of 15 pounds per square inch has a volume of gas
is
16 cubic
feet.
What
will be the
12 pounds per square inch ?
volume
if
the pressure
is
RATIO AND PROPORTION The number
69.
127
of miles one can see from an elevation of
very nearly the mean proportional between h and the diameter of the earth (8000 miles). What is the greatest distance a person can see from an elevation of 5 miles ? From h miles
the
is
Metropolitan
McKinley (20,000
Tower (700
feet
high) ?
From Mount
feet high) ?
168. When a problem requires the finding of two numbers which are to each other as m n, it is advisable to represent these unknown numbers by mx and nx. :
Ex. as 11
1. :
Divide 108 into two parts which are to each other
7.
Let then
Hence or
Therefore
Hence and
= the first number, = the second number. 11 x -f 7 x = 108, 18 x = 108. x = 6. 11 x = 66 is the first number, 7 x = 42 is the second number. 11
x
7
x
A line AB, 4 inches long, 2. produced to a point C, so that
Ex. is
(AC): (BO) =7: 5.
Find^K7and BO.
Let
AC=1x.
Then
BG = 5 x. AB = 2 x.
Hence
Or
2 x
=
4 '
r
i
A
B
4.
x=2. Therefore
7
=
14
= AC.
1
ELEMENTS OF ALGEBRA
128
EXERCISE 1.
Divide 44 in the ratio 2
:
2.
Divide 45 in the ratio 3
:
7.
3.
Divide 39 in the ratio 1
:
5.
4.
A line 24 inches
long
63
9.
divided in the ratio 3
is
:
5.
What
are the parts ? 5. Brass is an alloy consisting of two parts of copper and one part of zinc. How many ounces of copper and zinc are in 10 ounces of brass ?
consists of 9 parts of copper and one part of ounces of each are there in 22 ounces of gun-
Gunmetal
6.
How many
tin.
metal ?
Air is a mixture composed mainly of oxygen and nitrowhose volumes are to each other as 21 79. How many gen, cubic feet of oxygen are there in a room whose volume is 4500 7.
:
cubic feet? 8.
The
7 18.
total area of land is to the total area of
If the total surface of the earth
water as
is
197,000,000 square miles, find the number of square miles of land and of water. 9. Water consists of one part of hydrogen and 8 parts of :
oxygen.
grams
How many
grams of hydrogen are contained in 100
of water?
10.
Divide 10 in the ratio a
11.
Divide 20 in the ratio 1 m.
12.
Divide a in the ratio 3
13.
Divide
:
b.
:
m
:
7.
in the ratio x:
y
%
The
three sides of a triangle are 11, 12, and 15 inches, and the longest is divided in the ratio of the other two. How 14.
long are the parts ? 15. The three sides of a triangle are respectively a, 6, and c inches. If c is divided in the ratio of the other two, what are its
parts ? (For additional examples see page 279.)
CHAPTER XI SIMULTANEOUS LINEAR EQUATIONS 169.
An
equation of the
unknown numbers can be the unknown quantities.
first
degree containing two or more by any number of values of
satisfied
2oj-3y =
If
6,
2 y = - -. y = a?
,-L
then
(1)
5
/0 \ (2)
= 0, =,y=--|. x = 1, y = 1, etc.
x
I.e. if
If If
Hence, the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.
of values.
However,
there
if
different relation
is
given another equation, expressing a y, such as
between x and *
+ = 10,
(3)
unknown numbers can be found. From (3) it follows y 10 x and since
these
y
to be satisfied
by the same values of x and
y must be equal.
Hence
2s -5
= 10 _ ^
o
the equations have the two values of
y,
(4)
= 3. is x = 7, which substituted in (2) gives y both are to be the same satisfied Therefore, equations by values of x and y, there is only one solution. The
root of (4) if
K
129
ELEMENTS OF ALGEBRA
130 170.
A
system
a group of equa by the same values of the unknown
of simultaneous equations is
tions that can be satisfied
numbers. 6 and 7 x 3y = by the values x = I, y
are simultaneous equations, for they are 2 y = 6 are But 2 x 2. 6 and 4 x y not simultaneous, for they cannot be satisfied by any value of x and y. The first set of equations is also called consistent, the last set inconsistent.
x
-H
2y
satisfied
I
171. Independent equations are equations representing different relations between the unknown quantities such equations ;
cannot be reduced to the same form.
~ 50, and 3 x + 3 y =. 30 can be reduced to the same form -f 5 y Hence they are not independent, for they express the x -f y 10. same relation. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f- 3 y = 80. 6x
;
viz.
172.
A
unknown
system of two simultaneous equations containing two quantities is solved by combining them so as to obtain
one equation containing only one
unknown
quantity.
The process of combining several equations so as make one unknown quantity disappear is called elimination. 173.
174.
The two methods
By By
I.
II.
of elimination
to
most frequently used
Addition or Subtraction. Substitution.
ELIMINATION BY ADDITION OR SUBTRACTION 175.
E,X.
Multiply (1) by
Multiply (2) by
Solve
2, 3,
Subtract (4) from (3), Therefore,
-y=6x 6x
-f
-
4y
- 26.
= - 24, 26 y = 60. y = 2. 21 y
(3) (4)
SIMULTANEOUS LINEAR EQUATIONS
131
Substitute this value of y in either of the given equations, preferably
the simpler one (1),
3x
Therefore
In general, eliminate the
common
letter
coefficients
Multiply (1) by Multiply (2) by
+ 2.2 = 9 + 4 = 13, 3-7- 2 = 6- 14 =-8.
(3)
and
25 x - 15 y 39 x + 15 y
5,
3,
(4),
Therefore Substitute (6) in (1),
Transposing, Therefore
5
Check.
.
13
Hence
to eliminate
= 235. = 406. 64 x = 040. x = 10. 60 - 3 y = 47. 3y = 3. y = 1. x = 10. 10 - 3 1 = 47, 10 + 5 1 = 135.
(3) (4)
(6)
by addition or subtraction :
Multiplyy if necessaryy the equations by such
make
have the lowest
3. 8
2.
176.
whose
multiple.
Check.
Add
+ 4 = 13 x = 3. y = 2.
the coefficients
If the signs of these if unlike,
add
numbers as
will
of one unknown quantity equal. coefficients
are
like,
subtract the equations;
the equations.
EXERCISE
64
Solve the following systems of equations and check the
answers:
'
ELEMENTS OF ALGEBRA 5.
^ = ll.
13-
v
2/
= 24.
17.
-I
i
3
a;
6-1
l7a; +
7
'
1fi fl ,4.1ft
= 6.
is
+ 22/ = 40,
fl<>* r A
O
1
8. I
5y
oj
t
K
= 17. 19<
I
a;-f2/
= 50.
'
9-
1
r 20.
I
_.
,
= 41, [2o; + 3?/ { 3 x -f 2 y = 39. x 11.
I
12.
]
f
3#
i
3.9
?/
= 0,
~ y~~> 22. '
^
,v
23. 13.
f
14.
I
15
'
3X
J
7x
^
y = 1U, * + 3 y = 50. '
-
-60.
94 ^4
'
25
*
*- 3.5 y = -2.3.
SIMULTANEOUS LINEAR EQUATIONS
133
ELIMINATION BY SUBSTITUTION 177.
8,
Solve
Transposing
(1)
(2-7, I3ar + 2y = 13. and dividing by
7 y in (1)
Substituting this value in (2)
,
3
7 (
8
t
)
21 y
Clearing of fractions,
^""
x
2,
"
?/
(2)
24
+2y= +
4 y 25 y
Therefore
y
= 26. = 60. = 2.
This value substituted in either (1) or (2) gives x
178.
Hence
Find
in one equation the value of
to eliminate
terms of the other. tity in the
by substitution
and
:
unknown quan-
solve the resulting equation.
EXERCISE Solve by substitution
3.
an unknown quantity in
Substitute this value for one
other equation,
13.
65
:
f5aj
l3a;
= 2y + 10, = 4#-8.
ELEMENTS OF ALGEBRA
134
179. Whenever one unknown quantity can be removed without clearing of fractions, it is advantageous to do so in most cases, however, the equation must be cleared of fractions and ;
simplified before elimination
is
possible.
(1)
3
Ex.
Solve
2
(2)
7
Multiplying (1) by 12 and (2) by 14,
9 = 36. + 21-2y-4 = 14. 4* + 3y = 19. 7x_2y=-3.
43 + 8-f-3y + 7z
From
(3),
From
(4),
Multiplying (6) by 2 and (6) by
(7)
and
(8) ,
Substituting in (6)
1
+8 2
,
(6) (6)
3,
Sx + 6y = Adding
(3) (4)
3S.
(7)
21z-6y=-9. 29 x = 29. x = l. 2 y = - 3. 7 y = 6.
(8)
_
7
EXERCISE
66
Solve by any method, and check the answers:
+ 5(y + 5) = 64. f8(z-8)-9(y-9) = 26,
""^IT
(4t(x-\-)
'
\6(a;-6)-7(y-7)==18.
3.
\
SIMULTANEOUS LINEAR EQUATIONS 3x 4.
"25
6
'
tsjj
- 1) + 5(6 y - 1) = 121, r4(54(5 x l2(3'
'
15
8
8.
14.
9.
J
.
a;
+y 2
a;
,
"*"
ff
_13 ~
2'
4
15.
10.
16.
4 11.
;
10
2a?-5
17.
12.
2
4^
~ 3 =
3a?-2^4
13.
4~2v
3
1
18.
y-M a;-f-2
2,
= 3.
135
ELEMENTS OF ALGEBRA
136
<X +
-4_1 2' 19.
2
a;
-f y - Q ^
l_3
2/4-1
4'
20.
4
21.
22.
?~y
3x-\-" 1
,
23
24.
*
((* ((
,
{;
180. In many equations it is advantageous at first not to consider x and y as unknown quantities, but some expressions
involving
x,
and
y, e.#.
- and x y
SIMULTANEOUS LINEAR EQUATIONS x Ex.
Solve
1.
137
(1)
y
.
x
(2)
y
2x(2),
(3)
a;
33
Clearing of fractions,
Substituting x
=
11 x.
3 in (1),
y=4.
Therefore
Examples
=
of this type, however, can also be solved
by the regular
method. Clearing (1) and (2) of fractions, 15 y
+ 8 x - 3 xy. - 4 x = 4 xy.
(4) (6)
2x(5), (4)
(6)
+ (G),
(7)
Dividing by 11
y,
3
EXERCISE Solve
=
#, etc.
67
:
x
2' 3.
1.
1. 2*
* x 2.
y
4.
y
ELEMENTS OF ALGEBRA
138
4 K ---6 = 5, x y
10
5.
"
12
U
25
y
6.
253 7.
21 9 --=
x 8.
M-Oi y
x
a;
o 6,
y
13.
4
y
331 9.
LITERAL SIMULTANEOUS EQUATIONS 181.
Ex.
1.
(1)
Solve
(2) (1)
x
n,
(2)
x
6,
(8)
(4),
= en. 6w3 + bny = 6p. bmx = en anx anx + bny
(3) (4) ftp.
SIMULTANEOUS LINEAR EQUATIONS (an
Uniting,
bm)x
=
Dividing,
(2)
x w, x a,
amx + bmy amx -f any =
(7)
- W,
any
(1)
(an
Uniting,
bmy bm}y
en
bp.
an
bm
cm.
(6)
ap.
(7)
ap
cm.
ap-
cm.
W - cm y= an
EXERCISE
.
bm
68 -f-
ax
-f-
f
5.
ny =
fax -f
6^
= = 9a + 46.
fy/
1.
11.
= l,
12.
Find a and
13.
From
14.
c,
.y
x
of a, w,
=
x -f my = 1,
I sc
6.
6y
+ by = 2 a&,
nx -f my == m.
[
139
s in
terms of
n, d,
and
a
I if
the same simultaneous equations find d in terms
and L
From the same
equations find s in terms of
a, d,
and
I.
ELEMENTS OF ALGEBRA
140
SIMULTANEOUS EQUATIONS INVOLVING MORE THAS
TWO UNKNOWN QUANTITIES To solve equations containing
three unknown quantities three simultaneous independent equations must be given. 182.
By tions,
unknown quant iff/ from any pair of equasame unknown quantity froni another pair, the
eliminating one
and
problem
the
reduced
is
containing two
of two simultaneous equations
to the solution
unknown
quantities.
Similarly, four equations containing four unknown quantireduced to three equations containing three unknown
ties are
quantities, etc.
Ex.
1.
Solve the following system of equations:
= 8, Eliminate
(1)
-4,
(2)
l.
(3)
y.
=
Multiplying (1) by
4,
8B-12y +
Multiplying (2) by
3,
Oa + 12?/- 15z=-12
+
17 x
Adding, Multiplying (2)
Multiplying (3)
Adding,
16z z
32
=
20
by 3, x + 12 y - lf> z - by 2, 8 x - 12 y + 6 z = - 9z =11 x
(4)
12 2 10
(6)
Eliminating x from (4) and (5).
-(5),
100
Therefore
z
= 30. = 3.
3
-
20.
x
=
1.
(4)
Substitute this value in (4),
17 x
+
Therefore Substituting the values of x and z 2
Hence Check.
3y
(6)
(7)
in (1), -f
12 =s
8.
3y =
6.
y
=* 2.
2.1-3.2 + 4.3 = 8; 3.1+4.2-5.3=-4; 4.1-6.2 + 3.8 = 1.
(8)
SIMULTANEOUS LINEAR EQUATIONS EXERCISE
1.
10 x
+ y -f z = 15,
141
69
8.
4-
x
9. k
2/
-f 2
y
-M
?/
2?
=
4.
a;
-f
2
a?
a?
+ 70-9 = 26,
10.
4.
11.
~6?/
== 6,
12.
5.
x 13.
15 2
7.
+ 2 y -f 2 = 35, 4 = 42, 2z = 40.
= 45.
14.
2
-f-
2
i/ -f-
z
= 14,
ELEMENTS OF ALGEBRA
142
15.
23.
3x
?/
= 0,
60;
=
16.
5,
(3
_. 1510
x
4-
17.
_2
'
074-2!
3
J
18.
.2
a;
4- .3
y
+ .42 = 2, ^
19.
=s
20.
27.
84 21.
?
= llz, = 8*.
22. ;
32.
= 2.6 2.
SIMULTANEOUS LINEAR EQUATIONS
x
30.
z
y
x
M=i, y
29.
143
*
z
:
=1
:
2.
= 2 m, + z = 2p, z + x = 2 n. # 4- 2/
31.
2/
.
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS 183. Problems involving several unknown quantities must contain, either directly or implied, as many verbal statements as there are unknown quantities. Simple examples of this
kind can usually be solved by equations involving only one
unknown
quantity.
99.)
(
In complex examples, however, every
unknown quantity by
it is advisable to represent a different letter, and to express
every verbal statement as an equation.
Ex. 1. The sum of three digits of a number is 8. The digit in the tens' place is | of the sum of the other two digits, and if 396 be added to the number, the first and the last digits
Find the number.
will be interchanged. Obviously of the other
it is difficult
hence we employ 3
;
Let
y z
100
+
10 y
two of the required
letters for the three
+z-
the digit in the units' place. the number.
three statements of the problem can
x
+
y
+z-
1
+2+
now be
readily expressed in .
8.
+ lOy + z + 396 = 100* + 10y + x.
The solution of these equations gives x Hence the required number is 125. Check.
6
terms
quantities.
= the
symbols:
100s
digits in
unknown
the digit in the hundreds' place, 1 digit in the tens place,
x
and Then
The
to express
= 8;
2
=
= 1(1+6);
l,y 125
=
+
2,
396
2
= 6.
= 521.
(1)
(3)
ELEMENTS OF ALGE13KA
144 Ex.
If both numerator and denominator of a fraction be
2.
increased by one, the fraction is reduced to | and if both numerator and denominator of the reciprocal of the fraction be dimin;
ished by one, the fraction Let and then -
x y
= the
y
we
reduced to
Find the
2.
fraction.
nurn orator,
denominator
;
expressing the two statements in symbols,
By
fraction.
is
= the = the x
obtain,
+
2
I
(1)
and
x-
These equations give x
I
(2)
1
=
and y
4_2. 5_
3+1 5+1
Check.
3
5.
Hence the
fraction
is
f.
_4_
A, B, and C travel from the same place in the same B starts 2 hours after A and travels one mile per hour faster than A. C, who travels 2 miles an hour faster than B, starts 2 hours after B and overtakes A at the same How many miles has A then traveled? instant as B.
Ex.
3.
direction.
Since the three
men
traveled the
same
= xy + x xy = xy -f 3 x 2 y = 2. x 3x-4y = 12. = 8. y = 3. xy
Or
(4)-2x(3), From (3) Hence xy Check.
8
=
distance,
2y 4y
2.
12.
C4)
a:
= 24,
6
x 4
= 24,
4
x
(2)
(3)
24 miles, the distance traveled by A.
x 3
(1)
= 24.
SIMULTANEOUS LINEAR EQUATIONS EXERCISE
145
70
1. Four times a certain number increased by three times another number equals 33, and the second increased by 2 equals three times the first. Find the numbers,
Five times a certain number exceeds three times another 11, and the second one increased by 5 equals twice
2.
number by the
first
Find the numbers.
number.
Half the sum of two numbers equals 4, and the fourth 3. Find the numbers.
3.
part of their difference equals If 4 be
4.
Tf 3 be
is J.
to
L
added to the numerator of a fraction, its value added to the denominator, the fraction is reduced
Find the
fraction.
<>
numerator and the denominator of a fraction be If 1 be subtracted from increased by 3, the fraction equals .}. both terms, the value of the fraction is fa. Find the fraction. 5.
If the
6.
If the
numerator of a fraction be trebled, and
its
denomi-
nator diminished by one, it is reduced to J. If the denominator be doubled, and the numerator increased by 4, the Find the fraction. fraction is reduced to \-. 7. A fraction is reduced to J, if its numerator and its denominator are increased by 1, and twice the numerator What is the fracincreased by the denominator equals 15.
tion ?
The sum
8.
and
if
What
18 is
is
the
of the digits of a number of two figures is 6, to the number the digits will be interchanged.
added
number
?
(See Ex.
1,
183.)
added to a number of two digits, the digits will be interchanged, and four times the first digit exceeds the second digit by 3. Find the number. 9.
10.
sum
If 27 is
The sum of the first
of the three digits of a number is 9, and the two digits exceeds the third digit by 3. If
9 be added to the number, the last two digits are interchanged. Find the number.
ELEMENTS OF ALGEBRA
146
11. Twice A's age exceeds the sum of B's and C's ages by 30, and B's age is \ the sum of A's and C's ages. Ten years ago the sum of their ages was 90. Find their present ages. 12.
Ten years ago A was B was as
as old as
and 5 years ago their ages is 55,
how
old
is
B
old as
each
is
will be 5 years hence ; now. If the sum of
now ?
A man
%
invested $ 5000, a part at 6 and the remainder bringing a total yearly interest of $260. What was the amount of each investment ? 13.
at
5%,
14. A man invested $750, partly at 5% and partly at 4%, and the 5% investment brings $15 more interest than the 4 % investment. What was the amount of each investment ? 15.
A
sum
of $10,000
is
partly invested at
partly at
6%,
5 %, and partly at 4 %, bringing a total yearly interest of $530. and The 6 investment brings $ 70 more interest than the 5
%
%
investments together. How 6 %, 5 %, and 4 %, respectively ?
4%
16.
A sum
to $8000,
money and 17.
of
money
much money
at simple interest
and
in 8 years to $8500. the rate of interest ?
A sum
of
money
invested at
amounted
What was
at simple interest
to $090, and in 5 years to $1125. the rate of interest?
is
amounted
What was
in 6 years
the
sum
of
in 2 years
the
sum and
The sums of $1500 and $2000 are invested at different and their annual interest is $ 190. If the rates of interwere exchanged, the annual interest would be $ 195. Find
18.
rates est
the rates of interest. 19. Three cubic centimeters of gold and two cubic centimeters of silver weigh together 78 grains. Two cubic centimeters of gold and three cubic centimeters of silver weigh
together 69 J- grams. Find the weight of one cubic centimeter of gold and one cubic centimeter of silver.
SIMULTANEOUS LINEAR EQUATIONS
147
20. A farmer sold a number of horses, cows, and sheep, for $ 740, receiving $ 100 for each horse, $ 50 for each cow, and $15 for each sheep. The number of sheep was twice the number of horses and cows together. How many did he sell
of each if the total
number
of animals
was 24?
21. The sum of the 3 angles of a triangle is 180. If one angle exceeds the sum of the other two by 20, and their difference by GO , what are the angles of the triangle ?
22.
On
points,
/),
the three sides of a triangle E, and F, are taken so
ABC,
respectively, three
AD = AF, ED = BE, and CE If AB = G inches, BC = 7 inches, and AC = 5 inches, what is that
=
OF.
the length of NOTE.
Tf
AD, BE, and CF?
a circle
is
inscribed in the
An C touch ing the sides in D, and F '(see diagram), then AD = AF, BD = HE, and GE = CF. triangle
23.
7<7,
A
A
circle is inscribed in triangle
sides in D,
E, and F.
^
r
ABC touching the three sides if AB = 9,
Find the parts of the
BC=7, andCL4 = 8. In the annexed diagram angle a = angle b, angle c = angle d, and angle e angle/. If angle ABC = GO angle BAG = 50, and angle BCA = 70, B find angles a, c, and e. 24.
,
1
NOTE.
is
the center of the circum-
scribed circle.
It takes
25.
B
A two hours
longer
24 miles, but if A would double his pace, he would walk it in two hours less than than
B.
to travel
Find their
rates of walking.
CHAPTER
XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND EQUATIONS 184.
Location of a point.
two fixed straight lines XX' and YY' meet in at right angles, and PJ/_L XX', and PN _L YY', then
It'
P
the position of point is determined if the lengths of
P3f and 185.
Coordinates.
PM
lines
called
the
point P.
The
abscissa
The
and P^V are of
coordinates
PN,
or its equal
is the abscissa; and r or its equal OA is
OM, PM, the ordinate of point P. jr, the ordinate by ?/.
PN are given.
,
is
usually denoted by
line XX' is called the jr-axis, YY' they-axis, and point the origin. Abscissas measured to the riyht of the origin, and ordinates abore the x-axis are considered positive ; hence
The
coordinates lying in opposite directions are negative.
The point whose abscissa is a;, and whose ordinate is usually denoted by (X ?/). Thus the points A, B, (7, and
186. ?/,
is
Dare and
respectively represented
(2,
by
(3 7 4),
(2,
3),
(3,
-3). * This chapter
may
be omitted on a 148
first
reading.
2),
GRAPHIC REPRESENTATION OF FUNCTIONS The is
process of locating a point called plotting the point.
149
whose coordinates are given
NOTE. Graphic constructions are greatly facilitated by the use of cross-section paper, i.e. paper ruled with two sets of equidistant and parallel linos intersecting at right angles. (See diagram on page 151.)
EXERCISE 1.
Plot the points:
2.
Plot the points: (-4,
3.
Plot the points
4.
Draw
:
71
-2), (-4,
(4, 3), (4,
(-5,
2J-),
2),
1), (4, 0),
(0, 3), (4, 0), (0, 0), (0,
(-3, -3). (-2,
0).
- 2).
the triangle whose vertices are respectively
(4, 1),
(-l,3),and(l, -2). 6.
Plot the points
(6,
4)
and
(4,
4),
and measure
their
distance. 6.
What
the distance of the point
is
(3,
4)
from the
origin ? 7.
(4,1),
Draw
the quadrilateral whose vertices are respectively
(-1,4), (-4, -!),(!, -4).
8.
Where do
all
points
lie
whose ordinates
tfqual
4?
9.
Where do
all
points
lie
whose abscissas equal zero ?
10.
Where do
all
points
lie
whose ordinates equal zero?
11.
What
12.
If a point lies in the avaxis, which of its coordinates
is
the locus of
(a?,
y) if y
=3? is
known ? 13.
187.
What
are the coordinates of the origin ?
Graphs.
If
two variable quantities are so related that
changes of the one bring about definite changes of the other, the mutual dependence of the two quantities may be represented either by a table or
by a diagram.
ELEMENTS OF ALGEBRA
150
Thus the following
New
volumes
Y'ork City of a certain
1
to 8 pounds.
of
pound
tables represent the average temperature
from January 1 to December 1, and the amount of gas subjected to pressures from
The same data, however, may be represented graphby making each number in one column the abscissa, and the corresponding number in the adjacent column the ordinate of a point. Thus the first table produces 12 points, A, B, C, D, 188.
ically
each representing a temperature at a certain date. in like manner the average temperatures for every value of the time, we obtain an uninterrupted sequence etc.,
By representing
of points, or the curved line the temperature.
To
find
y
the so-called graph of
from the diagram the temperature on June
ure the ordinate of F.
15
ABCN
may be found
1,
we meas-
Thus the average temperature on May
to be 15
;
on April 20, 10
;
on Jan.
15,
1
.
A graphic
representation does not allow the same accuracy of results as a numerical table, but it indicates in a given space a great many more facts than a table,
and
it
impresses upon the eye
all
the peculiarities of
the changes better and quicker than any numerical compilations.
GRAPHIC REPRESENTATION OF FUNCTIONS
151
i55$5St5SS 3{utt|s33<0za3 Graphs are possibly the most widely used devices of applied matheThe scientist uses them to compile the data found from experiments, and to deduce general laws therefrom. The engineer, the matics.
physician, the merchant, uses them. Daily papers represent ecpnoniical facts graphically, as the prices and production of commodities, the rise and fall of wages, etc. Whenever a clear, concise representation of a
number
of numerical data
is
required, the graph
EXERCISE From the diagram questions
is
applied.
72
find approximate answers to the following
:
Determine the average temperature of New York City on (a) May 1, (b) July 15, (c) January 15, (d) November 20. 1.
ELEMENTS OF ALGEKRA
152 2.
At what date
New York is
is
(a) G
or dates
10
C., (1)
is
C., (c)
the average temperature oi 1 C., (d) 9 0. ?
-
3. At what date is the average temperature highest the highest average temperature?
?
What
4. At what date is the average temperature lowest? the lowest average temperature ?
What
5.
During what months
New York 6.
is
the average temperature of
above 18 C.?
When
is
the average temperature below
C. (freezing
point) ? 7.
From what
date to what date does the temperature
increase (on the average)? 8.
11
When
is
the temperature equal to the yearly average of
0. ? 9.
10.
What
is
How
much,
increase from 11.
June
on
the
average,
1 to
July
1 ?
does
the
temperature
During what month does the temperature increase most
rapidly 12.
the average temperature from Sept. 1 to Oct. 1?
?
During what month does the temperature decrease most
rapidly ?
on
13.
During what month does the temperature change least?
14.
Which month
is
the coldest of the year?
15.
Which month
is
the hottest of the year?
16.
How much warmer
May
on the average
is it
on July 1 than
1 ?
we would denote the time during which the temperaabove the yearly average of 11 as the warm season, from what date to what date would it extend ? 17.
ture
If
is
GRAPHIC REPRESENTATION OF FUNCTIONS
When is the average temperature the same as on April
18.
153 1?
Use the graphs of the following examples for the solution of concrete numerical examples, in a similar manner as the temperature graph was applied in examples 1-18. NOTE.
19. From the table on page 150 draw a graph representing the volumes of a certain body of gas under varying pressures.
20. Construct a diagram containing the graphs of the mean temperatures of the following three cities (in degrees Fahren-
heit)
:
Represent graphically the populations
21.
sands) of the following states
22. One meter equals 1.09 yards. transformation of meters into yards.
Draw
23.
Hour
.
Temperature
(in
hundred thou-
:
Draw
a temperature chart of a patient.
a
graph for the
ELEMENTS OF ALGEBRA
154 If
24.
then
C
C 2
is
the circumference of a circle whose radius
(Assume ir~
irJl.
circumferences of
all circles
>2
2
T
.)
from
R
is J2,
Represent graphically the = to R = 8 inches.
to 20 Represent graphically the weight of iron from cubic centimeters, if 1 cubic centimeter of iron weighs 7.5 25.
grams. 26. Represent graphically the cost of butter from 5 pounds if 1 pound cost $.50.
to
27. Represent graphically the distances traveled by a train in 3 hours at a rate of 20 miles per hour.
A
dealer in bicycles gains $2 on every wheel he sells. the daily average expenses for rent, gas, etc., amount to $8, represent his daily gain (or loss), if he sells 0, 3, 2 ... 28.
If
10 wheels a day. 29.
The
cost of manufacturing a certain book consists of the $800 for making the plates, and $.50 per copy
initial cost of
for printing, binding, etc. 1 to 1200 copies.
Show
graphically the cost of the
(Let 100 copies = about \- inch.) On the same diagram represent the selling price of the books, if each copy sells for $1.50.
books from
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE 189. An expression involving one or several letters a function of these letters. 2
2 xy
190.
x
+ 7 is a function of x. 2 8 y' + 3 y is a function of x and
is
called
y.
If the value of a quantity changes, the value of a of this quantity will change; e.g. if x assumes
function
successively the
values
1,
2,
3,
4,
assume the values 7, 9, 2 x -f 7 gradually from 1 to 2, x
tively
7 to 9.
x*
x
+7
will
respec-
x increases will change gradually from 13,
19.
If
GRAPHIC REPRESENTATION OF FUNCTIONS 191.
same
A
-A
variable is a quantity
155
whose value changes in the
discussion.
constant
is
a quantity whose value does not change in the
same discussion. In the example of the preceding article, x a variable, while 7 is a constant.
is
supposed to change, hence
it is
The values of a function for the be given in the form of a numerical table. Thus the table on page 1G4 gives the values of the functions x 2 x3 and Vsr, for x=l, 2, 3 50. The values of func192.
Graph
of a function.
various values of x
,
may
,
may, however, be also represented by a graph. E.g. to con struct the graph x of x 2 construct a series of -3 points whose abscissas rep2 resent X) and whose ordi1 tions
,
nates are the corresponding i.e. construct
values of x2
',
the points (-3,
(-1,1), and (3,
9),
(-
2, 4),
1
2
3
(0,0), (1,1), (2, 4), 9),
and join
the
points in order. If a more exact diagram is
required, plot points which
lie
between those constructed above, as
Ex.
To
1.
Draw the graph of x2 -f- 2 x
Q-,
-J),
(1^,
4 from x
=
2),
etc.
4, to
obtain the values of the functions for the various values of
following arrangement
may
be found convenient
:
x = 4. a*,
the
ELEMENTS OF ALGEBRA
156
Locating the points(
4, 4),
(-3, -1), (-2,4)... (4,20), and joining in order produces the graph
ABC.
(To avoid
very large ordinatcs, the scale unit of the ordinatcs is taken smaller than that of the x.)
For brevity, the function is
frequently represented
by a single letter, as y. Thus in the above example, 2 4 and if y = x -f- 2 x ;
01
r
+*
*/
.,-,
?/
=
rf
4
the
A
involving only
power of the variable. Thus 4x + 7, or ax + b -f c are funclirst
tions of the first degree.
It can be
194.
proved that the
of a function of the first degree
graph is a straight
line, hence two points are sufficient for the construction
of these graphs.
Ex.
Draw
2.
If If
by a
= 2x-3.
z x
= 0, j/=-3. = 4, y = 6.
7
3) and (4, 5), and join(0, straight line produces the required graph.
Locating ing
the graph of
y
.
>
if
/*
1i>
>
function of the
degree is an integral Y'
rational function
71
4J, etc.
193. first
,,
GRAPHIC REPRESENTATION OF FUNCTIONS EXERCISE
Draw
157
73
the graphs of the following functions:
1.
a?
+ 2.
4.
2x +
l.
7.
2-3x.
10.
a?
2.
x-l.
5.
3x
2.
8.
1
11.
xz + x.
3.
2
12.
4a?
13.
I.
a? 2
a; 2
4
+ 4.
16.
a;
the graph of
the diagram find (3.5)2;
Va25;
or
(c)
a
1.
a*
3. y = 2x = -4. a;
?/
2
ar.
from
2
#=
4 to
05
= 4,
and from
:
(ft)
(/)
(_
1.5)2;
Vl2^
;
22. Draw the graph of or from the diagram determine:
(d)
20.
-fa--
Draw
(6)
19.
6 -fa- -or.
2.
.
x+1.
2
21.
(a)
-Jar
17.
a;
(e)
2
18.
a;
15.
(a)
8
9.
a?.
-3 a -8.
14.
2
a;
6.
a?.
4
(C )
(-2.8)';
(0)
V5;
a?
+2
from x
2
(d)
(-If)
(^)
VlO-'S".
1 to
a;
= 4,
The values of the function if x = \, 1J-, 2J-. The values of a?, if a;2 4 # + 2 equals 2, 1, 1-J-. The smallest value of the function. The value of x that produces the smallest value
;
and
of the
function.
The values of x that make 2 4 a? + 2 = 0. 2 4 x -f 2 = 0. (/) The roots of the equation x 2 The of a x -f 2 = the 4 roots 1. equation (
(c)
23. Draw the graph ofy=2-j-2# and from the diagram determine
#2 from # =
2 to a?=4,
:
(a) (6) (c)
(d) (e)
The values of y; i.e. the function, if"a; = The values of a*, if y = 2. The values of a*, if the function equals zero. The roots of the equation 2 -f 2 a a*2 = 0. The roots of the equation 2 -{-2x a*2 = l. -J-,
1-J-,
2J.
ELEMENTS OF ALGEBRA
158
Degrees of the Fahrenheit
24.
(F.) scale are
expressed in
degrees of the Centigrade (C.) scale by the formula
Draw
(a)
the graph of
C = f (F-32) 4 F F=l.
from to
From
(b)
the diagram find the number of degrees of centi-1 F., 9 F., 14 F., 32 F.
grade equal to
Change
(c)
A body
25.
to Fahrenheit readings
10
:
moving with a uniform
second moves in
t
seconds a distance d
C.,
C.,
1
C.
velocity of 3 yards per
=3
1.
formula graphically. Represent 26. If two variables x and y are directly proportional, then this
y=
cXj
where
c is a constant.
Show
that the graph of two variables that are directly proportional is a straight line passing through the origin (assume for c
any convenient number).
27.
If
two variables x and y are inversely proportional, then y = - where x
Draw
c is
the locus of this equation
a constant. if c
= 12.
GRAPHIC SOLUTION OF EQUATIONS INVOLVING ONE
UNKNOWN QUANTITY Since we can graphically determine the values of x make a function of x equal to zero, it is evidently possible Thus to find to find graphically the real roots of an equation. what values of x make the function x2 + 2x 4 = (see 192), we have to measure the abscissas of the intersection of the 195.
that
graph with the o>axis, i.e. the abscissas of 3.24. Therefore x = 1.24 or x =
P and
Q.
GRAPHIC REPRESENTATION OF FUNCTIONS 196. tion x 2
To
solve the equa4 1, de-
=
+2x
termine the points where If the function is 1. cross-section paper
the points
by
may
used,
otherwise
inspection,
draw through
is
be found
1) a line parallel to the #-axis, (0,
and determine the
1
abscis-
sas of the points of intersection with the graph,
2 and 1.
viz.
197. An equation of the the form ax2 bx c 0,
+ =
+
where
known
a, 6,
and
c
represent
\-3
-2
1/2
--1
quantities, is called
a quadratic equation. Such equations in general have
two
Y'
roots.
EXERCISE
74
Solve graphically the following equations
4x_ 7
5
:
-a -5 = 0. = 0.
10.
or
2.
11.
a2 -2a;-7
3.
12.
(a)
1.
0.
4. 6.
6.
13.
2
8.
14.
2
(6)
z
(c)
a2
(a)
x2
7.
9.
a:
(a) (6)
-6a;-f 9
= 0.
4x
6
= 0.
159
ELEMENTS OF ALGEBRA
160
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO UNKNOWN QUANTITIES 198.
equations involving two
of
Graph
unknown
quantities.
=
we can
represent graphically equations of the form y function of x ( 1D2), we can construct the graph or locus of any
Since
unknown
equation involving two
quantities, that can be reduced
to the above form.
Thus
to represent
x
x
\
-L^-
-
-
=2
y=
and construct
x
-
graphically, solve for
?/,
i.e.
-
A
graphically.
(
Ex.1. Represent graphically JJ y.
Solving for
Hence
y ='-"-
?/,
x
if
x
if
-
2,
y
2,
y
Locating the points
X'-2
== 2.
(2,
and joining by a straight
4
;
4)
line,
and
(2, 2),
produces the
If the given equation is of the we can usually locate two
199. first
Thus
_
required locus.
7*
y.
3x
in
degree,
points without solving the equation for the preceding example:
- 2 y ~ 2. = 0, y = -l.
3x
s
If
Hence we may
join (0,
1)
and
(f
,
0).
Ex.2. Draw the locus of 4 x + 3 y = 12. If
x
=
0,
?/
=4
Hence, locate points
them by
straight line
;
if
(0,
AB.
y
=
0,
fc
= 3.
(3, 0), and join the required graph.
4) and
AB
is
NOTE. Equations of the first degree are called linear equations, because their graphs are straight lines.
T
GRAPHIC REPRESENTATION OF FUNCTIONS
161
200. The coordinates of every point of the graph satisfy the given equation, and every set of real values of x and y satisfying the given equation is represented by a point in
the locus. 201.
To
Graphical solution of a linear system.
find the roots of
the system.
By
method
the
of
the preceding article construct the graphs
AB
and
and
(2) respectively.
The
CD
of
coordinates
(1)
of
AB
point in satisfy the equation
every
but only one point
(1),
in
AB
also satisfies
equation
(2), viz.
By measuring
x=
3.15,
y
AB
and CD. P, the point of intersection of the coordinate of P, we obtain the roots,
= .57.
202. The roots of two simultaneous equations are represented by the coordinates of the point (or points) at which their
graphs intersect. 203.
Since two straight lines which are not coincident nor simultaneous
parallel have only one point of intersection, linear equations have only one pair of roots.
Ex.
3.
Solve graphically the equations
:
(1)
\x-y-\- 1=0.
(2)
ELEMENTS OF ALGEBRA
162
Using the method of the preceding para, AB the locus of (1), and
graph, construct CD the locus of (2)
.
Measuring the coordinates of intersection,
Ex.
we
Solve graphically the
4.
lowing system
Solving (1) for y,y~ Therefore, if x equals respectively
0,
3,
Locating the
+ 3), (4,
4,
points
3), etc.,
V25
+
y*
-
5,
fol-
:
=
25,
(1)
=
-C.
(2)
.
4,
2,
3,
4.0,
4.5,
1, 0, 1, 2, 3, 4, 5,
4.9,
5,
4.5,
4,
y equals 3, 0.
(5,0), (-4, and
obtain the graph (a circle) 2 equation x
x2
of P, the point
obtain
joining, we of the
AB C
= 25.
Locating two points of equation (2), e.g. (-2, 0) and (0, 3), and joining by a straight line,
3x
-
we obtain DE, the graph
of
2 y = -6. Since the two
points
graphs meet in two and $, there are two pairs of By measuring the coordinates of
P
roots.
P and Q we find 204.
:
Inconsistent equations.
The equations 2 4
= 0, = 0,
(1)
(2)
cannot be satisfied by the same values of x and y, i.e. they are inconsistent. This is clearly shown by the graphs of (1) arid (2), which consist of a pair of parallel lines. intersection,
and hence no
There can be no point of
roots.
In general, parallel graphs indicate inconsistent equations.
GRAPHIC REPRESENTATION OF FUNCTIONS 205.
'163
Dependent equations, as
2^3 and
3x
-f
2y
==l
=6
have identical graphs, and, vice versa, idengraphs indicate dependent equations.
tical
EXERCISE
75
Construct the loci of the following equations: 1.
a+r/=6.
3.
2x
2.
2x
4.
x~y=0.
y6.
Draw system,
3?/=6.
4.
5.
y
6.
y=x + 5. 2
7.
y=
8.
a2
the graphs of the following systems, and solve each If there are no solutions, state reasons.
if possible.
9.
17. 1
6*
+ 7 y = 3.
\
2x
+ 3^
10.
.,
a;
y
= 4. 19.
20.
16
22.
16
23.
ELEMENTS OF ALGEBRA
164
"~ 24.
U
#
26.
- 14 y = - 8.
4 a
= 3(6 - y).
25.
29
28.
30.
.
3 31.
Show
that the same values of x and y cannot satisfy the
three equations
:
x
-f
5y
=
5.
TABLE OF SQUARES, CUBES, AND SQUARE HOOTS
CHAPTER
XIII
INVOLUTION 206.
Involution
the operation of raising a quantity to a
is
positive integral power. To find (#(**&)" is a problem of involution.
a special kind of product, involution repeated multiplication.
Law
207.
According to 50, 3 -f a = -f a = +- a2
of Signs.
-fa- -faa a a a Obviously 1. 2.
3.
Since a power
may be
is
.
.
a
=
a3
,
etc.
follows that
it
powers of a positive quantity are positive. All even powers of a negative quantity arc positive. All odd powers of a negative quantity are negative. ^4/?
(
is
a)
2 aft ) 9 is
positive, (
negative.
INVOLUTION OF MONOMIALS 208.
According to 1. 2.
8.
4.
effected
(-
3
2
6 3 )*
52,
= a2 5 = 6 (5 )* n m n (a ) = a (a
2
3
)
5
= (-
3 a2 6 8 )
a2
a2 b5
.
6
on
.
(-
6
= ?>
fi
to in factors
3 a268 )
a 8 = _ (2m )
(8
+ 2 = a. = 6+ 5 + +fi =
2+2 5
____ 16 *)"" 27 n 165
(-
62.
by
ELEMENTS OF ALGEBRA
166 To find
the exponent
of the power of a power, multiply tht
given exponents.
To
raise
a product
to
a given power,
raise each of its factors to
the required power.
To
raise
a fraction
to
a power, raise
terms to the required
its
power.
EXERCISE Perform the operations indicated 1.
2.
(>y.
2 4
(-a )
5.
3.
-
76 :
2 5
(-a )
/2mV. (-277171
2 11 .
(afc )
24.
\ 3 J
4
6.
4.
.
)*.
-
'
M-W 10.
(-2ar).
27
'
'
11.
-
13.
^---
/
_4_V ' _4_
V
V/
/-2?n?A 4
30.
3
15.
am-Vy)
16.
(-|^^)
. '
2
V 3xy )'
.
INVOLUTION OF BINOMIALS 209.
210.
by
and
+
The
square of a binomial
The
cube of a binomial
&-
we
was discussed
63.
obtain by multiplying (a
= a + 3a 6 + 3a6 + * 6) = a - 3 a 6 -f 3 a6 - 6 (a 6)
(a
in
3
3
2
2
8
8
3
2
2
8
_j_
,
.
+ 6)
1
INVOLUTION Ex.
Ex.
167
Find the cube of 2 x -f- 3 y.
1.
=
(2s)
s=
8 a; 3
3 + 3(2aO*(Sy) + 3(2aj)(3y)> + 36 z2y + 54 xy* + 27 y3 .
n of 3 x* - y
Find the cube
2. 2
(3 x
.
- y) = (3 y?y - 3(3 a*)a(y = 27 a - 27 ay + 9 x y2n 6
2
EXERCISE
77
Perform the operations indicated: 1.
2.
(a
+ &)8
(a?-?/) 3
(a-fl)
4.
(m-2) 8 (w+w)
6.
8.
.
3.
5.
7.
.
8
.
3
(a-j-7)
Find the cube root of
3
+ 4aj)
(7 a
-
-I) 2
.
a;)
(l
8 .
3
-I)
.
3 .
+5a)
3 8
(1
.
8
13.
(3a-f26)
14.
(6m+2w)
15.
(3
a- 6
16.
(3a
17.
(a
18.
(4
2
or*
62
.
8 .
8 ft)
-l)
.
3 .
-
:
20. 21.
a8 -3a2 + 3a-l.
a
.
10.
+ 3a 6 + 3a& -f-& ^-Sx^ + S^ -^
19.
2
3
+
3
a
12.
.
(1
2
(3
lx
-
-a)
9.
.
8
(5
2
3
2
3
86
23.
.
w + 3 w + ra8 -126 + G6-l. 2
1 -f 3
22.
.
3
.
2
211. The higher powers of binomials, frequently called ex. pansions, are obtained by multiplication, as follows :
+ 6) = o + 3 d'b + 3 a6 + = a + 4 a?b + 6 a & + 4 a6 + b + (a 6) = a + 5 a 6 + 10 a*b + 10 a 6 -f 5 aM + 6 (a + 6) 8
8
4
4
5
5
2
b*.
(a
2
2
4
3
.
4
2
2
An
examination of these results shows that
1.
The number of terms
is
s
3
,
etc.
:
1 greater than the exponent of the
binomial. TJie exponent of a in the first term is the same as the expo2. nent of the binomial, and decreases in each succeeding term by L
ELEMENTS OF ALGEBRA
168 T7ie
3.
exponent ofb
1 in the second term of the result,
is
and
increases by 1 in each succeeding term.
The The
4. 5.
of the first term is 1. of the second term equals the exponent of the
coefficient coefficient
binomial 6. TJie coefficient of any term of the power multiplied by the exponent of a, and the result divided by 1 plus the exponent of b, is the coefficient of the next term.
Ex.
Expand
1.
=
5
ic
-f
5 x*y
(x
+
10
^V +
Ex.2. Expand (a??/)
+
10 x*y*
The
212.
+ y5
.
5 .
2
x5
5 xy*
+
x'2
10
(-
+
5 x4 y
signs of the last answer arc alternately plus y are positive, and the
minus, since the even powers of
and odd
powers negative. Ex.
Expand
3.
16
2 (2 #
*
-
3 y3 ) 4
.
- 4(2 * )'(3 *f) -f 6(2 ^) (3 y ) 8 - 4(2^(3 ^'+(3 y - 96 ^y -f 216 o?y - 216 a^ 4- 81 y
2 4
2
2
8
)
8
ic
9
EXERCISE
8 4
)
12 .
78
Expand: 1.
(p + q)
4
7.
*
4
2.
(w
3.
(tf-f-1)
4.
+
(
9.
(c-fd)
.
10.
4
4
5.
(m-J)
6.
(l-a&)
.
4 .
.
5
.
?i)
?/)
(1-for)
8.
.
4
(1
4
14.
.
&) 5
(?/i-~w)
.
:
(a-f 5)
12.
(a~^)
(2 4- a)
16.
.
17.
.
18.
5
25.
(l-fa
s
19.
.
3
(m -fl)
(m
2
5
.
(m
2 5
)
5 .
I)
+ n)
(?>i?i 2
4
mn
15. (l
.
8
11.
13.
6 2 ) 5.
8 .
-f c)*.
.
5
(mnp
.
I)
2
5
20.
(2w
21.
(3a -f5)
.
-f-l)
2
4
2 22. (2 a
4
.
.
5)
23.
(2a-5c)
24.
(1 -f 2
4
4 a:)
.
.
CHAPTER XIV EVOLUTION 213. tity
;
is the operation of finding a root of a quan the inverse of involution.
Evolution
it is
\/a
=
V
27
\/P 214. 1.
x means x n
=y
= x means
It follows
Any
means r'
= y
?>
a.
=
= 6-,
27, or y
or x
~
3.
&4 .
from the law of signs
in evolution that
even root of a positive, quantity
may
:
be either 2wsitive
or negative. 2.
Every odd root of a quantity has
the
same sign as
the
quantity.
V9 = +
3,
or
-3
(usually written
\/"^27=-3, (_3) = -27. and ( v/o* = a, for (+ a) = a \/32 = 2, etc.
3)
;
for (-f 3) 2
and
(
3)
2
equal
0.
for
4
4
,
a)
4
= a4
.
215. Since even powers can never be negative, it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers.
numbers, and
all
other numbers are, for distinction, called real
numbers. Thus
V^I is an imaginary number, which can be simplified no further. 109
ELEMENTS OF ALGEBRA
170
EVOLUTION OF MONOMIALS The following examples root
are solved
by the
definition of a
,
:
=
Ex.1.
v/^i2
Ex.
2.
3/0**
Ex.
3.
v^SjW 3 = 2 a
= am
= ^/gL^g * c*
Ex
A
82
for (a")"
,
Ex.4.
5
(a
a*, for
a
3
)*
= a 12
= a mn
&c*, for (2
a"
.
.
a 2 6c4 ) 8
=
2
To
216.
?*-
= .lL,for(*Siy 3 3 6 c* \ c*J
extract the root
2 b'
of a power, divide
ft^c20
243
the exponent
by the
index.
A root of a product equals the product of the roots of the factors. To extract a root of a fraction, extract the and denominator. Ex.
6.
\/18
.
14
63
= V2 3* = 2 32 6
25
.
.
Ex.
Ex.
7.
8.
VT8226
= V25
Find (x/19472)
Since by definition
Ex.
729
2 .
7
.
7
.
.
roots of the
82
.
62
= V2*
.
numerator
3i
.
6-
= 030.
7
= V26TIT81 = 5-3.9 = 136.
2 .
= a, we
( v^)"
have (Vl472) 2
= 19472.
9.
= 199 + (_ 198) - 200 - (- 201) = 2. EXERCISE 5
5
1.
-v/2
.
3.
-fy
2.
V?.
4.
-v/2^.
9.
V36
9
-
100
a
2 .
3 .
5.
V5
6.
-v/2
79 2
7 2.
3
33
10.
V25
7.
53
\/2
9
4
16.
v- 125- 64
8.
.
4
9
5
4 .
EVOLUTION
33.
34.
35.
36.
VH) + (Vl9) 2
(
2
(
VI5) x ( VT7)
2
2
171
- (V200) -f ( VI5)
2
(V2441) ~(V2401)
2
r
+ b\
28.
-\/d -\-Vab
29.
V8- 75- 98- 3.
30.
V20
31.
V5184.
32.
V9216.
2
-f
(
V240)
x ( V3)
.
45
9.
2 .
3 .
2 .
2
(Vl24) -{
EVOLUTION OF POLYNOMIALS AND ARITHMETICAL
NUMBERS
A
217.
trinomial is a perfect square if one of its terms is
equal to twice the product of the square roots of the other terms. In such a case the square root can be found ( 116.)
by inspection. Ex.
1.
Find the square root of a2 - 6 ofy 2 -f 9 y4
_ 6 ary -f 9 y = (s - 3 y2) ( vV - 6 tfif + 9 y = O - 3 ;/). 4
a*
Hence
8
.
4
EXERCISE
.
2
116.)
3
80
Extract the square roots of the following expressions 2
1.
a -f2
2.
l
+ l.
2y-h2/
2 .
3.
^-40^4- 4/.
5.
4
9^ + 60^ +
6.
-
2 2/ .
:
ELEMENTS OF ALGEBEA
172 7.
4a2 -44a?> + 121V2
10.
8
4a
+ 6 + 4a&.
11.
49a 8 -
12.
16 a 4
.
.
s
2
mV-14m??2)-f 49;>
9.
13.
2 .
- 72 aW + 81 &
4 .
#2
14. 15.
a2
-
16.
a2
+ & + c + 2 a& - 2 ac - 2 &c. 2
2
218. In order to find a general method for extracting the square root of a polynomial, let us consider the relation of a -f- b 2 2 to its square, a -f- 2 ab + b .
The
first
term a of the root
the square root of the
is
first
2
term
a'
.
The second term
of the root can be obtained
second term 2ab by the double of
2ab
a-\-b
is
the root
if
a,
by dividing the the so-called trial divisor;
,
the given expression is a perfect square. it is not known whether the given
In most cases, however,
expression is a perfect square, and b (2 a -f b), i.e. the that 2 ab -f b 2
=
and
b,
we have then to consider sum of trial divisor 2 a,
multiplied by b must give the last two terms of the
square.
The work may be arranged
as follows
a 2 + 2 ab 2
:
+ W \a + b
EVOLUTION Ex.
1.
173
Extract the square root of 1G 16x4
x*
- 24 afy* -f 9 tf.
__
10 x*
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y. By doubling 4x'2 we obtain 8x2 the trial divisor. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder, 8 /-, we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder. As there is no remainder, Explanation.
The square
of x.
,
*/''
,
,
,
,
,
4 x2
3
?/
8 is
the required square foot.
219. The process of the preceding article can be extended to polynomials of more than three terms. We find the first two terms of the root by the method used in Ex. 1, and consider Hence the their sum one term, the first term of the answer.
double of this term find the next
Ex.
2.
is
new
the
term of the
root,
by division we
trial divisor;
and so
forth.
Extract the square root of
16 a 4
- 24 a + 4 -12 a + 25 a8 s
.
Arranging according to descending powers of 10 a 2.
4
-
a.
3
+
24 a 3
4-
a2
-f
10 a 2
24 a
25 a 2
-
12 a
+4
-
12 a
+4
10 a 4
Square of 4 a First remainder. First trial divisor, 8 a 2 . First complete divisor, 8 a 2
8
Second remainder. 6 a. Second trial divisor, 8 a 2 Second complete divisor, 8 a 2
As
there
is
a.
\
a
-f 2.
no remainder, the required root
is
(4
a'2
8a
+
2}.
ELEMENTS OF ALGEBRA
174
EXERCISE
81
Extract the square roots of the following expressions
2a + a4
+ 1.
2.
3 a2
3.
a4
4.
+ 81 a 4-54 a + 81. 25 m 20 w + 34 m - 12 m 4- 9. 4-12 a& -f 37 a' 6 - 42 a -f 49 a 6
5. 6.
2 a3
x2
2 or 4-1 3
4-
16 a4
-|-
2x.
24 a3
2
3
4
2
J
2
3
3
4
4
>
4
40 afy 4-46 x
2
24 a^
8
25 x
8.
16x6 4- 73a4 4-40^4-36^4-60^.
9.
l
-f-
if 4-
4.2^4-3^4-2^ 4-
a;
4-
.
4
7.
9
.
i/
4 .
10.
1 4- 4 x 4- 10 x2 4- 20 o 4- 25 x 4 4- 24
11.
36a 4-60a 4-73a 4-40a 4-16a
12.
36it-
13.
6
4-36^?/4-69a;V4-30^4-25^ 4m 4- 12m 5 4- 9m 4 20m3 30m 4- 25.
14.
49 a 4
6
5
3
4
6
.
- 42 a*& 4- 37 a ^ - 12 a6 2
2
2
13#4 4-13ar 4-4a;6 - 14^4-4
4 0^4- 20
or
16 x
3
3
4
17.
ic
18.
729 4- 162 a2 60 a10 4- 73 a8
4-?/ 4-2x-
20.
46 a
22
16
4
4-
a?
2
4a;
XT
x*y
6 a5 4- a 6 4-
-f
_^ + 2JX
24.
44 a
8
2xif
j/
36 a
25 a
2
12
4-
-h
4-
4 64
.
4- 16.
4^
J
4
6
iK .
4
16.
4-
16
2
x
.
4-
.
6
15.
19.
5
or
2
.
- 54 a 40 a
12 a
12^.
2
6
4-
3
4-
9 a4 .
4-
16 a4
4
4-
.
25 a6 4- 40 a
:
EVOLUTION The
220.
175
square root of arithmetical numbers can be found to the one used for algebraic
by a method very similar expressions.
Since the square root of 100 is 10; of 10,000 is 100; of 1,000,000 is 1000, etc., the integral part of the square root of a number less than 100 has one figure, of a number between 100 and 10,000, two figures, etc. Hence if we divide the digits of the number into groups, beginning at the
and each group contains two digits (except the last, which may contain one or two), then the number of groups is equal to the number of digits in the square root, and the square root of the greatest square in units,
group is the first digit in the root. Thus the square root of 96'04' two digits, the first of which is 9 the square root of 21'06'81 has three digits, the first of which is 4.
the
first
consists of
Ex.
1.
;
Find the square root of 7744.
From
the preceding explanation it follows that the root has two digits, the first of which is 8. Hence the root is 80 plus an unknown number, and we may apply the method used in algebraic process.
A will
comparison of the algebraical and arithmetical method given below identity of the methods.
show the
7744 80 6400 1
160
+ 8 = 168
+8
1344
1344 Since a
Explanation.
The is
trial divisor
2 a
=
= 80,
160.
a 2 = 6400, and the first remainder is- 1344. Therefore 6 = 8, and the complete divisor
168.
As
8
Ex.
x 168
2.
=
1344, the square root of 7744 equals 88.
Find the square root of 524,176. a f>2'41 '70
2 a
a2 = +6=
41)
1400
+ 20 = 1420
00 00
341 76
28400 4
=
1444
57 76
6776
[700
6
c
+ 20 + 4 = 724
ELEMENTS OF ALGEKRA
1T6
off groups in a number which has decimal begin at the decimal point, and if the righthand group contains only one digit, annex a cipher.
221.
places,
In marking
we must
Thus the groups
in .0961
are
'.GO'61.
The groups
of 16724.1 are
1'67'24.10.
Ex.
3.
Find the square root of
6.7 to three decimal places.
12.688
6/.70
4
45 2 70 2 25
508
4064 6168 41)600
41344 2256
222.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator, or by transforming the common fraction into a decimal.
EXERCISE Extract the square roots of
:
82
EVOLUTION Find
177
to three decimal places the square roots of the follow-
ing numbers: 29.
5.
31.
.22.
33.
30.
13.
32.
1.53.
34.
37.
Find the
side of a square
1.01. J-.
35.
T\.
36.
JT
.
whose area equals 50.58 square
feet.
38.
Find the side of a square whose area equals 96 square
yards. 39. feet. TT
Find the radius of a (Area of a
circle
circle
whose area equals 48.4 square when R = radius and
1 equals irR ,
= 3.1410.) 40.
Find the mean proportional between 2 and
11.
CHAPTER XV QUADRATIC EQUATIONS INVOLVING ONE UNKNOWN QUANTITY
A
223.
quadratic equation, or equation of the second degree,
an integral rational equation that contains the square of 4x the unknown number, but no higher power e.g. x 2 7, 6 y2 = 17, ax 2 + bx + c = Q. is
;
A
224. complete, or affected, quadratic equation is one which contains both the square and the first power of the unknown
quantity.
A pure,
225.
or incomplete, quadratic equation contains only
unknown quantity. + bx -f c r= is a complete quadratic ax 2 = m is a pure quadratic equation.
the square of the axt
The
226.
absolute term of an equation
does not contain any In 4 x 2
7
equation.
x
-f
12
=
unknown
is
the terra which
quantities.
the absolute term
/
is 12.
PUKE QUADRATIC EQUATIONS
= a,
A pure quadratic is solved by reducing it to the form and extracting the square root of both members.
Ex.
1.
227. 2 ic
Solve 13 x2 -19
Transposing,
= 7^ + 5. 6#2 =
etc.,
x*
Dividing,
24.
= 4.
Extracting the square root of each member, x = + 2 or x
=2.
This answer Check.
frequently written x
is
13(
2)2
-
19
= 33
;
178
=
2.
7(
2)*
+
5
= 33.
179
QUADRATIC EQUATIONS Ex.2.
Solve
.=g x2
Clearing of fractions, ax
4 a2
Transposing and combining,
+ 4 ax = ax + 4 a 2 + x2 -f 2 x2 = 8 a 2 4 a2 x2 = x = V 4 a2 x= x = .
2, Dividing by Extracting the square root,
.
,
or
Therefore,
EXERCISE Solve the following equations 1.
2.
3.
-7 = 162. 0^ + 1 = 1.25. 19 + 9 = 5500. o;
2
2
a;
7. 8.
9.
10.
(a?-
6(--2)=-10(aj-l).
-?
x
+
s-3
oj
+3
= 4.
2 4fc -5'
=:
18. '
y?
b*
b
83
:
4.
16^-393 = 7.
5.
15^-5 =
6.
4 ax,
ELEMENTS OF ALGEBRA
180 on
__!_:L
a;
&
-{-
23.
If a 2 4- b 2
24.
If s
=
If
= Trr
25.
a
27.
If 2
28.
If 22
2
22
.
'
c#
=c
2
('
,
2 ,
2 ,
= 4w
2
If s
26.
r.
-f c
2
(
2a
and
-f-
1
c.
,
= 4 Trr
2 ,
solve for
r.
m.
sol ve for
G=m m '
solve for v.
If
29.
,
g
EXERCISE 1.
+a
.
solve for
= ~^-,
x
find a in terms of 6
solve for
-f 2 b*
9
4,
a
a;
solve for d.
84
Find a positive number which
is
equal to
its
reciprocal
144). 2.
A
number multiplied by
its fifth
part equals 45.
Find
the number. 3.
150. 4.
The
ratio of
two numbers
Find the numbers.
(See
2
is
:
3,
and their product
is
108.)
Three numbers are to each other as 1 Find the numbers. is 5(5.
:
2
:
3,
and the sum
of their squares 5.
The
sides of
two square
fields are as
3
Find the side
tain together 30G square feet.
:
5,
and they con-
of each field.
6. The sides of two square fields are as 7 2, and the first exceeds the second by 405 square yards. Find the side of each :
field.
228.
A
right triangle is a triangle,
_____ b
contains
c
one of
The side right angle. opposite the right angle is called the hypotenuse (c in the diagram). If the hypotenuse whose angles
is
a
units of length, and the two other sides respectively
a and b units, then Since such a triangle tangle, its area contains
c
2
=a
may
2
-f-
b2
.
be considered one half of a
square units.
rec-
181
QUADRATIC EQUATIONS The hypotenuse
7.
of a right triangle
other two sides are as 3
4.
:
Find the
is
35 inches, and the
sides.
8. The hypotenuse of a right triangle is to one side as 13:12, and the third side is 15 inches. Find the unknown sides and the area.
The hypotenuse
9.
two
The area
10.
sides are as 3
:
4.
of a right triangle is 2,
Find these
sides are equal.
and the other
sides.
of a right triangle Find these sides.
is
24,
and the two smaller
11. A body falling from a state of rest, passes in t seconds 2 over a space s yt Assuming g 32 feet, in how many seconds will a body fall (a) G4 feet, (b) 100 feet?
=
The area $
12.
the formula
whose radius equals r is found by Find the radius of circle whose area S
of a circle
= Trr
/S
=
.
-J-
2 .
equals (a) 154 square inches, (b) 44 square feet. 7r
=
-2
(Assume
2
7
13.
.)
Two
circles together contain
radii are as 3
:
Find the
4.
3850 square
feet,
and their
radii.
8 = 4 wr2 Find 440 the radius of a sphere whose surface equals square yards. 14.
If the radius of a sphere is r, its surface
(Assume
=
ir
.
-2 2
7
.)
COMPLETE QUADRATIC EQUATIONS 229.
ample
Method
of completing the
illustrates the
method
The following
square.
ex-
of solving a complete quadratic
equation by completing the square. Solve
- 7 x -f 10 = 0. x* 7 x=
or
Transposing,
10.
member can be made a complete square by adding 7 x with another term. To find this term, let us compare x 2 The
left
the perfect square x2 of
or
2m, we have
m = |.
to
add
2
(|)
mx -f m
2
Hence ,
to
2 .
Evidently 7 takes the place 7x a complete square
make x2
which corresponds
to
m
2 .
ELEMENTS OF ALGEBRA
182
Adding
2
( J)
to each
member,
Or
= f. = \ # = ff. or x = 2.
(*-i) x
Extracting square roots,
Hence x
Therefore 62
Check.
Ex.1.
-7
5
5
-|
+ 10 = 0,
22
-7
.
2
+ 10 =0.
80^69^-2 =
Dividing by
= 6. = | x |.
9 x2
Transposing,
sc
9,
Completing the square
(i.e.
15 x 2
Extracting square roots,
at
Transposing, Therefore,
Hence
Q) 2
adding
to each
(*~8) a =
Simplifying,
230.
2
member),
.
= x-\ = 2, |
\.
a;
or
J. J.
to solve a complete quadratic
:
Reduce the equation to the form x*-\-px==q. Complete the square by adding the square of one half the coefficient of x.. Extract the square root and solve the equation of the first degree thus formed.
Ex.2.
x
a Clearing of fractions,
x2
x x2
Transposing, Uniting,
s
a
+ 2 a2
-f
a
x
2 ax
- x(l
-f 2 o)
= 2 ax. 2 a*
a.
= - 2 a2 - a,
QUADRATIC EQUATIONS
183
Completing the square,
Simplifying,
Extracting square root, x
- 1+2?= "*"
-
-
Vl - 4
a2
Transposing,
x
= l+ * a
~
Therefore
*
= 1 +2
Vl
EXERCISE
85
<*
V IT -*
<
ELEMENTS OF ALGEBRA
184
45
46.
2x
3
4.
= 12.
48.
o^
3 ax == 4 a9
49.
or
7 wr
la
231.
Solution
.
x -}-
=8
r/io?.
=0.
Every quadratic equation can be
by formula.
reduced to the general form, 2
ao; -\-bx-\- c
Solving this equation we obtain
= 0.
by the method of the preceding
article,
2a The
roots of
any quadratic equation may be obtained by 6, and c in the general answer.
substituting the values of a,
185
QUADRATIC EQUATIONS Ex.
= 26 x-5.
Solve 5 x2
1.
5 x2
Transposing,.
Hence
a
=
+ 20
Therefore
4-
5
= -
b
5,
= 0. 26, c = 5. V^tT)* - 4
20 x
.
6
.
6
10 ==
2024 =6or
Ex.
Solve
2.
Reducing
2
x
p*x
j>o?
to general form,
p.
px*
Hence
a
=p
P + 2
Therefore
1
t
b
=
2. 3.
+ 2 = 0. 3 x -11 + 10 = 0. 2# 11 + 15 = 0. 2or
}
-5o;
2
2
(p
11. 12.
a;
13. 14.
6.
15. 16.
= 64-120?.
17. 18.
8.
9.
19.
10.
20. 21.
2 o;
-
1), c
p. -
86
a?
6.
+
:
4.
7.
2
VQ^+T? ^4^
EXERCISE Solve by the above formula 1.
l.
6
10
= 44 x - 15 x9 25x* = 21
.
= 12 - 25 x. 6^+5^ 56. 7^ + 9 x 90. 6m = 7 m + 12 = 64 7 x2
2
2
a;
.
2
a;
TIO;
?i
2 .
ELEMENTS OF ALGEBRA
186
Find the roots of the following equations places
two decimal
to
:
= 1 - x.
22.
x2
23.
3x?+x = 7.
24.
ar>-8o;
25.
4-2a; a=:i^-^.
26.
x(x
27
= 14.
<2
- 4) = - 2.
x==
2S-3x
2 .
1
28
7a-l=--
-
7s
a?
232.
Solution
by
or,
Let
factoring.
e(l uation:
it
be required to solve the
5^ + 5=26*;
transposing
all
terms to one member,
Eesolving into factors, (5
a?
Now, if either of the uct
is zero.
-!)(- 5) =0. Bx
factors
1,
or #
5
is zero,
Therefore the equation will be satisfied
such a value that either
_,
5x
1
a?- 5
or
Solving (1) and (2),
we
= 0, = 0.
the prodif x has (1) (2)
obtain the roots
x
=^
or x
= 5.
233. Evidently this method can be applied to equations of if one member of the equation is zero and the other
any degree,
member can be Ex.
1.
factored.
Solve a*=
7a? + 15x
.
=7
Clearing for fractions,
2 2*
Transposing,
2a^7x
Factoring,
2
+
2
--16rc
+ 3) (x 2x-f3=0, orz sc(2
a = 0, Hence the equation has three
Therefore
se
x
roots, 0,
16 x.
= 0. = 0. 5 = 0.
5)
},
and
6.
QUADRATIC EQUATIONS Ex.
-3x
Solve x?
2.
4x
-
x*(x
Factoring,
2
3)
187
+ 12 = 0.
4 (x
= 0.
3)
-4)(z-3) = 0. (*-2)(x + 2)(a-3)=0.
O
Or Hence the
roots are 2,
2
2, 3.
members of an equation are divided by an the unknown quantity, the resulting involving expression equation contains fewer roots than the original one. In order If both
234.
to obtain all roots of the original equation, such a common divisor must be made equal to zero, and the equation thus
formed be solved.
If or x
we
be required to solve
members by x But evidently the value x 3
=
- 3)(x + 3
Ex.
it
divide both
= 2.
equation x (x
E.g. let
is also
5) = 0.
Form an
3.
The equation
is
a root, for
Therefore x
we
=
3 obtained from the
2
9
5 (x
or x
= 2.
a:
=3
obtain x 4- 3
equation whose roots are 4 and
(x
evidently
I.e.
Or
3)
= 0,
6.
4)(x - (- 6)) = 0. (aj-4)(a; + 6)=0. x2 -f 2 x - 24 = 0.
EXERCISE Solve by factoring
=5
3,
87
:
= 0.
5.
0^
6.
ar>
+ 100;= 24. + 10 a = 24.
14.
7.
a?-10a=:-24.
15.
5 = 0. 3^ 25^ + 28 = 0. + 9 -f 20 x = 0. 4or + 18a -f 8a;:=0. 3# y 5 = 0. 3^ = 0(110-6). 0(0-2) = 7(0-2).
8.
aj(
16.
(5
-|-6 2. 3.
4.
0^
+
21
= 10
a?.
ar'-Sa^ -12. a* 10a=24.
+ 8=s:
7.
9.
10.
11. 12. 13.
2o3 -f9a; 2
3
or
a;
}
2
2
or
ELEMENTS OF ALGEKRA
188 f
17.
tt(3tt
18.
uz + u
+ 7tt)=6tt. 2.
21. 22. (2a?
3) (a 24. 25.
26.
ara +
(a
19.
w(w
20.
x2
2
w)=6tt. a 2 =(x
+ 2)=
(+ 3)(a?+2).
23.
3
or
-a -2
(y( j_ ?
ft
a)b.
+ 1) (a- 3) = (s + l) (3 -a).
+ c*.
27.
50.
'-3a!J -
2
a?
QUADRATIC EQUATIONS Form
the equations whose roots are
51.
3,1.
52.
3,
-4.
53.
-2, -5.
55.
54.
0,9.
56.
189
:
-2,3.
57.
1,2,3.
-2,3,0.
58.
2,0, -2.
1,
PROBLEMS INVOLVING QUADRATICS Problems involving quadratics have
in general two answers, but frequently the conditions of the problem exclude negative or fractional answers, and consequently many prob-
235.
lems of this type have only one solution.
EXERCISE
A
1.
88
number increased by three times
its reciprocal
equals
Find the number.
6J. 2.
Divide CO into two parts whose product
3.
The
difference of
of their reciprocals is
|.
two numbers is 4, and the difference Find the numbers.
Find two numbers whose product
4.
is 875.
is
288,
and whose sum
is 36.
The sum
5.
What
85. 6.
of the squares of
are the
numbers
The product
of
two consecutive numbers
is
?
two consecutive numbers
is
210.
Find
the numbers. 7.
Find a number which exceeds
8.
Find two numbers whose difference
product 9.
its
square by is
G,
-|.
and whose
is 40.
Twenty-nine times a number exceeds the square of the 190. Find the number.
number by 10. The
sides of a rectangle differ by 9 inches, and equals 190 square inches. Find the sides. 11.
A
its
area
rectangular field has an area of 8400 square feet and Find the dimensions of the field. feet.
a perimeter of 380
ELEMENTS OF ALGEBRA
190
The length
12.
AB of a rectangle, ABCD, exceeds its widtK AD by 119 feet, and the line BD joining
B 1
two opposite .
c equals 221
vertices (called "diagonal")
feet.
Find
AB and AD.
The diagonal
13.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
tangle as 5 4, and Find the sides of the rectangle. :
A man
14.
sold a
watch for $ 24, and lost as many per cent Find the cost of the watch.
as the watch cost dollars.
A man
15.
sold a watch for $ 21, and lost as many per cent Find the cost of the watch. dollars.
watch cost
as the
A man
16.
Two steamers
17.
of 420 miles. other,
and gained as many per Find the cost of the horse.
sold a horse for $144,
cent as the horse cost dollars.
and
is
ply between the same two ports, a distance One steamer travels half a mile faster than the two hours less on the journey. At what rates do
the steamers travel ? 18. If a train had traveled 10 miles an hour faster, it would have needed two hours less to travel 120 miles. Find the rate
of the train. 19. Two vessels, one of which sails two miles per hour faster than the other, start together on voyages of 1152 and 720 miles respectively, and the slower reaches its destination one day
before the other.
How many
miles per hour did the faster
vessel sail ?
If 20. A man bought a certain number of apples for $ 2.10. he had paid 2 ^ more for each apple, he would have received 12 apples less for the same money. What did he pay for each
apple ?
A man bought a certain number of horses for $1200. had paid $ 20 less for each horse, he would have received two horses more for the same money. What did he pay for 21.
If he
each horse ?
QUADRATIC EQUATIONS
191
--
On the prolongation of a line AC, 23 inches long, a point taken, so that the rectangle, constructed with and CB as sides, contains B 78 square inches. Find and CB. 22.
B
is
AB
AB
23.
A rectangular
24.
A
grass plot, 30 feet long and 20 feet wide, is surrounded by a walk of uniform width. If the area of the walk is equal to the area of the plot, how wide is the walk ? circular basin is surrounded
and the area of the path
is
-
by a path 5
feet wide,
Find
of the area of the basin.
=
the radius of the basin.
2
TT r (Area of a circle .) 25. A needs 8 days more than B to do a certain piece of work, and working together, the two men can do it in 3 days. In how many days can B do the work ?
26.
Find the side of an equilateral triangle whose altitude
equals 3 inches. 27. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost. How many eggs can be bought for $ 1 ?
236.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms, and the unknown factor of one of these terms is the square of the unknown factor of the other, as 0,
^-3^ = 7,
2
(tf- I) -4(aj*-l)
= 9.
237. Equations in the quadratic form can be solved by the methods used for quadratics.
Ex.
1.
^-9^ + 8 =
Solve
**
By formula,
Therefore
x
=
\/8
0.
=9
= 2,
or x
= \/l = 1.
ELEMENTS OF ALGEBEA
192 238. stitute
Ex.
In more complex examples it is advantageous to sub a letter for an expression involving a?.
+ 15 =
2.
x
<
J
Let
Then
=
or
r-f 15
or
y-8)=0. Hence
>,
or y
=
0,
8.
Le. Solving,
=
1,
EXERCISE Solve the following equations 1.
4 a; 4
-10a; 2 -h9:=0. 4-36
= 13.T
2.
a;
7.
3 a4
8.
16 a^-40
3.
2
4.
.
a4 -5o;2 =-4.
2
11.
4
6.
4
a -21or=100.
-44s + 121=0.
aV+9o
89
:
=0.
9.
10.
4
4
-8 = 2 a*
6.
-37aj 2 = -9. 2
2
(a:
4
+aj)
-18(x2 +a;)+72=0,
2 (^-Z) -
12.
"3 14.
1=2*. T
15
16.
^^
a?
17.
(a?-
18. 19.
^ 2:=Q>
~ 28
193
QUADRATIC EQUATIONS
CHARACTER OF THE ROOTS 239.
The quadratic equation
oa/*
2
bx
-f-
1.
2.
3.
it
follows 2
is
4c
is
a positive or equal to zero, the roots are real. negative, the roots are imaginary. a perfect square, the roots are rational.
4 ac
is
Iflr
kac
is 'not
4 ac
is zero,
4ac
is
2
(
:
4 ac
If b Ifb* 2 If b
Ifb 2 Jfb
has two roots,
2a
2a Hence
=
c
-f-
a perfect square, the roots are irrational. the roots are equal.
not zero, the roots are unequal.
240. The expression b 2 the equation ay? 4- bx 4- c
4 ac
is
called the discriminant of
= 0.
Ex. 1. Determine the character of the roots of the equation 3 a 2 - 2 z - f> = 0. The discriminant =(- 2) 2 4 3 (- 5) = 04. .
Hence the roots are
real, rational,
and unequal.
Ex. 2. Determine the character of the roots of the equation 4 x2 - 12 x + 9 = 0. 2
4
4
9
= 0,
the roots are real, rational, and equal.
Since
(
241.
Relations between roots and coefficients.
12)
the equation ax2 4- bx 4-
are denoted
c
__
b 4- Vfr 2
Tl
Vi
2
2a
Or
/ 1
4-r2
4 ac '
T* b
Hence
by
= a
,
4 ac
i\
If the roots of
and r2 then ,
ELEMENTS OF ALGEBRA
194
If the given equation is written in the form may be expressed as follows
these results
If the (a)
2 a?
+ a-x + -a =
0,
:
ofx
coefficient
2
in
a quadratic equation
The sum of the roots
is
equal
is
unity,
of x with
to the coefficient
the
sign changed. (b)
The product of the roots
is
2 E.g. the sain of the roots of 4 x
equal to theubsolute term, -f
5 x
3
=:
j, their
is
product
is-f.
EXERCISE
89 a
Determine without solution the character of the roots of the following equations
:
4.
= 0. 5a -26a? + 5 = 0. 2x* + 6x + 3 = 0. or + 10 + 4520 = 0.
5.
^-12.
6.
3a;2
7.
9x2 ~
1.
2
o;
-lla; + 18 2
2.
3.
a;
+ 4a: + 240 = 0.
2
+ 2-a;. = 5x. 12~x = x
8.
5aj
9.
x2 -7
10.
n
2
.
a?-3 '
12.
10 x
== ~
l
= 25 x + 1. 2
In each of the following equations determine by inspection sum and the product of the roots:
the
13.
14. 15.
= Q. -9a-3 = 0. 2a -4z-5 = 0. x2 -!i>x + 2
16.
z2
17.
2
18.
= 0. tfmx+p^Q. 5oj -aj + l = 0. Sa^ +
2
Ooj
2
Solve the following equations and check the answers by
forming the sum and the product of the roots 19.
20.
21.
a 2 - 19 #
= 0.
+ ^ + 2^-2 = 0. + 2a-15 = 0. 2
ar
60
:
22.
x2 -4 x
23.
0^
+
24.
or
j
-
205
= 0. = 0. + 12 2
CHAPTER XVI THE THEORY OF EXPONENTS 242. The following four fundamental laws for positive integral exponents have been developed in preceding chapters :
a m a" = a m+t1 . ~ a m -f- a" = a m n
I.
II.
m
mn . (a ) s=a m = aw bm
III.
IV.
The
first
,
provided
w > n.*
a
(ab)
of these laws
is
.
the direct consequence of the defiand third are consequences
nition of power, while the second of the first.
FRACTIONAL AND NEGATIVE EXPONENTS Fractional and negative exponents, such as 2*, 4~ 3 have meaning according to the original definition of power, and
243.
no
,
we may choose
for such
symbols any definition that
is
con-
venient for other work. It is, however, very important that all exponents should be governed by the same laws; hence, instead of giving a formal definition of fractional and negative exponents, we let these quantities be what they must be if the exponent law of multiplication is generally true.
244.
of
We assume,
m and n.
m therefore, that a
Then the law
*The symbol
>
an
= a m+n
of involution, (a m ) w
means "is greater than"
smaller than."
195
;
,
for all values
= a""
similarly
<
1
,
must be
means "is
ELEMENTS OF ALGEBRA
196
true for positive integral values of n, since the raising to a positive integral power is only a repeated multiplication.
Assuming these two 8*,
n 2 a, 4~ , a ,
laws,
we
try to discover the
In every case we
etc.
let the
meaning of
unknown quantity
and apply to both members of the equation that operation which makes the negative, fractional, or zero exponent equal
x,
disappear. 245.
To
find the
meaning
is
of
a fractional exponent;
x
Let
e.g. at.
a*.
The operation which makes the fractional exponent disappear evidently the raising of both members to the third power. Hence
^=(a^)
Or
3*
3 .
= a.
0?=-^.
Therefore
-
Similarly,
Hence we
we
find
a?
define a* to be the qth root of of.
Write the following expressions as radicals : 22.
m$.
24.
a\
26. (xy$.
28.
23.
a?*.
25.
A
27. 3*.
29. as.
(bed)*.
30.
'&M
31. ml.
THE THEORY OF EXPONENTS Express with fractional exponents 32.
33.
-\fi?.
36.
Vo5
37.
.
:
34.
-v/o&cT
ty?.
2'
= 4.
41.
a*
= 3.
43.
3*
40.
4*
= 2.
42.
*
= 2.
44.
27*
47.
4*
v'mT
-\/m.\/n.
:
39.
Find the values of
35.
38.
-\/xy-
Solve the following equations
197
= 27.
45.
5 a*
= 10.
= 3.
46.
7z*
= 49.
:
+ 9* + 16* + 25* + 36*.
48. 49.
64*
+ 9* + 16* + (-32)*.
50.
246.
To
find the
meaning
of zero exponent, e.g.
a.
a = a.
Let
The operation which makes the zero exponent disappear 2 evidently a multiplication by any power of a, e.g. a *
is
a2
a=l.
Or
Therefore the zero power of any number NOTE.
If,
however, the base
Indeterminate.
is
zero,
is
equal
to unity.
5L is indeterminate
a
;
hence
is
ELEMENTS OF ALGEBRA
198 To
247.
find the
meaning
of a negative exponent, e.g.
x=
Let
or".
Multiplying both members by
Or
a"#
248. Factors
may
cr n.
an x = a.
a",
= l.
be transferred
denominator of a fraction, or
from
vice versa,
the
numerator
to
the
by changing the sign of
the exponent.
NOTE.
The
fact that a
It loses its singularity
each
is
if
= we
1 sometimes appears peculiar to beginners. consider the following equations, in which
obtained from the preceding one by dividing both
a8
=
1
2
=
1
a1
=
1
a- 2
=
a
a
a2
a
a a
.
a
,
etc.
a
members by
a.
THE THEORY OF EXPONENTS EXERCISE
199
91
Find the values of:
Express with positive exponents 21.
or 5 .
22.
6 or 2
24.
7~ l a 2b 2
25.
:
^-^ ^. 3
27.
a;-
.
*
""^T"*'
.
Write without denominators 29.
<W*
* 31
'
arV
l>
30.
:
^L. c
32.
8
^?2 y'
34.
Write with radical signs and positive exponents 35.
mi
36.
m~^.
37.
3
40.
(2w)~i -.
2
,
38.
3
39.
2m~i
a?
*
.
a^
66
1
f
41. cci
44.
:
a;"*
45.
.
42.
m
43.
rfS.
2
.
1 -L ?>i""i
ELEMENTS OF ALGEBRA
200
Solve the equations
:
= l. ar = i. 2 =f 3* = f x~
46.
l
2
47.
z
48. 49.
Find the values
= 1. = 5.
= .1. = -^.
50.
17'
54.
10*
51.
z*
55.
5*
52.
5or*=10.
53.
10*
= .001.
of:
56. 57.
3-ll-
58.
4~*
59.
60.
61.
+ 1~* -f 21 - 9*. (81)* + (3f)*-(5 TV)*-3249 + 16 * - 81 -f (a - 6). - (.008)* + A. + A_. (.343)* + (.26)* 1
(I-)
2
.
5
-
75
USE OF NEGATIVE AND FRACTIONAL EXPONENTS 249. It can be demonstrated that the last three laws for any exponents are consequences of the first law, and we shall hence assume that all four laws are generally true. It then follows
that:
Fractional and negative exponents
may
be treated by the
same
methods as positive integral exponents. 250.
Examples relating
to roots can be reduced to
taining fractional exponents.
Ex.
1.
Ex 2
(a*&~*)*
+
(aVM = a*&~* +
V
'
=
'*&*
examples con-
THE THEORY OF EXPONENTS
201
Expressions containing radicals should be simplified as
251.
follows
:
(a)
Write
(6)
Perform the operation indicated.
(c)
Remove
all radical
signs as fractional exponents.
the negative exponents.
remove the fractional exponents.
(d) If required,
NOTE.
Negative exponents should not be removed until all operations of multiplication^ division, etc., are performed.
EXERCISE Simplify
2.
&.&.&.$-".$-*.
3.
72
4
25
26
5.
a- 3
-
6.
aj"
7.
6a-.5a.
8
6 *- 6 *' 6 *-
'
.
92
:
79
.
7~ 5
-
3 a- 4
a8
OA 20.
.
2~ 9
2~ 8
27
a- 4
7~ 6
.
.
2 a?
2 ar 1
.
22.
3-s-VS.
23.
/ 7-f--v 7.
25. 9.
7*.7i.7*.7W. .
26.
,
4 x^.
10.
#*
11.
V5.^/5-^5.
27
12.
95 -^9i
28.
13.
5-*-*- 5.
14.
S-'-s-S-8.
16.
a9 -i-a- 4
16.
14an-
17.
(4**-
18.
(Va)
'
a;
4 .
.
-
__ 29-
/m '-=V--
ELEMENTS OF ALGEBRA
202
V ra 4/
32.
-\/m
3
6
33. 34.
35.
40.
we wish to arrange terms according to descending we have to remember that, the term which does not contain x may be considered as a term containing #. The 252.
If
powers of
a?,
powers of x arranged are
Ex.
1.
:
1 Multiply 3 or
+x
Arrange in descending powers of
5 by 2 x
1.
x.
Check.
lix
=+1
2x-l
Ex.
2.
=
Divide
by
^ 2a
3 qfo
4- 2 d
THE THEORY OF EXPONENTS EXERCISE
203
93
Perform the operations indicated:
2. 3.
4. 5.
(7r-8Vr + r>)(9 Vr-7). 2 - 1 ). (a- + a -f 1) (a~ + a 2
2
2
6.
7. 8. 9.
10. 11.
12.
13.
14.
(4
a- 3
- 24 a- - 9 - 3 a~ )
1
2
1
-r-
(a"
- 3).
+ + 47i) + 35V5?)-*-(5Vp + l). VS" ^- ( Vo Vft) 1C H- (a~ -f 7 a- ^a~ a-*b~ 33 a6~ + 14 a+ (3 a _&)-*. (-^? + ^/-^ + */fr^ 15. 16. (a-6 + 2V6c c)-^-(Va+V6 Vc). 17. -y^TTOa; -f 13 - 12 *- + 4 aF*. (13Vp
l
(Va^-f aV^-&Va 5
3
)
3
l
2
2
^>~
2
3
1
1
)
(
1
18.
Vor
19.
V25 #
20.
^^
21. 22. 23. 24. 25.
l
2
2 x -h or 2
2 or
1
-f-
3.
- 2()"ar r+ 34 - 12 x -f 9 x*.
+2
(l+4^-flO^ + 20oT-f 25^T -f-24-\/i?-f 16 (1+V2)V2. (2+V2)(V2-2). (5+V3)(5-2V3).
a?
8
)*.
26.
(1-3VS)(2 + V5).
27.
(VU - V2)(Vn~3V2)
ELEMENTS OF ALGEBRA
204
Find by inspection 28. 29.
:
(x*
+ 3)(tf*-f 2).
a*
+ 3l-5.
35. 36.
8 (a;*
yi)
(5*-2*
.
2 .
30. 31.
V-
38.
2
32.
(3^
33.
(#* ^
2*)
.
39.
34.
(fl
-f-
5) (x*
5).
40.
(m
n)
-f-
11
(m*
-f-
n 5 ).
CHAPTER XVII RADICALS 253.
A
a quantity, indicated by a
radical is the root of
radical sign.
254.
The
radical is rational, if the root can be extracted
exactly; irrational, if the root cannot be exactly obtained. Irrational quantities are frequently called surds.
^9 4^
255.
The
+ V) *
are radicals.
= 2, V(a + 6) 2 are rational.
\/2,
root.
(*
V4a-f
b are irrational.
order of a surd /-
is
indicated by the index of the
va
is
of the second order, or quadratic.
\/2
is
of the third order, or cubic.
Vc
is
of the fourth order, or biquadratic.
.
256. A mixed surd is the product of a rational factor and a surd factor; as 3Va, a;V3. The rational factor of a mixed surd is called the coefficient of the surd.
An 257. factor.
entire surd is
one whose coefficient
Similar surds are surds 3v/2 and 6
3V2 and
is
unity; as
Va,
which contain the same irrational
av^
are similar.
3 V8 are dissimilar.
206
ELEMENTS OF ALGEBRA
206 258.
Conventional restriction of the signs of roots.
All even roots
may
be positive or negative,
VI = + 2
e.g.
or
2.
Hence 6. which results in four values, viz. 14, 6, To avoid 14, or this ambiguity, it is customary in elementary algebra to restrict
the sign of a root to the prefixed sign.
5 V4 4- 2 V4
Thus
= 7 VI = 14.
If the object of an example, however, is merely an evolution, the complete answer is usually given thus ;
=-
(oj- 2).
Since radicals can be written as powers with fractional
259.
exponents, all examines relating to radicals
may
be solved by the
methods employed for fractional exponents.
Thus, to find the nth root of a product ab we have T
1
1
(a6)"==a"6" I.e.
(242).
to extract the root of a product, multiply the roots of the
factors.
TRANSFORMATION OF RADICALS 260.
Simplification of surds.
A radical is simplified when the
expression under the radical sign is integral, and contains no factor whose power is equal to the index.
Ex.
1.
Simplify
= \/25~a~ Vb = 6 a*VS. 4
Ex.
2.
Simplify
-v/16.
-J/lB^^.
4/2
= 2^.
RADICALS
207
261 When the quantity under the radical sign is a fraction, we multiply both numerator and denominator by such a quantity as will make the denominator a perfect power of the same .
degree as the surd.
Ex.
3.
Simplify V|.
Ex.
4.
Simplify
EXERCISE
94
ELEMENTS OF ALGEBRA
208
/s
39.
37.
j
*x+y 38.
262.
An
\ 2m
n
imaginary surd can be simplified in precisely the as a real surd thus,
same manner
;
,
42.
V-16a
2
44.
.
2\-
Simplify and find to three decimal places the numerical values of 47. 48.
:
VJ.*
49.
Vf.
VJ.
50.
VA
263.
Reduction of a surd to an entire surd.
Ex.
Express 4 a V& as an entire surd.
EXERCISE Express as entire surds
95
:
1.
4V5.
3.
2-\/lL
5.
2.
3V7.
4.
3^5.
6.
7.
a VS.
8.
* See table of square roots on page 164.
RADICALS
209
264. Transformation of surds to surds of different order.
Ex.
1.
Transform -\/uW into a surd of the 20th order.
Ex.
2.
Transform
lowest order.
\/2,
V3, and
\/5 into surds of the
same
V2 = 2* = a* = '#64. |^ = 8* = 3A= ^gi. ^5 = 6* = 6* =^125. 1
Ex.
Reduce the order of the surd tyaP.
3.
Exponent and index bear the same relation as numerator and denominator of a fraction ; and hence both may be multiplied by
same number, or both divided by the same number, without changing the value of the radical. the
EXERCISE Reduce 1.
Va?.
96
to surds of the 6th order
-fymn.
2.
Reduce
3.
\/ v
:
4.
v'c?.
to surds of the 12th order
7.
V2~a.
8.
^v/mV
9.
10.
5.
\|
z
\
^-
6.
mn.
3
:
\/a4 6 2c.
11.
-\/oP6.
13.
-\/3ax.
12.
\/5a5V.
14.
a.
Express as surds of lowest order with integral exponents and indices :
5
15.
-v/o
20.
A/^
.
16.
\/oW.
22.
17.
VSlmV.
2
-v/IaT .
24.
18.
-\/
ELEMENTS OF ALGEBRA
210
Express as surds of the same lowest order
:
32.
25.
V3,
29.
2\
26.
A/2, s!/3.
30.
V2,
A/3,
^5.
33.
V3,
27.
-v/3,
^2.
31.
-v^S, -\/5,
-^7.
34.
^2, ^4,
28.
-\/7,
V2.
3*.
Arrange
in order of
35.
-v/3,
V2.
37.
\/7,
VS.
36.
-v/4,
-^6.
38.
V5,
^/IT,
magnitude :
^126.
39.
5V2, 4^/4.
40.
-^2,
^3, ^30.
ADDITION AND SUBTRACTION OF RADICALS 265.
form. terms their
To add or
(i.e.
add
proper
Ex.
subtract surds, reduce them to their simplest add them like similar
If the resulting surds are similar,
1.
their coefficients) ; if dissimilar, connect
them by
signs.
V| + 3 VlS- 2 V50.
Simplify
VJ + 3VT8 - 2 V50 = V2 + 9 V2 - 10 V2 = I
V2.
Ex.2. Simplify/a35 ~
o - 3-\|
Ex.
3.
+
3:
,
.
\/=^8
v~
^y
Simplify
V|~
8ft 2- s/a;
3 -
s/-
/
3ft 2
-
3
RADICALS EXERCISE
211
97
Simplify the following expressions
:
2.
2V8-7Vl8-f5V72-V50.
3.
VT2 + 2V27 + 3V75-9V48.
4.
V18+V32-VT28+V2.
6.
V175-V28+V63-4V7.
6.
VJ+V8-V1 + V50.
7.
4V80-5V45-.3V20 + 6V5.
8.
8VT8-J-2V32
+ 3V8-35V2.
9.
10.
11. 12. 13. 14.
V45c3 3 abv'ab
V80~c~3
V5a c + c 2
-f
+ 3 aVo^
3 Va^
;J
a6 V4
aft.
ELEMENTS OF ALGEBRA
212
.
23.
98 ab
^" fab
"
1
.fab
V
FW
\~\ jab
MULTIPLICATION QEJRABIQALS Surds of the same order are multiplied by multiplying product of the coefficients by the product of the irrational
266. the
for a~\/x b~\/y ab^/xy. Dissimilar surds are reduced to surds of the same order, and
factors,
then multiplied.
Ex.
1.
2
Multiply 3-\/25^ by 5\/50Y 3v
/
26^
.
5 4/6072
=
V2 by
16^6272.
.
6*. y*
=
Ex.
2.
Multiply
Ex.
3.
- 2 VS by 3 Vf + 10 VB. Multiply 5 V7
3\/l.
6\/7- 2v/6 + IPV6
8\/7
105- 6V35 106
4-60V35-100 - 100 = -f 44 VS6
6
+ 44\/36.
RADICALS EXERCISE
3.
V3 Vl2. V2 -V50. V3 V6.
4.
V
5.
Vr
1.
2.
213
98
6.
VlO V15.
11.
-v/18
7.
12.
8.
Vll.VSS. V20 V30.
13.
V5 Va
-VTO.
9.
-v/4.^/2.
14.
Va-
V42.
10.
-\/3
15.
V?/
16.
aVa; 6 V4
17.
V2a-V8^.
-\^).
-v"3.
fWa
a?.
18. 19.
25.
(V2+V3+V4)V3.
27.
(5V2-2V3-CVS)V3.
28
(3
20.
21.
10
40
Vm
1
.
30.
Vm) (Vm-f 1
(Vm-Vn)(Vm+Vn>
33.
(
34.
(Va
36.
(6V2-3V3)(6V2-|-3V3).
37.
(5V5-8V2)(5V5 + 8V2).
38.
(Vm-Vn)
39.
2
-\-
Va
(V3-V2)
-{-
Vm).
6(Va-f Va
8 .
.
40. 41.
+ VB)(2-V5).
6.
(V6 + 1)
(2-V3)
1 .
8 .
ELEMENTS OF ALGEHRA
214
S
42.
(3V5-5V3)
44.
(3V3-2Vo)(2V3+V5).
45.
(2
46.
(5V7-2V2)(2VT-7V2).
47.
(5V2+V10)(2V5-1).
43.
.
V3 - V5) ( V3 + 2 VS).
48.
49.
(3V5-2V3)(2V3-V3).
60.
51.
52.
Va
53.
-v/a.
-v/a
-
DIVISION OF RADICALS 267. Monomial surdn of the same order may be divided by multiplying the quotient of the coefficients by the quotient of the
surd factors.
E.y. a
VS
-f-
a?Vy
= -\/ -
x*y
Since surds of different orders can be reduced to surds of
the same order,
all
monomial surds may be divided by
this
method.
Ex.
1
Ex.
2.
268.
If,
(V50-f 3Vl2)-4-V2== however, the quotient of the surds
is
a fraction,
it
more convenient to multiply dividend and divisor by a factor which makes the divisor rational.
is
RADICALS
215
This method, called rationalizing the the following examples
Ex.
1.
Divide
divisor, is illustrated
VII by v7.
In order to make the divisor (V?) rational,
by V7.
VTL_Vll '
Ex. The
2.
by
:
~~" \/7_V77
,
we have
to multiply
/~
}
Divide 4 v^a by
rationalizing factor
is
evidently \/Tb
;
hence,
4\/3~a'
36
Ex.
Divide 12 V5
3.
Since \/8
=
+ 4V5 by V.
2 V*2, the rationalizing factor x
+ 4\/5 _ 12v 3 + 4\/5 V8 V8
12 Vil
g '
is
\/2,
V2 V2
269. To show that expressions with rational denominators are simpler than those with irrational denominators, arithTo find, e.g., metical problems afford the best illustrations.
- by the usual arithmetical method, we have
V3 But
if
we
simplify
1.73205
JL-V^l V3
^>
*>
Either quotient equals .57735. Evidently, however, the by 3 is much easier to perform than the division by
division
1.73205.
Hence
in arithmetical
work
it
is
rationalize the denominators before dividing.
always best to
ELEMENTS OF ALGEBRA
216
EXERCISE
99
Simplify : 1.
^/H 7.
.
V7 ~
V8?^
11 n
13
T
VH
V7 xy
-
8.
Vn
14.
Vf-f-V?.
5 -2-.
2V5
*
'
2 V3
V7
o
^
'
vfi* '
Va
Vll 212*.
12.
--.
V2 = 1.4142, V3 = 1.7320,
Given
and
V5 = 2.2361,
find to
four decimal places the numerical values of: 19.
-i.
20.
V2 22
.
12..
A.
21.
V3 23
V5
.
A.
V8 24
V8
.
JL.
25.
V48
4=V50
Two binomial quadratic surds are said to be conjugate, they differ only in the sign which connects their terms.
270. if
Va + Vb
,
and
Va
Vb
are conjugate surds.
271.
The product of two conjugate binomial surds
272.
To
rationalize the denominator of a fraction
is
rational
whose denom-
a binomial quadratic surd, multiply numerator and denominator by the conjugate surd of the denominator.
inator is
RADICALS Ex.
1.
Simplify
2V3-V2
217
'
V3-V2 ~
= 4 + V5.
Ex.2.
a;
s Simplify
- vffi^T _ - Vs2 - 1 x-Vtf a;
Ex.
3.
Find the numerical value of
V2 + 2 2V2-1
:
,
V2+2 _ V2+2 2\/2+l_6 + 6\/2.= 18.07105 = 7 7 2V2-1 2V2-1 2V2 + 1 e
EXERCISE Eationalize the denominators of
100 :
.
.
V8-2
2-V3
1-fVS
ELEMENTS OF ALGEBRA
218
13
6
~3
.
A
16.
V5-1
17
V3-V2
1-Va?
5V7-7V5 '
V5-V7
^-SVg.
15.
19.
V5-2
Vg+v/2
14
6V7-.W3.
18>
2V5-V18
m-Vm Va
22.
=
Given V2 1.4142, four places of decimals 23
_!_.
.
V3 = 1.7320, 25
.
V2-1 -=
24.
and
V5 = 2.2361;
find to
:
-J?_.
27.
Vo-1 26.
V3 + 1
v
2-V3
_
28.
3-V5
1+V5
'
V5+2 Find the third proportional
31.
V3-2* to 1
+ V2
and 3
-f-
2V2.
INVOLUTION AND EVOLUTION OF RADICALS 273.
By
the use of fractional exponents
shown that VcT = ( V) w Hence
.
V25~3 = ( V25) 3 - 5 3 = 125.
it
can easily be
RADICALS
219
274. In other examples of involution and evolution, introduce fractional exponents :
Ex.
1.
Simplify
Ex.
2.
Find the square
of
EXERCISE Simplify 1.
101
:
(3Vmw)
2 .
5.
V643
.
9.
7.
-\/l6*.
11.
8.
\/125" .
2.
3
3.
(V2~u-)
4.
V255
-
.
2
12.
SQUARE ROOTS OF QUADRATIC SURDS 275.
To
find the square root of a binomial square
According to
by
inspection.
G3,
V5 + V3) = 5 + 2 V5~^3 + 3 2
(
= 8 + 2 VIS. v8-f 2\/15, the If, on the other hand, we had to find problem would be quite simple if presented in the form v5-|-2V3 5 + 3.
it to this form, we must find 8 and whose product is 15, viz.
To reduce
two numbers whose sum 5 and 3.
is
ELEMENTS OF ALGEBRA
220 Ex.
l.
Find
Vl2 4- 2 \/20.
Find two numbers whose sum numbers are 10 and 2.
Ex.
2.
Find
is
12
coefficient of the Irrational
^TT- 6 A/2 = Vll Find two numbers whose sum numbers are 9 and 2.
Ex.
3.
^11 - 6\/2
Find
These
is 20.
Vll - 6 V2.
Write the binomial so that the
Hence
and whose product
is 11,
term
and whose product
= ^9 - 2 A/2 = V9-A/2 = 3 - A/2.
is 18.
+2
V4 + VJ8.
EXERCISE
is 2.
2 \/18.
102
Extract the square roots of the following binomials
:
The
RADICALS Simplify the following expressions 18.
Vl3-2V22.
19.
-+=. -
:
*
22.
*
4--
VT - V48
VT 4. V48
2 V6
r
221
4 20.
23.
.
V4 + V12 RADICAL EQUATIONS 276.
A
radical equation is
root of an
Vx =
an equation involving an irrational
unknown number. 5,
-\/x
+ 3 = 7,
(2x
xrf
1,
are radical equations.
277. Radical equations are rationalized,
i.e.
they are trans-
formed into rational equations, by raising both members
to
equal powers. Before performing the involution, examples to simplify the equation as
to
it
is
much
necessary in most as possible,
and
transpose the terms so that one radical stands alone in one
member. If all radicals do not disappear through the the process must be repeated.
Ex.1.
Solve
involution,
vV-f!2-a = 2.
Transposing
a;,
Squaring both members, Transposing and uniting, Dividing by Check.
first
+
12
x2
-f
12
=2
= x -f 2. = xa + 4 x -f 4.
4x
x
4,
The value x
Vsc2
reduces each
8.
= 2. member
to 2.
ELEMENTS OF ALGEBltA
222 Ex.
V4 x + 1 -f V4
Solve
2.
Transpose
V4 x
1
-f
Transposing and uniting, Dividing by 24, Squaring both members,
Vitf
-f
25
4x
-f-
25
4x
24 \/4 #-|-
25.
1
-f-
V4afT~l.
= 0. -f V/2TT25 = 5 + x
Therefore
V24~+~l
CftecAr.
Extraneous
7
=
12.
Squaring both members of an equaThus x 2 = 3 has only root.
roots.
new
tion usually introduces a
one
= 12.
25
= 12 = 144 24\/4# + 1 = 120. \/4 jc~+~l = 5.
,
Squaring both members,
278.
a; -f-
root, viz. 5.
4#-f 4 = 9, an equaSquaring both members we obtain or 1. tion which has two roots, viz. 5 and The squaring of both members of the given equation introSince duced the new root 1, a so-called extraneous root. radical equations require for their solution the squaring of both members, the roots found are not necessarily roots of
the given equation
The
279.
they
;
results
may
be extraneous roots.
the solution
of
of radical equations must be
substituted in the (jlren equation to determine ivhether the roots are
true roots or extraneous roots.
Ex.
3.
Solve -Vx
-f-
Squaring both members,
x
+
1
+ 2 Vx'2 +
1
x
+
(.
Transposing and uniting, 2 Vx^ Dividing by 2, Squaring both members, Transposing, Factoring,
Therefore Check.
member
It
= V2.
x
=
= 3 x - 3. = 9 x2 18 x + 8x 2 25x-f3 = 0. - 1) = 0. (x 3) (8 x x = 3, or = VzMx2
7
x
-f
-f 7
x
+
at
J,
the
first
member =|\/2
.
+ -jV2=|v^;
9.
RADICALS Hence x If a;
=
x
223
\ does not satisfy the given, equation it is an extraneous root. both members reduce to 5. Hence there is only one root, viz. ;
= 3,
3.
If the signs of the roots were not restricted, x root of the preceding equation, for it satisfies the equation .
NOTE.
VaT+T Ex.
Solve
4.
4-
Vz+T + V2aT+3 =
2 Clearing of fractions, V2x'
+ "b"x
Transposing, Factoring,
Therefore,
x
If
V,
Hence x
x
= 3,
tlie Jeft
=
3
8
would be a
-f 1.
A5_
4-2x4-3
15.
+ 6~ieT~3 - 12 - 2 r. 2 z 2 4 6 x 4 3 = 144 - 48 x + 2 x2 53 -f 141 = 0. - 3) (2 x - 47) = 0. (x x = 3, or x *j-.
Squaring,
Check.
-f
}
ViTie-
Transposing,
If
VxT~0 = \/8 x
=
is
both members reduce member = 12T V2, and
4 z2
.
to 5.
the right
member
=
|V2.
the only root.
Solve the following equations
:
= G.
* Exclude
all
solutions which do not satisfy the equation or which
the given radicals imaginary.
make
ELEMENTS OF ALGEBRA
224
280.
of
Many
radical equations
may be
238.
Ex.
1.
Solve
Factoring,
Therefore
af*- 33
af*
+ 32=0.
solved by the method
RADICALS Raising both members to the
x
Ex.
| power,
= 32~*
Adding 40 to both members,
then x2 - 8 x
Hence
2
y'
y
Therefore
y 2
= ^ or
1.
- 8 x + 40 = 36,
8 x -f 40
x*
$x + 40 = y,
Vz2
Let
or 1"*
8x
Solve x*
2.
225
2y
+ 40 =
= 35.
_ 2 y - 35 = 0. = 7,
- 8 z-|-40 = 7,
or y or
=
5.
Vi 2 -8a;-f40=
2_8z 4-40 = 49,
x
=9
or
5.
= 26.
x
1.
=6
or
3.
members of the equation were squared, some of the roots be extraneous. Substituting, it will be found that 9 and 1 satisfy the equation, while 6 and 3 are extraneous roots. This can be seen without substituting, for 6 and 3 are the roots of the Since both
may
+ 40 = 6. But as the square root is restricted to cannot be equal to a negative quantity.
2 equation Vx'
8x
positive values,
it
EXERCISE
its
104*
Solve the following equations: 1.
x + Vx
2.
a?
= 6.
2Va;
6.
3
= 0.
4.
4-12a* = 16. 45 14VJB =
5.
o;*-2a;i~24 = 0.
3.
.
* Exclude extraneous roots and roots which imaginaries.
Q
make
the given radicals
ELEMENTS OF ALGEBRA
226
8a
2
11.
or
12.
a^-
13.
x2
+x ;
14.
5
ar
15.
2
^
-f-
4
40
V
-fll x 3x
2
SB*
4-
V*
a;
2
8.a
+3=
12 V5l?
4-40
6.
+1 1^7-^30 =
+ G V2^"-^I + 2 =
16. 17. 18.
19.
a;
20.
6
2
7a;-f 18
7a?H-V^
Va?~3o~
3
= y?
= 35.
= 24.
3 x -f
2.
1
4.
CHAPTER
XVIII
THE FACTOR THEOREM 281. If x* - 3 x~ + 4 x + 8 is divided by x remainder (which does not contain a?), then or*
3 x2 -f- 4 a;
-f
Or, substituting Q " and
Remainder,"
8
=
ani^
2)
(a?
^
x Quotient
-f
and there
is
a
Remainder. "
respectively for
Quotient
"
and
transposing,
R = x* - 3 x + 4 + 8 2
a?
As
-2
(a?
- 2) Q
.
we
does not contain a?, could, if Q was known, assign any value whatsoever and would always obtain the same answer for R. = 2, then (x 2)Q 0, no matter If, however, we make a? what the value of Q. Hence, even if Q is unknown, we can find the value of R by making x = 2. 72
to x
# = 2 -3- 2 + 4- 2 + 8-0 = 12. 2
3
Ex. 1. Without actual division, by dividing 3 x* -f- 2 x 5 by x 3. Let then
z
find the
remainder obtained
= 3,
^ = 3-81+2.3-6-0 = 244.
Ex. 2. Without actual division, find the remainder when m. ax4 4- bx? + ex2 4-
E = ax + &z +
Let then
2 4 8 ca: -f (to + e (x = w, R = am* + 6m3 + cm2 + tZw + e.
227
m) Q.
ELEMENTS OF ALGEBRA
228
If an integral rational expresm, the remainder is obtained
The Remainder Theorem. x is divided by x
282.
sion involving
by substituting in the given expression
The remainder
E.g.
(4x
of the division
- 4x4-11)^0 +
6
m in place of x.
is
3)
4
(-
3)
5
- 4(- 3)-f
11
=- 949.
The remainder obtained by dividing (x
+ 4)4 _ (3 + 2) ( X -
1)
by x -
+7
EXERCISE Without actual division dividing
-3
.
+ 7 = 632.
105
find the
remainder obtained by
:
3.
+ 3x3 -2x* 32x12 by a?-3. x*-x + 4x -Tx + 2\)y x + 2.
4.
a100 -50 a47
5.
x5 -
6.
a^
x*
2.
283.
1 is 6*
s
b
2
48 a2
4-
by x
5
-}-
2 by
a-1.
b.
+ ^by x + b.
7
7.
a -f b 7 by a
8.
^-14y
j
+ 6.
~132/
If the remainder
2
--
is zero,
the divisor
is
a factor of the
dividend.
The Factor Theorem. ing x becomes zero
If a rational integral expression involvm is a is written in place of x, x
when
m
factor of the expression. E.g.
43
-3
00 *. fora?.
if
x8
42
-
3 x2 2 4
2 x
- 8'= 0,
8
is
divided by x 4, the remainder equals 8 2 x - 8. 3 x2 4) is a factor of x
hence (x
-
-
-
Only factors of the absolute term need be substituted
TEE FACTOR THEOREM Ex. The
Factor a?
1.
a?
2
7a?-f 15.
-f
factors of the absolute term,
+ 15, _
5,
-7
229
i.e.
15, are -f 1,
1,
-f-
3,
8, -f 5,
15.
Let x = 1 then 7 x + 7 a; -f 15 does not vanish. Let x = - 1, then x8 7 x'2 4- 7 x + 15 = 0. Therefore x ( 1), or x -4- 1, is a factor. ,
By
2
x8
dividing by x a?8
-
7
-f 1,
x2
+
we obtain 7
x
+
16
= (x +
EXERCISE Without actual 1.
4x
2.
2
3.
x*
j
or
5
+3x 2
34
a?
ar
2
5
as
+ 3^ - 7
4
5
-f
225
is
Resolve into factors 4. 5. 6.
7.
is or
-
by
a;
8
a;
-f
16)
106
show that
division,
2
2
l)(x
2
divisible
-5a
divisible
18
is
by x
1.
divisible
by x
2.
5.
:
2o? m -6ra -fllm 6. 8. a 5x 6. a -2a + 4. 9. 2m -5m - 13m + 30 10. a -8a -f 19 a -12. p -5^ + 8p 4. 11. & p*- 9^ + 23^-15. 4m p~m p + 16m^ 12. m 4 n4 25 mV + 19 ran 13. m -f m n 14. a + 32. 3
8
2
-}-
3
2
3
2
s
3
3
4
4
8
2
2
3
s
-t-
.
5
Solve the following equations by factoring 2
+ lla;-r-6 = 0. + tt-t-15 = 0.
15.
ar*-f 6aj
1ft
o?-5ar
17.
^-10^4-29^-20=0.
18.
oj
19.
20.
l
3
5x2 -f3a;4-9 = 0.
a^-8^ + 19a;-12 = 0. 7 4-6 = 0. 3
a;
a?
21.
2 2. 23. 24. 25.
:
+ 27 + 27. - 7 + 16 - 12. ^ + 7y + 2y-40 = 0. x -4o8 + 2a^ + 4a?~3 =0 4^ or*
-f
9
or*
a?
2
aj?
a?
2
4
a?
ELEMENTS OF ALGEBRA
230 285. If n is a Theorem that
positive integer,
it
follows from the Factoi
xn y n is always divisible by x y. For substituting y for x, xn y n y n y n = 0. 1.
xn -f- y n is divisible by x -f ?/, if n For ( y) n -f y n = 0, if w is odd.
2.
By
actual division
is
odd.
we obtain the other
any positive integral value of
If n
is
for
+p=
e.g.
z6
-
y
=
5
-
(x
can readily be seen that #n -f either x + y or x y, if n is even. 286.
It
287.
Two
importance,
1.
:
not divisible by
= (x +/)O - xy +/), 2
Factor 27 a* -f
8.
+8=
27 a 6
The
y is
special cases of the preceding propositions are of viz.
x* -f-/
288.
and have
odd,
ar
Ex.
factors,
n,
2 8 (3 a )
difference of
+
two even powers should always be
considered as a difference of two squares.
Ex.
2.
We may
Factor
m
consider
m
6
6
n9
.
n 6 either a difference of two squares or a
* The symbol
means " and so forth to."
dif-
THE FACTOR THEOREM The
ference of two cubes. leads
more
3.
method, however,
-f
n)(m
2
mn
-f
is
preferable, since
Hence
directly to the prime factors.
= (m Ex.
first
231
w 2 )(wi
;i
mn
-f
w 2).
Factor a 12
EXERCISE Resolve into prime factors
107
:
Solve the following equations: 25.
x3 -8=0.
26.
y
3
+8=0.
27.
as -27=0.
28. a;=
it
CHAPTER XIX SIMULTANEOUS QUADRATIC EQUATIONS 289. The degree of an equation involving several unknown quantities is equal to the greatest sum of the exponents of the unknown quantities contained in any term. xy x*y
y = 4 is of the second degree. + 6 a?V - y4 is of the fifth degree.
-f
290.
Simultaneous quadratic equations involving two un-
known
quantities lead, in general, to equations of the fourth few cases, however, can be solved by the methods degree. of quadratics. *
A
I.
EQUATIONS SOLVED BY FINDING
x
+y
AND x-y
291. If two of the quantities x -f y, x y, xy are given, the third one can be found by means of the relation (oj-j-y) 2 4 xy
Ex.1. Squaring
Solve
==5
x
(2)
& + 2 xy +
(1),
(2)
(1)
>
1^ = 4. = 25. 4 xy = 16.
4,
x-y-
Hence,
Combining (5) with (1),
2
(3)
2/
(4)
3.
(5)
we have
= 6, Hence
/ |
"
*The
X y
= } = 4.
graphic solution of simultaneous quadratic equations has been
treated in Chapter XII.
232
SIMULTANEOUS QUADRATIC EQUATIONS In many cases two of the quantities x -f y, x
292.
233 and
y,
xy are not given, but can be found.
F* Lx
'
2
(1) '
(2) (3) (4)
-2 + 3 = 293.
The
1.
roots of simultaneous quadratic equations must be e.g. the answers of the last example are
arranged in pairs,
:
r*=-2,
b=-3. EXERCISE
108
Solve: 1.
2.
'
3.
r-
10.
1 = 876.
("
" 8.
I I
"' {
r
"'
x + y=7.
12.
^, =
ELEMENTS OF ALGEBRA
4 [
x
-4- i/
=
r
13.
6
"I
14.
,o 18.
x+y
I
19.
I* Jj
^ [.
= a.
=^ 18*
ONE EQUATION LINEAR, THE OTHER QUADRATIC
A system of simultaneous equations, one linear and ne quadratic, can be solved by eliminating one of the unknown 294.
uantities
Ex.
From
by means
Solve 2 x
(1)
we
+
of substitution.
3y
= 7,
(1)
- ~ y = 5.
Substituting in (2) Simplifying,
49
Transposing, Factoring,
etc.,
7 ,
-
( \
42 y
2
~^V\ + 2
(3)
/
+
9 y2 17 y 2
(y
2y
-
1
+
8
2 ?/'
40 y )
(17 y
Hence
-
4 y = 20. + 29 = 0. - 20) = 0. or y = 1
Substituting in (3),
aj
EXERCISE Solve
"
x
have,
,
= 2,
f J.
or JJ.
109
:
r^ 2
as
-47/ = 0.
3.
]
la;
'
-f- a;?/ -
= 6, -
.
^
5.
i
'
f
or*
-f
4 xy
= 28,
SIMULTANEOUS QUADRATIC EQUATIONS
-
y
235
'
>
1
lla
7.
8-
10
12~
13. 9.
10.
III.
HOMOGENEOUS EQUATIONS
A
homogeneous equation is an equation all of whose terms are of the same degree with respect to the unknown 295.
quantities. 4^ 3 x 2 y
3 y3
and # 2
2 xy
5 y2
are
homogeneous equations.
one equation of two simultaneous quadratics is homogeneous, the example can always be reduced to an example If
296.
of the preceding type. '
Ex.
1.
Solve .
Factor (2),
x*- 3 2x
(x
Hence we have
to solve the
2
y*
+ 2y = 3,
7 xy
(1)
+ G if = 0.
2t/)(2 x
3y)
two systems
=
(2)
(
:
(3) (1)
From
x-2y.
(3),
Substituting in (1),
4 f-
Hence
3 y2
y
=1
+ 2 y = 3,
8
y ,
1
3 3,
':il -e :)
=
V-~80
ELEMENTS OF ALGEBRA
236
297. If both equations are homogeneous with exception oi the absolute terra, the problem can be reduced to the preceding case by eliminating the absolute term. =
Ex. 2
Solve
.
(1)
2,
Eliminate 2 and 6 by subtraction. (1)
x
5,
(2)
x
2,
15 x2
15 y 2
= 2 x 5.
(3)
(4)
= 0. = (rc-2/)(llx-5y) 0.
11 a2
Subtracting,
Factoring,
Hence
- 20 xy +
solve
16 xy -f 5 y 2
:
(3)
(2)
From
(3),
j
Substituting y in (2),
109
^ EXERCISE
a;2
VI09, y
=
110
Solve: 6ar --7aK/4-27/2 ==0, }
f
10^-370^ + 7^ =
16^-7^
SIMULTANEOUS QUADRATIC EQUATIONS
m
<""
U.
^
-=m
'
_
'
f
1
14
'
237
&- 3^4-2^=43.
15.
150 a?- 125 ay = - 6, 150 */2 - 175 ay = 12.
"
IV.
SPECIAL DEVICES
Many examples belonging to the preceding types, and others not belonging to them, can be solved by special devices, which in most cases must be left to the ingenuity of the 298.
student.
Some
of the
more frequently used devices are the following:
299. A. Division of one equation by the other. Equations of higher degree can sometimes be reduced to equations of the second degree by dividing member by member.
E,!.
Dividing (1) by (2),
Squaring (2), (4) -(3),
* + '-*
Solve
{
y? a?
-f
- xy 4- y = 7. 2
2 xy
+ y2 = 10. Bxy-9,
(3) (4)
ELEMENTS OF ALGEBRA
238
EXERCISE Solve
111
:
faj-y=152,
*
^
i
*>.
f^ +
3 7/
= 133,
= 189, '
*
Some simultaneous
B.
300.
considering not x or 2
-, xy,
x
x
,
x
+y
y
etc., at first
more complex examples letter for
Ex.
1.
quadratics can
be solved by
but expressions involving x and
?/,
it is
as the
unknown
quantities.
?/,
as
In
advisable to substitute another
such expressions. Solve
i"
<---
-'
(1 > (2)
Considering
V# +
y and
Vx
y as
we have from (1),
Vx
from
V^^y = 3
(2),
But the negative
-f
4 or
y
x
y
and
solving,
2.
we
= 16, jc~ y = 9. = 12 J, y = 3|. 4-
quantities
6,
or
roots being extraneous,
x
Therefore
unknown
obtain by squaring,
SIMULTANEOUS QUADRATIC EQUATIONS
239
,
Ex.
2.
(1)
Solve
(2)
Let
Then __
r
I e.
=
or
x
two systems
:
U)
+
*/
=
[2x +
17.
solution produces the roots
y=
17.
:
EXERCISE Solve
4.
4
x
to solve the
!
|,
7
V
Hence we have
The
17^ + 4-0.
=
Hence
112
:
5.
36* 2.
6.
M-6. 4.
7.
F+y+
ELEMENTS OF ALGEBRA
240
Solve by any method : far'
9
+ a^lSG,
'
**
= 198.
'
5x+ 7y =
'
1
15.
16.
2
or
5
CCT/
25.
18.
13
f- 21 ^ =
+ 3 f + 3 - 4 y = 47 a;
(
xy
f
(7
m
2
+ o5)(6-hy) = 80, =
,
19
'
26.
=34, '
1
6 xy
= 15. 27.
*
x
n*.
1
1
x2
y*
.
20'
y
=
41 400'
.
SIMULTANEOUS QUADRATIC EQUATIONS i
y
.
Q
~\
30.
OK OO.
7,
%
y
241
9
36.
f*K
31.
32.
3 a2 38.
33.
25 34.
39.
7'
.
j/
= 48-
ix
Solve graphically (see
y
201, 203):
40.
INTERPRETATION OF NEGATIVE RESULTS AND THE FORMS OF 5 .
,
oo
301. The results of problems and other examples appear sometimes in forms which require a special interpretation, as a --,
^
-,
,
etc. etc
oo
302.
Interpretation .of
division, finite
-
=x
y
value of
indeterminate.
if
a?,
=
-
But this equation
x.
hence
According to the definition of
--
may
be any
is satisfied
finite
by any
number, or ~
is
ELEMENTS OF ALGEBRA
242
Interpretation of ?
303.
creases;
The
number, however
comes
If
The symbol 304.
oo is
made
larger than
called infinity.
The
fraction
- decreases
in-
if
x
is
:
infinitely large.
I,i
x
if
X
and becomes infinitely small, or infinitesimal) This result is usually written
creases,
be-
zero,
= QQ.
QO
solving
a problem
unknown quantity
or oo indicates that the
the result
If in an equation
problem has no solution. the
any * assigned
customary to represent this result
Interpretation of
305.
By making x
a.
x approaches the value
It is
infinitely large.
by the equation ~
de-
TO^UU"
- can be
great.
= 10,000
~~f
ToU"
sufficiently small,
x
if
x
^-100 a,
e.g.
fraction - increases
cancel, while the
all
terms containing
remaining terms do not
cancelj the root is infinity.
The
306.
x
solution
=-
indicates that the problem
is indeter-
If all terms of an minate, or that x may equal any finite number. equation, without exception, cancel, the answer is indeterminate.
Hence such an equation
is satisfied
by any number,
i.e. it
is
an
identity.
Ex.
1.
Find three consecutive numbers such that the square and third by 1.
of the second exceeds the product of the first
Let
2,
as
+ l,
x
-f 2,
be the numbers.
Then Simplifying,
+
I)
2x
+
(a:
x2
-f
2
1
-x(x + 2)= - x'2 2 x =
'
Or,
Hence any number will satisfy equation the given problem is indeterminate.
1.
(1)
1.
= 0. (1),
i.e.
(1)
is
an
identity,
and
SIMULTANEOUS QUADRATIC EQUATIONS Ex.
Solve the system
2.
243
:
(1)
(2)
From
z
(2),
=
1
Substituting,
Or,
1=0.
Hence
y
numbers can
/.e.
no
1.
One half
finite
QO,
and
a;
=
oo.
satisfy the given system.
EXERCISE of a certain
number
113 is
equal to the
sum
of its
Find the number. third and sixth parts. Find three consecutive numbers such that the square of 2. the second exceeds the product of the first and third by 2.
~K x 6
~o x 3 v
3.
Solve
4.
Solve
6.
x
4
a;
-3
Solve
x
6.
Solve
x
6
x-5
a2 - 8 x
+ 15
- 2 y = 4. *
Solve |
7.
r
(aj
9
+ 1)
:
(x
+ 2) = ( + 3)
EXERCISE
:
(a?
+ 4).
114
PROBLEMS 1.
The sum
squares
is
2890.
of
two numbers is 76, and the sum of Find the numbers.
2. The sum of two numbers Find the numbers.
is
42 and' their product
their
is
377.
ELEMENTS OF ALGEBRA
244 3.
The
two numbers Find the numbers.
difference between
of their squares
is
325.
4. Find two numbers whose product whose squares is 514.
is
is
sum
17 and the
255 and the sum of
5. The sum of the areas of two squares is 208 square feet, and the side of one increased by the side of the other e.quals 20 feet. Find the side of each square.
6.
The hypotenuse
the other two sides 7.
The area
hypotenuse
is
of a right triangle is 73, Find these sides.
103.
and the sum of (
228.)
of a right triangle is 210 square feet,
Find the other two
is 37.
and the
sides.
8. To inclose a rectangular field 1225 square feet in area, 148 feet of fence are required. Find the dimensions of the
field. 9.
The area of a
nal 41 feet. 10.
rectangle is 360 square Find the lengths of the sides.
The diagonal
perimeter
is
of a rectangular field
146 yards.
Find the
and the diago(Ex. 12. p. 190.)
feet,
is
53 yards, and
The mean proportional between two numbers sum of their squares is 328. Find the numbers.
11.
the
its
sides. is 6,
and
The area of a rectangle remains unaltered if its length increased by 20 inches while its breadth is diminished by 10 inches. But if the length is increased by 10 inches and 12.
is
the breadth
is
diminished by 20 inches, the area becomes
the original area.
Find the
-f%
of
sides of the rectangle.
13. Two cubes together contain 30| cubic inches, and the edge of one, increased by the edge of the other, equals 4 inches. Find the edge of each cube. 14. The volumes of two cubes differ by 98 cubic centimeters, and the edge of one exceeds the edge of the other by 2 centimeters. Find the edges.
SIMULTANEOUS QUADRATIC EQUATIONS
245
The sum of the radii of two circles is equal to 47 inches, their areas are together equal to the area of a circle whose radius is 37 inches. Find the radii. irR *.) (Area of circle 15.
and
=
16.
The
radii of
two spheres
differ by 8 inches, and the equal to the surface of a sphere Find the radii. (Surface of sphere
difference of their surfaces
whose radius = 47T#2.) 17.
If a
its digits,
is
is
20 inches.
number
1
of
two
the quotient
digits be divided
is 2,
and
the digits will be interchanged.
by the product of 27 be added to the number, Find the number.
if
CHAPTER XX PROGRESSIONS
to
307.
A series
some
fixed law.
The terms
a succession of numbers formed according
is
of a series are its successive numbers.
ARITHMETIC PROGRESSION 308. An arithmetic progression (A. P.) is a series, each term of which, except the first, is derived from the preceding by the addition of a constant number.
The common
difference is the
number which added
to each
term produces the next term. Thus each
of the following series is 3,
17, a,
7,
10, 3,
a
an A. P.
11, 16, 19,
+
d,
-4, a
+
:
....
11,
2 d, a
-f
....
3d,
....
The common differences are respectively 4, - 7, and d. The first is an ascending, the second a descending, progression. 309.
To
find the
nth term
/
of an A. P., the first
common difference d being given. The progression is a, a -f d, a + 2 d,
term a and
the
Since d
a
-f
3
d.
added to each term to obtain the next one, 2 d must be added to a, to produce the 3d term, 3 d must be added to a, to produce the 4th term, (n 1) d must be added to a, to produce the nth term.
Hence
is
/
= a + (n - 1) d.
Thus the 12th term of the
series 9, 12, 15 is 9 -f- 11 246
(I)
3
or 42.
PROGRESSIONS To
310.
term
find the
sum s
the last term
a,
of the first
247
n terms of an A. P., the
first
and the common difference d being given.
19
= a + (a Reversing the order,
Adding,
2*=(a + Z) + (a + l) + (a + l)
Or
2s = n
Hence Thus from
*
(a
+ + (a + l)
= (+/). 2
to find the
sum
I
Hence
.
= I + 49 = *({ +
.
odd numbers,
1, 3,
6
we have
= 99. 99) = 2600. 2
EXERCISE
Which
115.
of the following series are in A. P. ?
(a) 1, 3, 5, 7, .-; (6)
2,4,8,16,...;
(c)
-3,
(d) 1J,
2
5,
9,.-.;
-|, -24, -4^....
Write down the
first
(6)
a = 5, d = 3; a = 2,' cZ == - 3
(c)
a = -l, d
(a)
3.
1,
l).
(II)
of the first 60
'
'
(I)
1.
.-
6 terms of an A. P.,
if
;
= -2.
Find the 5th term of the
series 2, 5, 8,
4.
Find the 10th term of the
5.
Find the 7th term of the
6.
Find the 21st term
7.
Find the 12th term of the
8.
Find the 101th term of the
9.
Find the nth term of the
.
series 17, 19, 21,
series
of the series 10, 8, 6, series
....
.
1-J, 2, 2J,
.
-4, -7, -10,
series 1, 3, 5,
series 2, 4, 6,
.... .
ELEMENTS OF ALGEBRA
248 Find the
last
10.
11.
Sum
term and the sum of the following series :
3, 7, 11,
to 8 terms.
,
4,
2,
',
6,
to 7 terms.
12.
8, 12, 16,
,
to 20 terms.
13.
3, 2J, 1|,
,
to 10 terms.
the following series
:
to 20 terms.
14.
7, 11, 15,
15.
33, 31, 29,
16.
15, 11, 7,
17.
1,
18.
2-f
19.
2.5
20.
(x +"l) 4- (#
21.
1
22.
1+2+3+4H
23.
Find the sum of the
1,
,
to 16 terms.
,
-,
to 20 terms.
,
to 15 terms.
1J,
H + i-f
>
+ 3.1 -f 3.7 -f -f-
,
to 12 terms.
2) -f (x -f 3) H
+ 2-f-3 + 4 H
Q^) How many times
to 10 terms.
,
to
a terms.
hlOO. \-n.
first
n odd numbers.
does a clock, striking hours only, strike
in 12 hours ? (&fi)
For boring a well 60 yards deep a contractor receives yard thereafter 10^ more How much does he receive all
for the first yard, and for each than for the preceding one.
$1
together ?
^S5 A bookkeeper accepts a position at a yearly salary of $ 1000, and a yearly increase of $ 120. How much does he receive (a) in the 21st year (6) during the first 21 years ? j
311. In most problems relating to A. P., Jive quantities are involved; hence if any three of them are given, the other two may be found by the solution of the simultaneous equations .
rf.
(i)
(ii)
PROGRESSIONS
24ft
Ex. 1. The first term of an A. P. is 12, the and the sum of all terms 1014. Find the series. s
=
1014, a
last
term 144,
= 12, = 144. I
Substituting in (I) and (II), (1)
l)e?.
1014
= ^(12 + 144).
(2)
2
From
= 1014, or 144 = 12 + 12
78 n
(2),
Substituting in (1),
.
series
Ex.
2.
is,
d.
12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122, 133, 144.
Findn,
if s
= 204, d = 6, J = 49. = a + (w- 1) .6. 204 = ^ (a + 49). 49
Substituting,
From
= 13.
d=ll.
Hence
The
n
a = 49 -6(71 - 1). 204 = ^ (98 - ~n~\
(1),
Substituting in (2),
(1) (2)
6).
= n(104 - 6 n). 6 n2 - 104 w + 408 = 0. 3 n2 52 n + 204 = 0. n = 6, or 11 J. Solving, But evidently n cannot be fractional, hence n = 6. 408
312.
When
three numbers are in A. P., the second one mean between the other two.
is
called the arithmetic
Thus x
is
the arithmetic
b form an A. P., or if
Solving, I.e.
the arithmetical
half their sum.
x
mean between a and a=b
x=
6, if a, #,
and
x.
-
4
mean between two numbers
is
equal
to
ELEMENTS OF ALGEBRA
250
EXERCISE
116
Find the arithmetic means between 1.
2.
5.
a x
-f-
b
and a
3.
b.
y and #-f-5y.
Between 4 and 8
that an A. P. of 5 terms 6.
7. 8.
Between 10 and 6
How many terms How many terms
:
and
m
n
a+
-
and
4.
a
b
insert 3 terms (arithmetic is
insert 7 arithmetic
has the series
^
j ,
T?
,
^,
Find
12.
13.
14. 15.
,
has the series 82, 78, 74,
17.
11.
so
means
16.
10.
means)
produced.
= 16, s == 440. Find a and Given s = 44, n = 4, = 17. Find a Given a = 7, = 83, n = 20. Find d. Given a = - 3, n = 13, = 45. Find d. Given a = 4, n = 17, = 52. Find d and Given a = 1700, d = 5, = 1870. Find w. Given a = |, = ^ 3 = 1. Find n. Given a = 1, n = 16, s = 70. Find?.
9.
b
Given d = 3, n
f? ,
6?
I.
f
J
1
1
.
/
I
I
in terms of a, n,
and
s.
A
man saved each month $2 more than in the pre 18. ceding one, and all his savings in 5 years amounted to $ 6540. How much did he save the first month? 19.
$300
is
divided
person receives $ 10 did each receive ?
among 6 persons
in such a
more than the preceding
way
one.
that each
How much
PROGRESSIONS
251
GEOMETRIC PROGRESSION 313.
A geometric progression
multiplying
it
-2, +1, -I,....
2 a, or,
ratios are respectively 3,
To
find the
....
4, 12, 36, 108,
4,
314.
a series each term of
first, is
E.g.
The
is
(G. P.)
derived from the preceding one by by a constant number, called the ratio.
which, except the
|,
nth term
ar8
and
,
r.
a G. P., the first term a and
/ of
the ratios r being given.
The progression is a, ar, a?*2 To obtain the nth term a must evidently be multiplied by .
,
Hence
l
Thus the 6th term
= ar
.
= a + ar -for ar -f ar Multiplying by r, rs =
2
s
.,
Therefore
1) 8
NOTE.
If
n
is less
the following 8 form
of a G. P., the first
(1)
arn
,
(2)
a.
= ^ZlD.
(II)
JL
6 terms of the series 16, 24, 36, fl
lg[(i)
4-
= ar" 7*
8 =s
,
or 81
(2),
s(r
first
16(f)
4
-- arn ~ l .
2
of the
.
(I) is
315. To find the sum s of the first n terms term a and the ratio r being given.
Thus the sum
n~ l
n~l
of the series 16, 24, 36,
Subtracting (1) from
r
-l]
than unity,
==
32(W -
it is
g==
*.
q(l-r") 1
= 332 J.
convenient to write formula' (II) in
nf +
:
1)
.
r
ELEMENTS OF ALGEBRA
252
316. In most problems relating to G. P. Jive quantities are in. volved ; hence, if any three of them are given, the other two be found by the solution of the simultaneous equations :
/=
(I)
,_!=!>. Ex.
l.
To
insert 5 geometric
Evidently the total
=
a
=
Substituting in
I,
Hence n
7,
number
9, I
may
(it,
means between 9 and 576.
+ 2,
of terms is 5
or
7.
= 670. 676 t
= r6 = 64.
.
r^2. Hence the
series is
or
0,
And the
required
0, 18, 36, 72, 144,
- 18,
36,
means are
- 72,
144,
Which (a)
2
3.
term
i 288.
117
of the following series are in G. P. ?
2,6,18,54,-.;
(b) 1, 4, 9, 25,
term
576.
72, 144,
18, 36,
EXERCISE 1.
288, 676,
- 288,
(c)
...
f,l,,4,
(d) 5,
;
....
- 5, + 5,*- 5,
Write down the first 5 terms of a G. P. whose and whose common ratio is 4.
first
Write down the first 6 terms of a G. P. whose and whose second term is 8.
first
is 3,
is 16,
4.
Find the 6th term of the
series J, f, 1,
5.
Find the 7th term of the
series
6.
Find the 6th term of the
series 6,
7.
Find the 9th term of the
series 5, 20, 80,
8.
Find the llth term of the
9.
Find the 7th term of the
10.
....
series
series
^,
ratio is
.
+-f%9 %
-fa,
4, 3,
^, |,
Find the 5th term of a G. P. whose
whose common
.
-fa,
....
\
.
t
,
,
first
term
.
is
125 and
PROGRESSIONS Find the sum of the following
series
32, 48, 72,
243, 81, 27,
-,
to 6 terms.
13.
14, 42, 126,
.-.,
to 8 terms.
14.
1,
15.
81, 54, 36,
2, 4,
16 - nV> i*> !7-
M,i
18.
a9
,
a^,
Given r =
to 7 terms.
,
..-,
>"> 7 ,
,
to 6 terms.
to 6 terms.
12 terms.
-, to a;
:
to G terms.
11.
12.
.-.,
25S
to 5 terms.
= 3, == 160. Find a and n = 4, = 3. Find a and Given r = Given r = 2, n = 5, s = 310. Find a and Given r = 3, n = 5, s = 605. Find a and
19. 20.
4,
n
Z
Z
s.
5.
-J-,
21. 22.
J.
I.
23.
Find the geometric mean between
24.
Prove that the geometric mean between a and b equals Vo6.
7,J-
and 270.
INFINITE GP:OMETRIC PROGRESSION 317.
If the value of r of a G. P. is less than unity, the value The formula for the sum may if n increases*
of r n decreases, be written
= _ fl
flf
n
taking n sufficiently large, r , and hence ~ r , may be than any assignable number. Consequently the sum of an infinite decreasing series is
By
made
less
-r^Ex.
1.
Find the sum to
Therefore
8^
infinity of the series 1,
=1
i
=
'- .
1
+
1
4
J,
-J,
ELEMENTS OF ALGEBRA
254 Ex.
Find the value of .3727272
2.
.;)7?7272
The terms afteAhe
=
...
first
.3
+
form an a
+
.072
= .072, 1
Therefore
.37272
-
=A+
. . .
10
1.
2.
1,
1,
i i
J,
1,
.99
i. 65
= 1L
9, 6, 4, ....
If
a
9.
10.
15. is J.
16. 8.
4.
= 40, r = j.
Find the value
= .72. = 990
.
66
.
110
118
16, 12, 9, -..
3.
....
8.
^
.01
infinity of the following series
-.
7.
....
= .Ql.
r
EXERCISE Find the sum to
.00072 -f
infinite G. P.
.= _4Z* - =
Hence
....
3,
- 1,
i,
-.
Find the sum to
:
5.
5, 1, I,
6.
250, 100, 40,
....
infinity.
of:
.555....
11.
.191919-...
13.
.27777
.717171-...
12.
.272727-..
14.
.3121212-..
The sum
of an infinite G. P.
Find the
first
The sum
of an infinite G. P.
Find
...
is 9,
....
and the common
ratio
term. is 16,
and the
first
term
is
r.
17. Given an infinite series of squares, the diagonal of each equal to the side of the preceding one. If the side of the first square is 2 inches, what is (a) the sum of the areas, (6) the sum
of the perimeters, of all squares ?
BINOMIAL THEOREM EXERCISE
Expand
2.
the following
(x-y)
6 .
119
:
+ xy.
3.
(1
4.
(a-2)
7 .
5.
(s
+ i).
7.
6.
/2a+|Y-
8.
9.
l
(z2
4
\
Simplify
257
:
4 (1+V#) + (1
Va)
4 .
10.
(\
+ b) w (a b)
9
11.
Find the 5th term of
12.
Find the 3d term of
13.
Find the 4th term of (w
(a
.
.
12
-f
.
ri)
+ a)
11
14.
Find the 5th term of
15.
Find the 4th term of
16.
Find the 6th term of (x - a2) 25
17.
Find the 5th term of
18.
Find the 3d term of fa -f
(1
7 (a -f 2 b) .
f
Find the
20.
Find the
.
Vx + -^r
-^Y
-
Va/
V 19.
.
u 13 coefficient of a?b in (a -f 5) .
21.
a4 b 12 in (a -f 6)16 Find the coefficient of a5 b 15 in (a - 6) 20
22.
Find the coefficient of a?V" in (a
coefficient of
.
8
16
100
.
2
2
23.
Find the
24. 25.
Find the middle term of (x + y) 4 Find the middle term of (a b)\
26.
Find the middle term of
27.
Find the middle term of (m ri) 16 Find the 99th term of (a + b) m im Find the 1000th term of
coefficient of
a6
in
.
f f
x
}\8 :
)
28. 29.
.
.
(a
.
- 6) - b ). (a
+ b)
.
-^
ELEMENTS OF ALGEBRA
258
REVIEW EXERCISE Find the numerical values 1.
~
27 x*
27 x-y
*=M
or
y=2j 2.
-
16 x*
9 xy~
-f
32 afy
4 *2
-
4 xy
-
-
13 a a b
1
]
]
1
1
1
lj
2j
3}
4j
2J
4J
a 8~T +
a2
+
3 2 ft'
ft
c
8
y
,
3,
4.
=
3,
2,
3,
4,
3,
4,
5.
2,
^+ = = =
2,
3,
3,
3,
4,
4,
5.
4,
2,
3,
4,
3,
2.
2,
1,
1,
2,
2,
1,
3,
6.
38
aft
+
24
= =
3
ft
-
4
)
2 (2 a
2,
2,
3,
3,
4,
4,
5,
6.
1,
2,
1,
2,
2,
4,
2,
3.
5.
a 2^
+
3 a l} 2
-r
ac
aft
-
-
3
aft
2,
4,
4,
5,
5,
5,
2,
1,
5,
1,
3,
5,
6.
3,
3,
2,
1,
2,
4,
2,
7.
+
a)(a
3,
4,
1,
2,
2,
1,
2,
2,
1,
l,
2,
1,
3,
3,
ft)
- a(a
4-
4,
2,
2.
1,
3,
3.
ft
-
c)
+
c(a
-3, -5, -6. c
7 a 6
c
= = =
2,
-2,
1,
1,
3,
-3,
if
i
,
(a-ft)(a-c)
(ft- c )(ft-a)
-
-f-
4
2 ft
), if
,f
1,
-
-
be
2,
(c
5J
if
,
2,
31 a 2 ft 2
3,
2
1,
a
)
^+^
2
2,
+
= =
+
2 ?/
M.
if
3,
4
= = =
+ 2,
c
a
8
2,
;]
4-
^
2,
3 -r C T + + + c2 + 2
c)(c
8
3
1,
ft
(ft
-
3
2
2
x^l,
a
6.
2
24 afya
-f
ft
5.
if
,
M
?/
4 (2 a
#
3J
a:
4.
:
8
2
y 3.
of
-
(c-a)(c-ft)'
-4 -
1,
-1,
3,
4,
2,
-
2,
2,
4,
-3, -1, -1, -3,
2,
2,
+
1.
1.
2.
-|-
c), if
259 x
c)
(b
6-)
= 1, = 2, c = 3, x = 4, a
1,
/>
9. a, by
The
and
Find
Add
2, 5,
(c
+
(5,
g)(x
c(x
,
-
-
a) (c
3,
2,
4,
6,
5,
1,
2.
2,
3,
1.
-
'
b)
2.
radius r of a circle inscribed in a triangle whose sides are by the formula
= = c =
a
3,
10,
8,
6
4,
21,
17,
24,
5,
26,
15,
7,
r, if
41.
29,
25,
9.
21, 20,
40.
the following expressions and check the answers
-
11. x 2
+ -
x
4
2 ax*
y
+
4
-f
a zx
z8
-
3
4x y
xy -
C
+
+
6 x
a8
4 x2
,'
12 xy*
2
+
2 ?/
x3
,
-
-
5 z3
4 xy
G y4
,
8
,
+
8
4 xy*
2
.c'
-
4 ?y
-
,
4
-
2 a3
3 ax'2 ,
-f
7 y4
4 x 4 /
,
-f
-
3 //
y
ax'2
-
10 z 8
2
x'
:
2 x 8. 6 y4
,
zy +
12
-
6 2 8.
12 xy*
-
4 y4
4 .
+ x/y 2 + + y'2z + 2 3 x 10 y'2 + 5 z2 - 7 ys, - x 2 + 4 2 ~ 10 z 2 + z 2 + 11 yz + 8 2:2 - 2 x?/, 4 z - \ yz + xz, 2 2 x2 + and 9 2:2 y' xy. 1 + 3 x + 2 x 8 - x 5 4 - 2 x2 - 8 3 + 7 x4 - 4 x'2 -f 12 x and 5 2 + 7 x8 - 11 x 5 12
13. x 3
14.
3,
a}
~c)(b- a) - 1, - 2, 3, + 1, -f 8, 4, - 2, - 4, 5, + 2, + 4,
c is represented
10. x 3
12.
c)(x
b(x
.
-
.r
z
3
7/
ary,
2
15.
16.
11 z 4
-
-12
x4
-
-
2
9
11 x 8
.
4 2 */, 7 xy 3 - 2 a?y + 3 aty - 8 y y 3 4 8 5 3 5 4 * + + xy a?y y, 7y 4
,
*y
+ 12 a 8 - 10, a 4 + 11 a - a 5 ,
14
6 a4
4 a8
-7xy* +
x^ij
+
8 x4
.
a
+
,
or
,
17. 4 a 5
18.
a:
,
r>
a;
- a8 - 7 + - a 4 - 5.
z 3,
x3
-
2 x 2//
2 a2
+
3
,
4a
2 x?/
+
-
7 y3
+ 3 y 2* - 2 z8 4 x- 8 + 2 // - 11 z 3 4 4 ?p 2 - 3 xyz, and 3 y 8 -f 12 z 8 - 7 y 2* 4- 4 xyz + 4 xy'2 - 4 yz\ ,
9 a2
-
3 a5
,
,
ELEMENTS OF ALGEBRA
260
4-X-5V14- #4-8, - 4\/i + x 3Vl 4- x 4- 4 Vl 4- 4- 3, and 2 Vl 4-
19.
6 VI
20.
Take the sum
2VT+7 - ?> x
*/
6 x8
4x
4-
-2
a 2x
From
G x 4y
2 y5
-x
2
4-
the
2 x2
4-
2
xy
From
sum
26.
of 2
From sum of 2
-
29.
1
from
Add
0" 30.
find
(a) a ft
Simplify
34.
12 x 5
4-
2 xs
4
take the
4
- 2 x 8y2 44 - x - x2
of
of
-
2 c
the
-
ft
G x2
sum
G x
3 a,
4- c 4-
ft
2 c
a,
of
-
2 c
5 10 ,
6
4-
c 4-
2 a
4-
3
4- c
ft
-
2 c
a,
- 5 10 -
x4-y4-2, -f
-
ax 2
and
a'2x,
2
4-
4-
x2
2 x6
,
7 x
x'2 . 4
4 x 4 ?/
/-
2 x2
-f
and 4
4- 5,
3
4-
5 y/
-2x
x5
,
3
4-
x// 5 ?/
4
,
#
.
-
4
-
5
54-2 x 2 and
x*,
,
7 x
b
=
7 12
.
x
c,
(
-\-
a
,
G 11
3
y
(c) a 4-
ft,
3
to 3 x 2
-
ft,
and a
3
ft,
3 x2
3 c take
ft
4-
c.
ft
2 x
;]
and a
3 will give
4-
4-
3 c take
ft
- 2 - c. - 3 x - 1 and x 8 -
and 2 a
ft,
4-
2
5 x
c 4- a
a,
2 a
a
and 2 a
ft,
ft
3 x2
+
5 10
+
x.
G 11 4- 3
4-
and
5
y
of G x 5
3 y5
,
x
and 5 x 3
4-
x?y
sum
-
x8
,
.
2 x8
2, 4-
3 ax 2
4-
4 x8
11 x.
4-
sum
ft
-
G x5
4-
sum
4 x2
2
=
Ifcc
-
5 10
3
2
4-
z,
-
c
ft
4- c,
ft
+
c,
7 12
5
G 11
4-
y
z,
7 12
4
-
4
,
.
.
=x
-}~
(*) a
(/) a
and
+ -
d=
x4-#4-z
ft
4-
c
4-
6
+
c
-
rf.
:
x - (5 y - 6T - 4 ft) 4- 2 a - (a - 2 _[5ft-{^ (5 c - 2 2 8 4 x* - [4 z 8 - {G * 2 - (4 * - 1)}] - (x 4- [4 x 4- 6 x ]
31. 2 x
33.
^V
the
7 12
9
(ft)
32.
sum of - 3 .n/ 4
4 a 2x
Subtract the difference of x 8
28. 3 x
4vTT~y 3. - x 2 4- 4 x 8 from
4-
and 4 x - 2
9
4-
Find what expression added
27.
the
//
the
-
take 4
3
4
,
From
25.
the
.
4-
2 a 2x
4-
23.
24.
7
G a8
of
x 3 from G a 8
Take the sum of 3 x 4- 8 3 4- 4 x from the sum of 9 x
and
+
sum
the
ax'2 4- 3
2 x2
22. 4-
4 x2
4-
4- 7.
Take
21.
of 2 x 8
- [3
if
-
(3
a
6-
ft
_^ ft
4-
-f-
4
c)}] a:
.
4- 1).
13-3ft-[l7a-5ft^[7fl-3ft-{4fl~4ft-(2a-3ft)}]]. - (x* - 4) - x - 5 - {2 x 2 - (7 x 4- 2) - (4 x 2 - 27~-~7)}].
35.
3 x2
36.
I 2a4-7c-(7ft4-4c)-[6a-3ft4 2~c4-4c-{2a-(ft-2T-2)}].
[4
REVIEW EXERCISE 37. 7 a 2
38. (5 a 39.
2x
40.
a
42. 43.
44. 45. 46.
-
2 a
-
2
[2
-
i
2
{3
j-
- [4 x -
+ {4
2 z
-
-f [3 c
ft
7 a - 5)} + (3 a 2 - 4 a - 12). - (2 a + 5 a - 0)} - (2 a 2 - 7). - (5 y - 3T~2~s)} + 5 2]. - JT^T+1)} + (2 - 3 c)].
-2 a + (2 a 2 -
a2
- [3 y -
-
41. 3 x
-{5 2
261
5
(3 a
ar
ft
- (2 x2 - (7
a;
+ 2) -
(4 x
2
-
2 x
-7)}].
5a-(7ft+4c) + [6 a- 3~ft -f 2 c + 4 ^ - {2 a -(ft - a~^~c)K]. a -{- b -(c - rf)} + a -[- & + {- 2c-(V/ - e -'/)}] -(2a + 2b - 3c). 5a + 2 c + 4 c - {2 a - (6 - 2a - 6-)}]. [0 a (7 i + 4 r:) 13 a - 96 -[17 a- 56- [7 a 36 -{4 a 46 (2 a 3 ft)}]]. a - [2 - {3 c - (4 d - 5 )}] + {4 c - (2 ,Z - 2
13 ft
ft
_[&-{2c-(3d + Perform the operations indicated 47.
48.
49. 50.
7e)-a}].
:
51. (1 -ar+a; )(l-z a ). + *+!){> + 2). 2 52. (.r 2 + !>ar + 3)(^ 2 - 2x + 3). (;r -2:c+ l)(ar- 1). 2 53. (2 x 2 -3 ar+ 1)(3 z -2 x+ 1). 4x + + 3). (x 5)(j; 2 5 54. (/> 4 - 2 2 + 1)(7, 4 + 2 2 + 1), x x' ) (2 - 3 *). 6 + (1 55. (4 + 3a 2 - 2)(1 - 4 a 2 + a 4 ). 56. (a 2 + 2 + c 2 - ab - ac - 6c) (a + -f c). - c). 57. (a 2 + 2 + c 2 + aft + ac - be) (a 58. (x 2 + 4 y 2 + 3 z 2 ) (.c 2 - 2 2 - 3 z 2 ). 59. (a 2 + 2 + 9 - 3 a + 3 + aft)(a + 3). 60. (4 z 2 + 9 2 + ^ 2 - 6 xy - 2 zz - 3 yz)(2 2
a
(*
2
ft
ft
ft
ft
ft
ft
?/
ft
ft
ft
a:
?/
61.
+
(ar
+
7)(ar
+
5)(a:
3).
62.
(x- 3)(*-5)(* -7).
63. 64.
(a:-2)(r-4)(a:-9). 2 2 x + !)(* - * 2 + (x + x + l)(a:
65.
(z
66.
(r
67.
68.
(1 (a;
2
+
a) (2:
+
^+ y)(x
2
2 7/
-*)(! +
-
a)(x )
(x
ar)(l
2
+ a 2 )(a: 4 + a 4 ). - ary + 2) (^ 4 ?/
+
^ 2 )(1
+
**).
1).
*V +
ELEMENTS OF ALGEBRA
262 69. 70.
(a
71.
(a
72.
(2
75.
(a
76. 77.
78. 79.
8
-
2
-
(a
-
+
2
(:r
"
ft
4- c
6
+
n
+
ft)(a
(??2
Simplify
p(p +
-f-
2
?)
2
(x
2 y) (2 ^
84.
(p
4-
3 y)2(/
85.
O (a
(x
89.
(x
90.
(
--y
I)(a m
(x
+
2
1).
).
A 2 *).
4-
- ac b + n~ + /? 2c -
20
- 2ft) 8 ( + 2 ft). - 3y) a (* 2 4- 6*y - 9y2).
n ft
ft
n
+ + c). - m np c - n pc). ft"
c)(a"*
?n
an b
l
ft
4-
88.
n
-f
).
1).
+ 4 A) (a - ft) + 4(2 - a) (2 + 7(7> ~ 'y) 2 4-
83.
87.
4-
p ) (w
+
(x
)OK
2 am
c
82. a(2
86.
2m
2a
-f
12
(
74.
2
a:
:
80. 4 (a
81.
4-
a
73.
y).
2
+ +
12
l)(a
2 6)(a^+ - am&t
2n
(a
)
.
l)(u
a;y'*4-y
(rtP+i 4-
(a
4-
-2a +
6
+
2a
-f
a:
2
4- ft)
b)*(a
2
l)(a
- y)\x
ar
a
2
+
2 a
+
z 3 )(a 6
+
z 3) (a 8
2
+
44-
y y)
3ft)
a
2
J
-(2a -
- (x -
- (^
4
3 V)
b 4- c)
2
2
4-
y)
y
- (a
4-
-
O
)
,v) 4- (a? 4-
-
y) (^
4-
4 ^/(.r
ft- c)
y)*(x
+
3 v)^(;> 2 2
-
y
4-
xy
). 4-
y
2
).
2 .
+ *) - 2 (y 4- z) - z\x 4- y). 4- y + z)(x + y - z)(x -y + z)(- x + y + z). _ ft) (a: + a)(x + b) + (b-c)(x + ft) (a: 4- c) - a) (a? 4--(c 4-
y) (y
4-
j;
2) (s
95.
Prove the following
93.
y).
2
4-
94.
92.
a 2 ).
3 9).
- (ft 4- c) 2 - (c 4- a) 2 - (a 48 8 8 4- c) 4) (ft 4- O (a 4(a 43 a 2 ft}) -f (3 a 5 {3 a (4 2 (a 3 (ft c)]. 3[a{2 a c)}
91.
+
.
^)
- (/> 2
2
2
8
4-
y) (^
4-
/;
(a 4-
ft
4-
2
2
c)
ft
ft
ft
ft
^>)
8 ft)
c)(ar
4-
a2
4-
ft
4-
a8
4-
ft
+
a).
2
2 4- c .
8
4- c
8 .
a}.
ft
identities,
by multiplying out each
side
of the equality. (a) (a (ft)
8 4- b 4- c) rr
a8
4-
ft
8
4- c
8
4-
3(6
(.:-y)( a;-2y)(.r-3y)4-l)y( a
4-
c)(c
4-
a)(n
+
ft).
2
:-y)^-2y)4-18 // (2r-y)4-6
8 //
REVIEW EXERCISE Simplify
:
96.
5(a
3*- 5 a 21 (10 a
99
O3a
4 (6 x
4-
3m n
(3'
4-
42
23 x s 33 z
3
4-
a;
2
-
5(a
4-
&)
5 a*.
-=-
2a
35
4a;
3".
-T-
3")
41 x
4-
72 x 2
]-
Qa-f-l^
3 3n
~*~
6)
5 a*)
4-
O4a
i
4~
n
-
(20 x*
O2a
2
4-
n
4- b)
98.
100.
103.
-
8
4- &)
[10(
6 (a
102.
263
20)
(3 a*
-*-
4-
4
~ (4 ^ -
4- '30)
a?
+
5 x
5).
4-
10).
1O4. 105. 106.
(2<
107. 108.
109. 110.
-
(x*
(2 y
4
111. (80 a 112. 113.
114.
9 ax 8
44-
3 a
4-
12
4-
02 y
-
4
.
3
23 a
16 y a 50
4-
4-
-
- (y 2 2 ~ ) (a
48)
5 y
2
6 a
5 a
-
12).
10).
25 4 - 16 a 6 4- 40 />) - (2 a 2 - 4 aft - 5 b*). 6 y 4 4- 27 x* - 35 x 2 2 ) - (7 xi/ - 9 x 2 - 2 y 2 ) 3 xy (25 4 - 2 2% 4- y - 2 xy 8 - 21 x*if) (4 ^ 2 - y 2 4- 5 xy). (8 x* 115. ( y 8_o7)^^2 + 3 y + 0). 116. (.r 4 4- 2y 2 4- y 4 ) - (x 2 - xy 4- y 2 )
(4
4
-
2 y
2
8
a,v/
4 a 2// 2
4-
3
/>
~
//
-=-
a:
4
117.
(a
118.
(a
119.
(a
8
8 8
120.
Cr
121.
(z
122. 123.
124.
4-
4
4-
-
.
16 a 2 ^4
8 68
256)
4-
+
& 8)
-s-
8 4- c 4-
2
~-
(a 4
(
-
6 afo)
4a
4-
16).
+ ^ 4 ). -26 (a
2 2
^
-f-
4-
c).
-27y -l-9a:y) -(a:-3y-
1).
8
y
6
)
-r-
2 (a:
4-
xy
4-
y
2
).
- *) -(x 8 - 1). 2 (a+ - 3 a"+ 4- a"- ) - (a 1 2 8 a*- 6 ) (a** (a &). (x
4-
10
3
J
1
-r-
a
-
1).
ELEMENTS OF ALGEBRA
264 125.
126. 127.
128.
-
(1
-
a8
8 z 8)
-5-
-
(1
-
a
2 x).
- 3 a#z) (ar + y + s). y (* n - x+ l x+ x a ) ~ (xa + + x). (*+ + 3" -3a 18 *&) 86 27 a (1 (1 +
3
+
z8
*-
l
l
3
1
What is the 2 by a*-ab + 26 ? 129.
130.
-
G ax
By what
2
-f-
remainder when a
expression must a:
+
B
4
3 a b
12
-
8 a*b + 4 a 131. By what expression must 3 a 2 ab + & 2 ? be divided to give the quotient 3 a 2
132.
+
By what expression must
x*
+
-
G x2
4
1
a:
2
6
2
-
divided
b* is
8
x*7
ttfc
+ 2187? - 12 M
8
be divided to give
remainder?
9 as quotient, with 8 as
5 #
-
3 be multiplied to give
-f
4
x2
&).
a 2 6'2
Solve the following equations and check the answers:
- 4(0 x - 5) = 12(4 x - r>) - 22. - 3) = 12 - (x -f 9). 5(2 x 3)- 2(j: 2) = 3 - 2(5 - 9) + 3. 7(2 x - 9) 4- 7(4 * - 19) + 5 = 4 - 3(2 z - 3). 10(2 x 5 x + 3(7 x - 4) - 2(10 x - 7) = 4 - (x - 5).
133. 3(2 x 134. 135. 136. 137.
-
4(ar
+?+4=
139. 1 o
13.
o
140. 10(2 x
143.
a:
a:
2
142.
3)
a:
138.
141.
.
1)
-
9)
-
7(0 x
-
32)
+
5
=
+ 5 + 1=15.
4x
o
o
-
3(2 j
-
3).
- 3 a:). - G) - 2 {3 8)} ^ 5(13 4(j = 5{2 x - 3(* + 4) + 9} - (1 - 3 x). 2(3 x + 4) 8 [2 (a: - 1) - (x + 3) ] - 5{.r + 7[or - 2(4 - a:)]}. a?
- (j -
144. 4-2(3ar
145. 5
- (3 a?
4} = 2(3 x -1) = 2(* 2) (a: + 3).
2 [2 x
- 2 7^~5] +
+ (x 4- 1) (a? - l)(ar + 2) (a: (ar
146. x 147.
a?
2
148.
(2ar-
149.
(5a:
150. (4 x
- 3) (3 x 4- 7) =
3)
= x\x - 2) +
(7 x
-
1 1)
(3 x
2(ar
1).
+ 4).
- 4) - (9 x +
10)
(a:
-
3)
.
REVIEW EXERCISE 151.
152. 153.
154. 155.
156. 157.
158. 159. 160. 161.
162. 164.
265
+ 4) (2 x + 5)- (* + 2)(7 z + 1) = (* - 3) (3 - 5*) + 47. - 2) (j? + 1) + (x - 1) O + 4) = (2 * - 1) (s + 3). - 3) (a: - 4) (a - 5) = (3 - l)(z - 14) (a: + 3)- 24. (a - 2) (7 -*) + (*- 5)(.r + 3) - 2(x ~ 1) + 12 = 0. (a; - 7) (a; + 5) = (9 - 2 x) (4 - a:) + 229. (2 - 3) (3 - 2). 6 x) (3 - 2 x) = (1 (7 - z) (4 - 5 x) = 45 x - 76. 14 3) (j; + 2) + (5 5(x (x
(x
a;
a:
ar
a;
+ 5) 2 -(4-a:) 2 =r21a:. - 2) a + 7(x - 3) = (3 x - 7) (1 x - 19) + 42. 5(ar - 17) 2 + (4 x - 25) 2 - (5 x - 29) 2 = 1. x (3 O + ;T)O - 9) + (a; + 10) (ar - 8) = (2 x 4- 3)(* (ar
2
+ ?=13 + ^ 10 o 2o
163.
.
7)
-
113.
+ ^s-O. f-^ 2 4 ;j
Write down four consecutive numbers
of
which y
is
the
greatest.
By how much does 15 exceed a ? How much must be added to k to make 23? 167. A man is 30 years old how old will he be in x years? 168. Find five consecutive numbers whose sum equals 100. 165.
166.
;
169. There are 63 sheep in three flocks. The second contains 3 first, and the third twice as many as the first.
sheep more than the
How many
sheep are there in
eacli flock Y
The sum
The second of the three angles of a triangle is 180. angle of a triangle is twice as large as the first, and if 15 were taken from the third and added to the first, these two angles would be equal. 170.
What
are the three angles?
A picture which is 3 inches longer than wide by a frame 2 inches wide. If the area of the frame inches, how wide is the picture ? 171.
is
is
surrounded 108 square
172. The formula which transforms Fahrenheit (F.) readings of a thermometer into Centigrade readings is C. = | (F 32). (a) If C. = 15, find the value of F. (b) At what temperature do the Centigrade scale and the Fahrenheit scale indicate equal numbers? (c)
How many degrees C. transformed into
F. will produce F.
= 2 C.?
ELEMENTS OF ALGEBRA
266
A
173.
number increased by
multiplied by
A
174.
number divided by
diminished by
An
175.
3 gives the
same
result as the
numbet
result as the
number
Find the number.
3.
3 gives the
same
Find the number.
3.
express train runs 7 miles an hour faster than an ordinary trains run a certain distance in 4 h. 12 m. and 5 h. 15 m.
The two
train.
What
respectively.
is
the distance?
A
square grass plot would contain 73 square feet more Find the side of the plot. side were one foot longer.
176.
A
177. sister
;
as old as his father
is
boy
the
sum
and
3 years
of the ages of the three is 57 years.
if
each
younger than his Find the age of
the father.
178. A house has 3 rows of windows, 6 in each row the lowest row has 2 panes of glass in each window more than the middle row, and the middle row has 4 panes in each window more than the upper row there are in all 168 panes of glass. How many are there in each window ? ;
;
179. Four years ago a father was three times as old as his son is now, and the father's present age is twice what the son will be 8 years hence.
180. is
What are their ages ? Two engines are together
16 horse power
181.
more than the
The length
power one of the two Find the power of each.
of 80 horse
other.
;
of a floor exceeds its width
by 2
feet;
if
each
increased 2 feet, the ana of the floor will be increased 48 square feet. Find the dimensions of the floor.
dimension
182.
is
The age
three years ago of each.
183.
A
is
boy
father; the
two boys is twice that of the younger; was three times that of the younger. Find the age
of the elder of it
sum
5 years older than his sister
of the ages of all three is 51.
and | as old as his Find the age of the
father.
Resolve into prime factors
184. x* 185.
+x-
y-y
186. z 2
2.
-42.
-92;-36.
:
187.
2
188.
2
+
a
_ no.
-ll?/-102. 189. aW + llab-2&. 7/
13 a + 3. + 11 ~ 6. z 2 + x - 56.
190. 4 a 2
+
191. 10x 2 192.
a;
REVIEW EXERCISE - 77 y + 150. 2 a 2 - 19 a - 10.
193. y 2 194.
195. a 2
+
3a
196. 6
-
2
3y 2
197.
- 28.
a;
198. x*
+
8
5 xy
- 6 y2
13 y
+
a;
2
+
205.
14x 2 -25ary + Gy 2
206.
3 x* -x -
12 x
+4.
-
+
13
6 2.
219.
120.
28
ary
//
48.
+
a:
66 y.
2
.
if-W-y+b.
210.
-11 2 + 10 20 x 4 - 20 z 8 -
ar.
221.
5 x 2.
222. x*y
a:
.
223.
+ G *2#2 + 9 x*y\ 6 x* + 5 a:y - 6 2
230.
15 x 2
231.
9a-4a6
2
232.
(a
229. *2
233.
211.
212. 3 x
- 21
2
a:
-
54.
224.
.
+ 30 x. 2 2 2 2 2 x + ) (a + z ) (a 2 3 + + y a;y) (x (r y) 6 a 2 + 5 a - 6.
218. x
220.
8
- 22 z +
217. 2 afy
209.
a:
+
29 y
a:
216. 2 x 2
.
207. 16x 4 -81. 208. 2 a 8 - 8
-
201. # 2
15.
2
-
204. 5
2
200.
4.
a;
+ 1 1 a*b - a 2/A 214 12 x*y - 14 2 - 10 xy. 215. z + 5x 2 - 6s.
+
?/
+ ary - 10 y a x* - 12 * - 64.
199. 2
.
213. 60 a 2
202. z 2 -2;r?/-f y 2 -9. 203. x 5 - 19 z 4
267
.
?/
7x 2
225. a^a
226.
a;
4
-f
yx*
+ z*x + z*y.
227. 7a 228.
a:
+
8.
+ b - c) 2 -
(a
+
26 x a .
- c) 2
.
234. 235.
a;
8
236. 24
+ 2 - 1. -23 -12.
-a; a:
2
a
238.
a:
239.
(13z
2
240. 4a 2& 2
242. xm+l
237.
a:
241. (a + - xm y + xym -
5# 2) 2 -
-
2 2 (a 6
+
(a
m +^. 2
3y
244. 2a
3% te
ly
a:
2
-
4
a:
.
+ la
a:
V
-
+ 4y2) 2
2
- x + 1. 2 2 y -f
a:
1.
.
c 2 ) 2. 2
c)
-
(c
+
2 rf)
245. 3 ap
243. 4
248.
(12
a:
246. 3 x
247.
mx +
- (b + rf) 2 - 6 aq - 3 c/> + 6 cq. - 3 xf + 3 * 2y - 3 xy. .
V
a+a* + o a +l. wiy
aw.
ELEMENTS OF ALGEBRA
268
- 6 by. 3 ay 4- a + 2 4- 14 bx a%% 8 - 3 abc - a 2 />c 2 -f 3. 2 8 - 2 ax 2 + 2 for 2 - 2 aft*. 2 a.r + a# + az -f 2 6z fry 4- &z.
249. 7 ax
250. 251.
a;
252.
Find the
II.
ft
-I-
C. F. of:
253.
G(x+
9(x
l)'\
+
-
2
1). 2
+ 23 x -f 20. + 20 x 4- 8. 2 2 + 39 xy 4- 15. x*y* 4- 18 xy + 5, 18 x 2 - 11 a 2 - 10 a 4- 9. a 4- 10, a 3 a 2 2 - 5 ab -f 2, 3 a% 2 - 4 ab + 1. 10 x 2 - 23 + 12, 30 ^ - G7 x -f 33. 2 2 - (55. x 16 x 7 -f 71 x 413, 28 12 2 2 - 1 9 ;ry -21,48 afy 2 - 73 xy - 91.
254. 3 #2
-|-
10
255. 5 x 2
+
7 r
256.
a;
a;
8,
-f 2,
15 # 2
z/
257.
258.
/;
259.
a?
260.
a:
261.
a:
//
+ 8 x + 5, x 2 -f 9j: + 20. * 2 - 9 x + 14, 2 - 11 x -f 28. x 2 + 2 x - 120, x* - 2 z - 80. x* - 15 + 30, * 2 - 9 x - 36.
262. x 2
263.
1
x-
264. 265.
ar
Find the L.C.M. of:
+ 3 x + 2, x 2 + 4 + 3, x 2 + 5 -f - 3 x - 4, * 2 - x - 12. * a - 23 x -f 20. 2 x2 - 7 -f 5, 2 z 2 -f 13 x + 1 5, 8 2 + 10 x - 3. x 2 - 18 ry + 32 y 2 2 - 9 xy + 14 y 2
266. z 2
ar
ar
6.
267. z 2
268.
a:
269.
;r
270.
,
Reduce to lowest terms
a:
.
:
271. 2-2-
272
x2 a;
273
-f
2 -f-
4
a:
!8a:
- 77 + 77
P a -5y>+4. ^2-7/7 + 12
'
2?5
x2
~ +
5
a;
'
5
2?6
a:
2
8 xf <
3 xy
-7 - 17
2
+ a;
x
_
40 y 2
+
6 *
14
12 Jr 2__7^/_ J/ 2 + 3 .ry - 2/ 2
28 x 2
REVIEW EXERCISE 8
277
278.
-
agg
9
_
m ~n w 4 + 2 7w% 2 -f sa
*2
281
-
a;
2Q4
'
279.
280.
6
m
2
2 q^(
z4
w mp - n 2
2
2
-
a:
z2
(a
+
1
*
)
m
+ ac
c)a;
-
a;
2
294
295
296
'
297
'
'
289
'
290
+
y
2
+
22
_
+ 2
2 yz
fr
4-
_ 22 _
8
- -
1
ar
2*
-f
*2
3 x
+
ar
a
- 2* + 3 x*
- 2c a:
2
- (y -
z)
-
+
2
y*
+
z2
ary
2
'
y
2 2
+ 2 cV +
2 a 2^ 2
-
4
-
ft*
~
* t
z2
y)'
x'2
'
2 zx
293
?/.rL.!/...
^
288
a;
2
0;2
292
"
+
(j;
.
*2
291
287
+
2
t-
283
1
286
-
282.
'
285
.
)P
2
4- J' 4-
-8x+8
'
n*
+ Og-e. -9
"
"*
4 *2
269
c4
+ 2 0:2
ELEMENTS OF ALGEBRA
270
Find the value of
298
23.
:
-23. + 19)
19
*
23
Lnl +
300.
2
ar
a;
+4
3
__ + -*_ + -la?-la? +
l
+
l)(ar
2)
l)(ar
+
2)(*
+
3)
*
a
1
1
303.
+
(a
x x
~~
+
c) (a
a
+^
~
i
(a
ct)
^
^
a
x
b
+
c)(a
a
~ ''^
a) (x
(:
2.
7.
-f e)
.
6) _
i
305.
O(c-a)
i_ a 20
306. X2
+9
.
+
(c-a)(a-i) 1
_L
x2
12 x
4-
2
308.
+ 7 _ 44
35
3.
^_2*-
^-
m+n
n
"*"
309.
310
+
1
1
307.
x
a:-2
x
*-3
^. 4 +
a?
304.
+ 2Lz| 3 x
(x
+
(a:
i
'
x
1
301.
302.
2 99
"
19(23
(m
+ 3
a:
+ n) 2
^ ""
-2
ar-3
g
a:
2
-2
a:
-
17
x 2 -5a:-i-6"
(a-
a
+7
BE VIEW EXERCISE i
311.
+
a2
b
x8
x2
1
-.-
+ + a;
ft
2
_. -
_
312.
+
1
1
a:
2ft 2
313.
a8
314.
}
.
.,+ a
(1
+
-
2
*2
3
-9*+
1 -f
:
1
(a?
nl
1a:
+
g(jL+ 2 )
y
a;
2
+
y
8*
+
-
2 x'
-
2)
^^^_
4-
Dx
*
8(1-*)
4(1
15
2
319.
+*)
2
8(1
+
x(l *)
4(1
-
321
(
6_
c)
2_ (a._
- *) + * 2)
c) 2
- (a - ft) 2
g~ft
ft-c
322.
323.
-
1
*2
316.
317.
*2
20
,2
10 z 2
315.
318.
271
(
'(a-6)*-(a:-r)a
ELEMENTS OF ALGEBRA
272 Simplify:
-
z2
325.
~
2 x*
324.
-
+
10
-lOx
2 a
8
-
2
^ ""
4
a;
5x- 2
2
1B x
+
10 o# 2
?/
331 g gy
6y 3
a:
2
a
+
3y
-f
'
12 a
40
-
-
y
~
4 xy -8
*2
a;
_
^2
2
l5rt~+~54*
1
6 q
+
-
a
5 x
4
-
a
t
~
3;B
1037
.y
x
-
6
2 y/
+
5
?/
-
(a?-4y)
x
2
3(2 x
^/
- ll.y-20
G
9
27^-12^7 - 5 a - 6*
6~7**
- 9 *// + 27 - 7 xy + 12
2
x'
x 3y
-
+
6 a*
x
24 y 2
a;
3
4 y2
y
+
2y
G
fl
4
2 ;/
-
-
-
3 y
y~4-y+
' _
3 y) 2
6 (
15
^e
-
10
8
a;
- 15 -33
a:
?/
a:
2
a:
2
2
-f
5 sy
zy
2 + 0^ + ^2^7 a...
.a: '
. '
a:
~"
+ 8 2 - 4 x?/ 2 ^_ G x 2 + 13 gy_+ - 19 xy + 6 y 2 8 x* '
10
+
2 y
5 x8
2
*2 '
333.
~
- 7 acy + 12 y2 + 5 a:y + y 2
2
2
42
4 a
y
x*
a:
+lOar
+lly-10
2 z6
333.
250
J?_ x fl^-^ffjje _ 2
+
2
11
-
-
8 .V
332
-
2
i^+^T-
03
4g~0yg
329
330
^ "" 12 *.
2 a;
'
-
^ - 28
8 '
2
334
5
-*
a2
328.
-i-
-
3a; 2
327
4 r8
8 x2
4 x
+ 4 y* - 2 y2
20 44
'
6
y
REVIEW EXERCISE ~
336.
~
c*
c*
337.
q2
a2
-
(a-f2/,)
2
338.
.
+
342.
343.
344.
\
+
-
1
q
{
fl-.1V. \x
c
340.
(
+ IV.
5
347.
xi
348.
349
**
_ o;
350.
2
-i. '
-"l
a>74
5
a:
76
Find the numerical values of 351.
1+
\.
2"
'
;
.;r
w
~ ~ x2 + 8 + -5 2 .13
13 s
:
i--, if a
=
3.
352.
:
353
^-3
pE+1
*
L
2
354.
a
- +
?_2 ^
a:
. r
y
+^
x
/2x~l V
4
5a:~2 10
6
1
-
\
f
V.
\5yl
-
346.
-
|
b
345.
-?-f!?.y.
Simplify
q
|
yj
7
(
C
c
a
339.
ni + -.
l
~
f
aj
(a-Wi + iJ. (ar \
:
+ lV.
(a \
341.
*|
c
b
5
5
h
\
a
278
11
ELEMENTS OF ALGEBRA
274 355
___ _/| 2(*-l)J
5
f
x
U<>-3) 356
357. ar
1-10*W*-1 1-** JUa-l
1-* YTx
fl V
2
xy
+ ya
xl
+
xy
+ yl
x*
_
358.
+
a
l
359.
y-x
y360. (a
a
yabc
c
b
361.
1+2
i+5
1+1
362. 9 x2
-f
363.
(~
364.
+ 1 + W?* _ 1 + (* 2x \3a 2x) \3a
365. 1
a
,
-f
I -
366. a2
4-
I
+
x
x
REVIEW EXERCISE
^\
2
f
367.
\b*
+ c*
+ b
b
a
b
(b* -f c*) }
b*-c*)^ c
c
,
b
b
4-
c
c
368. (1
+ab)(l+bc)
369. a
a
370
b
+ .
6
b
a
b
c
1
~^ _
'
'
(-/')(&-o)
1
1
a
372.
2
-
m
373
"1*7 374.
.
275
ELEMENTS OF ALGEBRA
276
375.
-3 Solve the equations
376.
a:
2
8 (a;
45
379. 4(* .
+
+ 6)-
380. 5 {2 x
+
2
+
r
~
-
-
O
-
1
5"^
r
J !__7. a:
1)
-
-
:
5*-8,*-2 = 15.
378. 2(3 x
381.
or
iLf-5 + !*=! = 2 J.
377.
<3
|(x
+ 4) +
-(* +
10)
20
+
= 1
J
1)}
x-f x -
10
|(x
j(* v/ O
o
!
7)
= 0,
+ 5)-
^ - ^-^ + _j_
+
3(*
+
10 x
^-\:)
-f
51)
+2J
=
382.
. '
k
#
1
_j-
j
a:
2
3_ = !.
383. __4 2x
5
3
7
vC
a:
10
385
17
387
*
-7ar =
L*J> _ 14
(5 ar
ear-7 + i3JTo^ 1
,
389
1
2
10ar
+
15
,
"""
28
+ 16ar_63 -24 g
2T~~~ia
2J
'
7
12f
-
8 a'
a
-
5
390.
,
+
0,
/
14(ar-l)
7
18
105
a:
3
REVIEW EXERCISE x
391.
_x
4
3737-0 ^ x
a:
~
^
"i
~r;
x
_
5
2
_
i
-
'2 a;
+
2
^ (a -
(8 x
6
-
3) (x
1
8 9*
-
a?
R -
7 ~r
1
"
-
a)
^
2
-
(a:
2
3)
1) = (4 x
1(5
a;
+
~
^
H- 1
4-
3(4 *
2
1) (4 x
397. 1
398.
.5
x
-
399.
.5
x
-f .6
400. 3*
401. y
rt
^=
=r
=
.25 x
x
-
.8
+
=
.2
a:
.75
x
-
408. (x
-
~
5).
402.
c c
-}-
-q.
a
-f
1
404.
b
b
a:-
2)(ar
1.
4O5
40,.
-
-f .25.
.
n
-(a:
.
-f 1
c
JLg:== 7wa:
f
-
a;*
177,147.
~
&
2
a:
a;
403.
4
m
x
~
*
u
a;
7
*2
l)(x
2
+
x x
7
.
396.
7 _ x -8~a; -
a:
a;- 6~a:
ar- 5
277
a)
+
a)(a:
a)(x
a-(a;
-
x
b
-
&)(>:
ft)
=
2(ar
-
) (a;
-
J).
+ 2a +2&) = (a: +
-f 6) -f c
=
(z
+
a)(a:
____-_ ^
a:
2
a:
5
i).
2 a)
1
+
a
+
1).
ELEMENTS OF ALGEBRA
278
-
410.
(x
-f
411.
(x
- a)(x -
a)(z
a
412.
x
b)
b)
x
-f
a
x
a
x
a
x
b
1
417.
mx ~
a
nx
c
(a
x1
~
-
~
-
-2
a)
6 2a.
2 .
)
-f c
ab 1
x
c
b
mx
d
nx
b)(x
b)(x
a-b
+
I2x
a
-
~
b
x
c
b
c
d
b
- c) - (5 -
a
418 ~j-o.
b
a
-f
b
-f
x
(x
1
x
416
b
c
1
x 415.
a
x
b
=
-
(x
_a
-f
1
414.
2 alb
=
b
a
b
x
-
c)(:r
-
a)
lfi:r
- (c -
a)(x
-
b)
=
0.
rt
~r l
a
2
x
419. )
a
ar
IJ a;
x
2 a
-f c
a
c
+
~ a
A
4x
a
a
x
2 b 6
a
+
Qx 3 x
2 c 6
c
a
+
6
-f
walks 2 miles more than B walks in 7 hours more than A walks in 5 hours. Find the number of miles an hour that A and B each walk. 420. Tn 6 hours
in 9 hours
B walks
;
11 miles
number of two digits the first digit is twice the second, 18 be subtracted from the number, the order of the digits will be inverted. Find the number. 421. In a
and
if
422.
A man
drives to a certain place at the rate of 8 miles an
Returning by a road 3 miles longer at the rate of 9 miles an hour, he takes 7 minutes longer than in going. How long is each road ? hour.
A
person walks up a hill at the rate of 2 miles an hour, and at the rate of 3^ miles an hour, and was out 5 hours. far did he walk all together ?
423.
down again
How
REVIEW EXERCISE A
424.
lowed
279
steamer which goes at the rate of 264 miles a day is foldays by another which goes 286 miles a day. When will
in 2
the second steamer overtake the first?
425. Find two consecutive numbers such that the sum of the fifth and eleventh parts of the greater may exceed by 1 the sum. of the sixth and ninth parts of the less.
Find the fourth proportional
-
426. x 427.
,
y,
i,
-
z2
y\
:
-xy + y*.
x*
+
428. a
|.
5,
a
-
t>,
a8
-f
2 ab
-f
6 2.
Find the mean proportional to
and
429. 3
431. Find the ratio x
=
5x
432.
A
7y
:
22
-I
22
.
a
y, if
wi*
;
+ y=
ny; ax
10 inches long
line
- iand
430. z 2
1J.
-\-
-
by
ex
+
dy.
divided in the ratio m:n.
is
Find
the length of the parts.
433. The sum of the three angles of any triangle is 180. angle of a triangle is to another as 4 5 and the third angle to the sum of the first two, find the angles of the triangle. :
434. If a b
=5
435. Solve
n
:
436.
:
Which
:
and
7,
m
ratio
:
is
n(n
b
:
c
=
x)
14
15, find
:
=p
greater, 5
:
a
m n(p :
7 or 151
:
:
If is
one
equal
c.
x).
208?
437. Prove that the number of miles one can see from an elevation of h feet
438. a. b.
is
very nearly equal to ^-
Which
-
miles.
of the following proportions are true?
+ 4ft):(Oo + 86)= (a-26):(3o-46). - 46 2): (15a 2 - 31 afc + UV ) = (15 a 2 + 31 ab + H 6) (25 a2 - 49 63), 2 2 8 2 2 (a + &*) (a -h & ) = (a - ) (a -6). 8 8 5 ~ a*b + a*b* - a 2^ 8 + aft* - & 5 ) (a 8 - 6 8 ), (a + 6 ) (a + ft) = (a (3a
2
2
(9
:
c.
d.
fc
:
:
:
:
ELEMENTS OF ALGEBRA
280
439. Find the value of a.
29(a
+
&)
c.
2 (3 a
+
2 ab
:
x
=
-
x, if
551 (a 3 -
8 ft)
:
-
2 ft
2 (5 a
:
)
-f
19(a
4 ai
-
&).
12
ft
2
)
=
a?
:
(5 a
The volumes
-
6
ft).
of two spheres are to each other as the cubos of a sphere 2 inches in diameter weighs 1:2 ounces, what is the weight of a sphere of the same material having a diameter of 3 inches ?
440.
their diameters.
If
Solve the following systems:
441. 7
a:
-2y=
1
3
;
+
a;
5y
=
59.
443.
x + 17 # 53; 8 x + y = 19. 33 x + 35 y = 4 55 * - 55 y = -
444.
7jr-9y =
445.
7a?-y = 3; 5x+4y=lQ. = 25. 7 a: - 3 y = 3 5 -f 7
442.
446.
447. 448.
449. 450. 451. 452.
453. 454. 455.
456.
457. 458. 459.
=
;
17;
;
9ar-7# =
a;
16.
71.
?/
x + 5 y = 49 3 x - 11 y = 95. ax + ly = 2 a*x + & 2# = a + b. 5z-4:# = 3;r-f-2# = l. ox -f &// = 2 + y) = a + 8a + 21+3ft = 0. 28 = 5 a - 4 12 - 89 = q. 5j + 7 7 = = 2; 42 = 15y + 137. 20y + 21 18a = 50 + 25y; 5#+ 10 = -27 a. 56 + 10y = 7a;; 15ar = 20 + 8y. - 11 7; 21 7 = 27 + Op. 9/> = 2 - 7 y = 25; 4 = 5 y + 29. 3 - 35. 8 - 59 = 3 z; 5 2 = 7 ;
;
;
ft.
/>(.*;
ft;
;
ar
a:
/)
a;
a:
+ 5y)- (or |-l(*-2y)=0; 1(3 | a;
REVIEW EXERCISE 460.
3 x
+
7
3
_
y
28i
~~~^ = 5;7;c=56-3y. 8
461.
a?
7
3 y
a?
_
1
~~10
15
12
4
__ "10
10
463.
465.
~
4 g
7g
2
-
467.
468. ^
+
+
3
.?/
+
1
a:
. '
= 2;
i^
2 g
+
47O
469.
}
cte -
by
=
c
\
472. ax cx
ey =/.
475.
474.
dx+frj-
c\
= 2J.
(2
--i = 5; i-
4*
by
= m;
+ ey-n.
-_ &
_
_
_
3~12
471. ax
3 y
- 2y)-
(or
= 7;- + -=2.
+
+y
3
473.
car
x
= 4-
rf
y
ELEMENTS OF ALGEBRA
282
476. In a certain proper fraction the difference between the nu merator and the denominator is 12, and if each be increased by 5 the Find the fraction. fraction becomes equal to |.
477. What is that fraction which becomes f when its numerator is doubled and its denominator is increased by 1, and becomes when its denominator is doubled and its numerator increased by 4 ? j|
478. to
,
If 1
it becomes equal becomes equal to ^. Find the
be added to the numerator of a fraction
be added to the denominator
if 1
it
fraction.
479.
The sum
of three
numbers
by 4, and the other number least. Find the numbers. least
is
is
The greatest exceeds the sum of the greatest and
21.
half the
480. There are two numbers the half of the greater of which exceeds the less by 2, also a third of the greater exceeds half the less by 2. Find the numbers. 481. Of the ages of two brothers one exceeds half the other by 4 is equal to an eighth of
years, and a fifth part of one brother's age that of the other. Find their ages.
482. If 31 years were added to the age of a father it would be also if one year were taken from the son's age
thrice that of his son
;
and added to the father's, the Find their ages. age.
A
483.
latter
and B together have $6000.
B
spends \ of his. had each at first?
B
then has
J
as
would then be twice the
A
much
son's
spends } of his money and as A. How much money
484. Find two numbers such that twice the greater exceeds the by 30, and 5 times the less exceeds the greater by 3.
less
485. A sum of money at simple interest amounted in 10 months to $2100, and in 18 months to $2180. Find the sum and the rate of interest.
486. A sum of money at simple interest amounts in 8 months to $260, and in 20 months to $275. Find the principal and the rate of interest.
487.
sum of
A
number
consists of
two
the digits be multiplied by
the number.
digits 4,
whose difference
is
4;
the digits will be inverted.
if
the
Find
REVIEW EXERCISE 488. There
number
283
two
digits which is equal to seven times the digits be transposed the new number Find the will exceed 10 times the difference of the digits by 6.
the
sum
a
is
of the digits
of
also
;
if
number. 489.
Find two numbers whose sum equals
and whose difference
s
equals d.
490. The sum of two numbers squares
Solve the following systems
491. x
-f
492. ,
+
493.
a;
5
y
-f
2
=
2
+
z
29|
41; *
x
;
+
$x
a:
+z=
x
+
499. 3 x 500.
and the
difference of their
:
=
z
y
18J ; x
-f z
y
=
13|.
;
+ 5=84.
*i, , 3
G; 2
y
a:
ar
+
-f-
3 y
2 z = 8 - z = 20; ;
2y + 2z =
a;
7
a:
-4#+
a:
497. y :
+
=
4
4a;-5#+2z =
Solve
-
62
3 y
494.
498.
,
425
s
495.
496. 4
is
Find the numbers.
is b.
=
//
+
5 y
=
i-f-i
x
11;
=
a;
;/
-f
101
;
z
7
x
y
=a;
+
1+1z = 6; y
y
502. 3ar
5; z
1;
2 z
25 ;
a;
- z = 12. = 209; 2
+
#
+z=
35.
a:
a: -f
i-fi =
z
= 79.
e.
x
z
+ 2y = 8; 4z+3z = 20; = 15; 2y + 3a = ll; = -2^ 20; 2/>-3r = 4; 30
503. 2a:-f 7;/ 504. 7;?
506. 2
3^ =
a; '
507. --\
'
~
8;
5^
9z
=
10;
a:
+
4r=-9. 4
?/
2z
= 15.
2.
3z
=
35.
ELEMENTS OF ALGEBRA
284
516.
517.
+ + 3579 2+?.
!f
== 2800, ra?
524
523.
x
+
9
+
:
ll"
1472.
+
= 3a-f& + r, 36 + c, i=a + 6 c, z
t
\
jx [y +
=
y
+y + = + 2
z-
=3a-&-c.
REVIEW EXERCISE
285
525. When weighed in water, 37 pounds of tin lose 5 pounds, and 23 pounds of lead lose 2 pounds. (a) How many pounds of tin and lead are in a mixture weighing 120 pounds in air, and losing 14 pounds when weighed in water? (b) How many pounds of tin and lead are in an alloy weighing 220 pounds in air and 201 pounds in water ?
A
and B together can do a piece of work in 2 days, B and C and C and A in 4 days. In how many days can each alone do the same work? 526.
in 3 days,
sum of the reciprocals of of the reciprocals of the first of the reciprocals of the second and
527. Throe numbers are such that the the
first
and second equals
;
the
sum
and third equals \\ the sum third equals \. Find the numbers.
M
N
528. A vessel can be filled by three pipes, L, M, N. Tf and run together, it is filled in 35 minutes; if and L, in 28 minutes; if L and Af in 20 minutes. Tu what time will it be filled if all run
N
t
together? 529. A boy is a years old his mother was I years old when he was born; his father is half as old again as his mother was c years ago. Find the present ages of his father and mother. ;
530. A can do a piece of work in 12 days B and C together can do the same piece of work in 4 days A and C can do it in half the time in which B alone can do it. How long will B and C take to do ;
;
it
separately
?
531. A number of three digits whose first and last digits are the same has 7 for the sum of its digits; if the number be increased by Find the number. 90, the first and second digits will change places. 532. In circle
and
A ABC, AB=6, BC = 5,
touches
AC
F respectively.
and CA=7. An (escribed) and the prolongations of BA and BC in Find AD, CD, and BE.
in /),
E
533. Two persons start to travel from two stations 24 miles apart, and one overtakes the other in 6 hours. If they had walked toward each other, they would have met in 2 hours. What are their rates of travel?
ELEMENTS OF ALGEBRA
286
534. Represent the following table graphically
:
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES, FRANCE, GERMANY, AND BRITISH ISLES
Draw
535. One dollar equals 4.10 marks. formation of dollars into marks. 536.
The number
work
of
How
long will
537. If is
to
t /
D
in
days it
of
workmen
take 11
men
to
=
3.3
down
2 t' .
the graph
from
D
D=
1 to
12.
do the work?
feet is the length of a
I
seconds, then / = 3 and write
required to finish a certain piece
Draw
is
a graph for the trans-
Draw
pendulum, the time of whose swing a graph for the formula from / =0
the time of swing for a
pendulum
of length
8 feet.
Draw
the graphs of the following functions 2
+
x.
:
-
-
538. 3 x
+
5.
542. x
539. 2 x
-
7.
543. x 2
-
x
-
5.
547.
3 x.
544. z 2
+
x
-
3.
548. x 8
-
2
x.
545. x *-x
+
2.
549. x*
-
x
+
540. 2 541.
-
-
3 x.
550. Draw the graph of y 2 and from the diagram determine
+
2 x
546. 2
x*,
from x
=
:
a. b. c.
d. e.
The values of y, i.e. the function, if x = f 1, 2|. The values of x if y = 2. The greatest value of the function. The value of x that produces the greatest value of y. The roots of the equation 2 + 2 x x z = 1. ,
x
x2
.
x*.
2 to x
1.
= 4,
REVIEW EXERCISE The formula
551.
a.
2
Represent
] f/f
287
for the distance traveled
by a falling body
is
to t = 5. graphically from t = (Assume g = 10 scale unit of the t equal to 10 times the scale
meters, and make the unit of the b.
How
c.
In
\
^
2
.)
far does a
body fall in 2^ seconds? seconds does a body fall 25 meters?
how many
Solve graphically the following equations
x*-"2x-7 = Q. 2 ~0a: + 9 = 0. 553. 2 554. + 5 - = 0. 555. x* - 5 x - 3 = 0. 556. z 2 - 3 x - = 0. 557. x 2 ~ 2 - 9 = 0. 558. 3 x* - 3 - 17 = 0. 566. x 4 - 4 x 2 + 4 - 4 = 0. 2 567. x 5 - 4 - 11 x* + + 2 8 569. If y +5 10, a. Solve// = 0. = 5. J. Solve .r
561.
-1
562.
563. 564.
a;
565.
a?
-
4 x - 15 = 0. + 10 x - 7 = - 13 = 0. 3 x - G - 3 x - 1 = 0. 3 + 3 z - 11 = 0. 2 8 - 6 + 3 - 0. z 4 - 10 x 2 + 8 = 0.
560. 2
a:
a;
:
559. 2 x 2
552.
j;
2
1
a;
a:
3
.r
a:
.r
a:
a;
x-
18 x
-
4
=
0.
568.' 2*
+
Z
-
4
= 0.
a;
//
e.
f.
c. r?.
Solve y Solve y
=
5.
15.
Determine the number of real roots of the equation y Determine the limits between which m must lie, if y
2.
=m
has
three real roots. g. h. i.
Find the value of m that will make two roots equal if y = m. Find the greatest value which ?/ may assume for a negative x. Which negative value of x produces the greatest value of y ?
Solve graphically
570
:
'
571.
572.
.
'
=
8.
ELEMENTS OF ALGEBRA
288
+ ,-4, j^ x -f = 3.
y =10,
|4,-5 xy = 0. 577.
=
4'
^
3
(f-,
?/
[
[
581.
{f_7l j?
582.
|''- o2
= 578.
579.
2 \*> a: + [
=
: y*
or 25.
f-MV --
585 594. 595;
596. 600.
586
^
'
+
jf:ji f-
:
6)T
'
587.
(a-iy.
588.
(a
589.
(1-a:)
-f ?>)
(a; + ^) + (air-%)8. 4 - %) 4 (aa; + %) (a* 5 6 + *) (1 a:) (1 2 2 -f (2 + 3 x + 4 ) (2 -3 x +
3
(a
8 ft)
590>
(2
591.
(3
592.
.
(1
593.
597.
(1
+
598.
(1
+
599.
.
a:'
-
3.
.
-f-
-
> 3'
a:.
+
583.
jj+;frf
Perform the operations indicated 584.
2* + ?/
[
4
a;
2
(1
2
)
.
(#
-
4
x) (l
+ + -
2
3 ,
a:)
xY. 4
2)
.
4
^)
+ x' )'2 - x + 2 )'2 601. (1 +
.
2
x
.
a;
.
x
+
z2) 8
.
Extract the square roots of the following expressions:
-
602. 64 a 12
128 a 10 6
603.
4-8 xf +
604.
a:
16
-2
-
605. a 8 606. 9
+ 2
-
2
-4
?/ '
14
100 a 8
4
608.
a:
/;
-
2
100
3
?/
a:
1
2
?y
~+ x
10
[
a:
1
6
.
a:
a:
V 6x
4
611. a
2
612.
2
a:
-f
2
aAa:
+
- 2 6a: +
2 ?/
-
+
9 6
-f
943
2
+
25 c
+-+
2
30 &c
+
a;
6
+
--+
2
-
3
2
10 ac
-~-bx.
?/
a:
7
//
fe
.
1) . 5 a*.
609.
610. a 2a; 2
10 6. 1
9(5
//
a:
^i -
aW + 100 aW- 48 a*h +
10 2 + x^f - 128 a*^ + 04 aty 4 6 2 10 3 5 4 3 2 + y. -f zy x*y* + ^s_ 14 a 4/,4 + 4 a 8^6 + 9 a a^e _ 6 aW + 8
04 aty 6
+ + 4 a 6& 2 + x -f 13 2 - 4 8 -f 4 4 + i 2 ) 2 -f (a 1 - 2 & 2 ) (4 a:
a;
2 (2 a ft
-f
a:
4 fSb
607.
6
30
aa: 2 .
-
a6.
.
REVIEW EXERCISE Find the fourth root 4
613.
4-
of
:
a2
+
4 a*b
614.
10:r 4
+ 9G* 3 +
615.
HI x s
-
108 afy
616.
10 a 4
-
32
Find the eighth root
289
fe
2
2 /;
-f
4
aft
+
8
4 ft
.
+ 81. + 54 'x*y* - 12 a?y + y*. + 24 a 2 4 - 8 aft 6 + 8 +
21G.*; 2
21Ga;
/;
ft
.
of:
617. a 8
-
8 tvb
+
28 a 6 //2
618. a 8
-
10 a*
+
112 a 8
-
50 a c ft 8
+
-
70 a 4 ft 4
50 a 8 ft 6
+
28 a 2ft
- 448 z +
1120
Find the square root of
:
619. 942841.
621. 0090.2410.
620. 25023844.
622. 4370404.
625-
VOIOOD + V582T09.
a:
4
-
1792 x*
8
aft
7
+ 1792 2 - 1024 x +
+
ft
8 ,
a:
256.
623. 49042009. 624. 44352.30.
V950484
626.
Find to three decimal places the square numbers
- V250 - \/4090.
roots of the following
:
627. 49.871844.
629. 035.191209.
631. 494210400001.
628. 371240.49.
630. 210.15174441.
632.
633. 21.
634.
V
32
635. 4J.
636.
2.
637. 40.
9g.
638. GGff.
639. According to Kepler's law, the cubes of the distances of the planets from the sun have the same ratio as the squares of their periods of revolution about the sun. If the distances of Earth and Jupiter from the sun are at 1
5.2,
:
and the Earth's period equals 3G5J
days, find Jupiter's period.
Solve the following equations
= 70. + 2 -21 x = 100. *+* = 156. x 2 - 53 x ~ - 150. 8*' + 24* = 32. 9a; 2 + 189 z = 900. 651. (x -
640. x 2 641.
642.
643. 644. 645.
:
646. x 2
9 x
a-
647t x
2
-f
2)
-f
(x
~ -
+ 9x _ 5x _
16
= 0.
22
= 0. =
648
x2
649.
* + 9? , = 87.
650. 3a; 2 2
x
+
5)
2
66
+x = 14.
= (x +
7)
2.
.
290
ELEMENTS OF ALGEBRA
"""
ar
a
x
b
ab
REVIEW EXERCISE ~
X+
1>
x -
+
x
a
+
a
~
"
b
x
~T~
+c ~ a
i~ i
c ~ b
rj* 2
*
4 x + + Ox + 4
=
690.
291
4(5
691.
692.
693.
694.
1 + V* -2bx + a 2 + 2 ax a a )jr 2 a(l + & )z -fa 2 (1 ax + to -f ru: 2 - ax - bx - c = 0. .
2
2
695.
ft
2
696.
698. 699.
a:
2
- 2V3:r
-f
+ fa + 1 = 0.
2
2 V5
+
4
a:
=
ex
-a-b-c= fx
702. 2(4 :r
704.
706. 707. 708.
-
2
/'r'S
__
2
+
(:r
2
:
'
0.
4
^^
0.
701. (x 2 +3a:) 2 -2a; 2
7r\O
fi
2
697. ax 2 a;
2
3
2
a:)
)'*' _i.
:r)O
2
-
7^^
+
-6a:-
28
x2
+
21
a:
^^ ^T
^3"
:c-f 1)
=
8
=
42.
0.
+
5 '^
=
0. 1
=
0.
ELEMENTS OF ALGEBRA
292
+36 = 0. 16 x* - 40 a 2* 2 + 9 a 4 = 0. 2n n 2 2 -f-2aar + a -5 = 0.
*2
709. **-13a: 2 710.
711.
25
16 |
-16 - 44#2 + 121 = 0.
25 713. 3or
a:
___ _ 2* -5 3*2-7
714
+
'
4
a2
i
2
715.
716. Find two consecutive numbers whose product equals 600. 717. What number exceeds its reciprocal by {$.
sum is a and whose product equals J. needs 15 days longer to build a wall than B, and working together they can build it in 18 days. In how many days can A build the wall? 718. Find two numbers whose
719.
A
The area
of a rectangle is 221 square feet and its perimeter Find the dimensions of the rectangle. 721. Find the price of an apple, if 1 more for 30/ would diminish
720.
equals CO feet.
the price of 100 apples by $1.
722. 217
;
The
difference of the cubes of
two consecutive numbers
is
find them.
723. Find four consecutive integers whose product is 7920. 724. Find the altitude of an equilateral triangle whose side equals a. 725. A man bought a certain number of shares in a company for
$375;
he had waited a few days until each share had fallen $6.25 might have bought five more for the same money. How shares did he buy ? if
in value, he
many
726.
What two numbers
are those whose
sum
is
47 and product
312? 727.
A man
bought a certain number of pounds of tea and
10 pounds more of coffee, paying $ 12 for the tea and $9 for the coffee. If a pound of tea cost 30 J* more than a pound of coffee, what is the price of the coffee per
pound ?
Find the numerical value of
-4*+
728.
12
729.
(J)-*
-
:
8- l
+
(3|)*
+
8
(a
-8 +
ft)'
+
64-
+ i.
REVIEW EXERCISE
293
implify : 30.
(y*
31.
(a*
32.
(a*
33.
+
y*
-
x*)(a*
+
/^
-f
+w
(rrr
+ y*+l)(y*.-l). +
c^
5 n*
a*x*
M
-
+w
+ x*). - aM -
ft*c*)(a*
n
3
+
?n^n^
n )(m*
-f
35.
36.
(a-2
37.
(a-
38.
(1
39.
(i*
+
&-2)( a
-2_
4 d*).
j-2).
+
(64 x~
l
41.
(^ -
42.
(a*
+
(a^
+
6*
-
-
2 a
M+
44.
(x*
46.
(4
48
^i?
1
1
(4 x~*
a:
2
3 ar 2
+
2
+
27
y
ary*
+
x^y
M
-
x T
^
n. (v/x)- X
2
)
6)
c*
+
-f
^2?
a;-
a8)
l)(>r
*
U*") -4-
sT~
-
a*6^
-*
-i
(
+
&*).
aM"
9 x*
-
42 x*
.2?
1).
^ + cb-
2
x
+
y*). -f-
-
(x*
+
+
-
2
3 y"*).
(x*
+
(a*
28 x
x
+
y*)
-f-
2
1
(4 x~^
-r-
-
-f
12 x*
50.
52.
n^).
-6- 1 + c- )(a-i + &- 1 -f c" ). + a^ 1 + a 26" 2 )(l - aft* 1 + a 2*- 2 ). 1
40.
43.
6*
3
- 2 d*m* + 4 d-)(w* + 2 rfM + 1 + x- 2)(x 2 4- 1 -f ^ 2 ). (x*
34.
+
'
s
-f
a).
+
49)*.
+
c*).
ELEMENTS OF ALGEBRA
294 753. 754.
755.
~
r*._ x
1 4j "r
[1r^ T 1
i
;.
O/lf ^ ^ II r* *
4"*"
1 "1 A 1.
JU.1+J
756.
757.
2^3(^-2^21 + 4^-3^:0. + V22
758. 759.
776.
760. 4\/50
+
12 V2b8
761.
\/G86
+
v/lG-v/128.
768.
vff +
2-V2
4-
SVlOOO.
V^~ 4^ -2^/2
2-V3
-
IIEVIEW EXERCISE 780.
295
y/a
-f
x
+ Va
x
y/a
+
x
Va
x
782.
781.
783.
Find the square roots of the following binomial surds: 784. 10 785.
786. 787.
+ 2V21.
789.
38-12VIO.
790. 14 791.
103
94-42V5. - 2 V30.
792.
75-12V21.
793. 87
799. a
c
+ 2 Vab
- VlO.
3J-
- 4 V(j.
16 + 2V55. 9-2VI5.
788. 13
794.
- 12VIT.
-
12 ^
ac + 6t
Simplify
801.
806.
7
+ 3 V5
7
-
7
+ 3V5
Va
z
2
3 V'5 (
(
7-3V5
4
+ V3 + V3
4
- 2V3*
^
|
5
807.
Va 809.
-f
x
Va
-f
x
+ Va
a;
ELEMENTS OF ALGEBRA
296
810. Find the sum and difference of (ar
811.
x/aT+l
+ V2y-x 2)* and
- V? =
(x
- \/2y -
813.
1.
812.
a:
2
)
VaT+lJ
4
.
-f
814. 815.
\/2(r+
816.
817. 818.
819.
/3
1)
-
a:
+ v/2 x +
+ V3
5
a:
-f
15
12
=
13.
= 17. = 1.
+ 10-3Var- 1 V* + 60 = 2 Vx~-K5 + V5. 2\/^"+~5 + 3Vor-7 = V25 - 79. /9ar
a:
820. 3
836. Va:
- V-c^lJ - V2 -
x
+
2
829.
Va:
+
28
830.
V14 a;
-f
9
-
3
831.
\/12
832.
V2a:
833.
Va:
+
834.
V3
ar
a;
+ 3
ar
10.
+ V9 x - 28 = 4 V2 ar
14.
+ V3a:+ = 0. + ViTli + V7 - 13 = 0. -f
2 VaT+1
1
a:
7
-2 Vx
-f 1
-f
1
= V5x + 4. + Vx - 4 =
V4a;-f 5
0.
^l - g.
KEVIEW EXERCISE 838. 5 x*
839. 4 a;
840.
a:
2
36. 6.
V4 x 2 - 10 x -f 1 = 10 x + L * 2 - 3 - 3 Va: 2 - 3 x - 10 = 118. a:
843. x 4 6
844. x
:
+
2
-f
x4
a;
+
8
+
5
a:
8
-f
- x*y +
8
847.
a;
848.
a:
7 a8
3x
+3 -4
8
a:
851. x*
4-
2
a;
8
854.
a;
855.
a;
a^
8
-
13
a;
8
-8a: 2
857. 4 x 8
-f
858. 16
8
859. 8 a; 8 860. 8
a:
8
8
-J-
+
a;
+
13
a/
y
4.
15. 4o.
a;
19 x
-
2
40 x 2
14.
3x
-
7
a;
9.
-f
49.
868. a 8
-
b**.
-64.
870. a*"
-f
& 6n .
8
1
872. a 8 873.
- 1000 6. + 512 y8
8^-27^.
+ 1. + 216 rt aty a 10 - ab9
871. a
.
a:
3.
19a;-12.
2
864. 275 8 -l.
866. 729
64 y*.
-f 12.
l-64a. +
865. 64 a
80.
4 xy 8
869. a* *
a;
863. z*y 8
+ 16.
-f 1.
.
+ a8
861. 27
x
-f
2 a#*
27 y 8.
4-
8 a;
a:
2
2 x
-
11
4
+
^
-f"
856. x 8
18a:
-f
a:
4.
8
a;
-2a; 2
853. x
-
2
4 ar 2
852.
+
6 2 -f 3 6 s.
+
a;
2
-f 2.
8
850.
a;
- 28 a -
845. 5 a 4
846. 4 x*
849. a 8
867.
=
9)
-f
Resolve into prime factors
862.
- 12\/(ar4-4)(5z~
+ 4\/3^~- 7x + 3 = 3ar(a; - 1)+ + Vo: 2 + 3 x -f 5 = 7 - 3 a:.
841. 4 a: 2 842.
11 x
-f
297
.
874.
875. a 18
ft*.
.
4- a.
876. a l0m
- 1.
ELEMENTS OF ALGEBRA
298 877.
Show
that 99
878.
Show
that 1001 79
879. For
by x
-
880.
+
what value
1 is divisible
-
by 100.
1 is divisible
m
of
2 #3
is
by 1000.
mx*
5x
3 exactly divisible
3 ?
What must be the value of m and n to make 8 + mx 2 + nx -f 42 exactly divisible by 2 2 and by a;
Solve the following systems
:
+ 2y=\2, xy + y = 32. 2 3 2 z3 xy + y = 7. y = 28, x 8 3 + y = 13:3. -f y = 7, 2 + xy = 10, y*+ xy - 15. 4 2 2 + afy 2 + -f ary + y = 37,
881. x
882.
-f-
883.
a:
884.
a;
885.
a;
886.
a:
a:
888.
a:
889.
a;
890.
a:
2
a:
a:
1
892. 1 x
,
+
? + p"ia-
=
L+L=13.
1
1
5;
y
a:
a:
a:
8
-
8
+
y y
9 9
= =
37
1
2
y
;
152,
y
a:
2
5
2
l-I = xz
y
895.
_
1_3.
1-1-21; x
894.
?/
a;
-;Vi'
896.
481.
+ y 2 = 34, 2 - y 2 + V(j; 2 - y 2) = 20. 2 + y 2 - 1 = 2 a#, xy(a:y + 1) = 6. 2 -f ary = 8 + 3, 2 + ary = 8 y + 6. 2 = 2 + 5, a:y - y 2 = 2 y + 2. xy
a:
893.
=
--.
887.
M1 891
4 ?y
2
8f.
+ ary + 2 = 37. - xy + y 2 = 19. - 18. -f ?/
a:
2
x*-xy- 35, a# f + xy = 126, y 2 4- sy = 198. 2 + 3 y 2 = 43, 2 + xy = 28. 2 + 2 f = 17, 3 x 2 - 5 xy + 4 f = 13. 2 2 = 16 y, y(a:2 + y 2 ) = 25 x. ar(ar + y ) 2 2 2 2 xy - y = 2 ay + a a# = 2 aa: + 6
897. z 2
898. 899.
900. 901.
a;
a;
a:
a:
,
.
x
3?
REVIEW EXERCISE 902. xa
-f
y
903. x 4
+
a;
904. 905. 906.
907. 908.
909.
2
=
299
+ 2 a:y + 2 = = x* 9. -f + 243, y # y ary 2 2 -f -f y = 84, Vary + y = 6. - y) (a? - y) = 33. (0 x + y}(x + y) = 273, (!) * Vx -f 10 -f v^+T4 = 12, * + y = 444. x + y 2 = aar, y x 2 = by. + y = 9, ^ 2 - #y + 2 = 27. 23 x 2 - y2 = 22, 7 y - 23 = 200. a:
2
x
2
-f
2 y
xy 2
1,
2 x 2y
xy
z
4
or
ar//
-f-
a;
?/
^:
910.
^-f!i^2, ny
911.
L/ay =
-
+
ma:
=
5?
ft-
*
-
a
m*.
+ g. g=^ a o
+
o-
j
2
2
+ a:y = 2, 3 y 2 + xy = 1. 2 913. + 2 ary = 39, xy + 2 y 2 = 65. 2 5 xy = 11, y 2 + 3 ary = 2. 914. 2 915. x -f 2 a:y = 32, 2 y 2 -f ay/ = 16. 2 2 2 916. x 2 ry + y = 3, # + xy + y = 7. 2 - 3(* + y) = 6. 917. (* 3)2 = 34, *y 3) -f (y 918. (3 x - y) (3 y - x) = 21, 3 :r(3 - 2 y) = 49 2 919. (a; + 2 y) (2 + ?/) = 20, 4 (a; + ?/) - 16 y 3 8 = 920. (o; + y)(a; 3 -|-y 8 ) =1216, y 49(x y). 2 2 = 6 2 (x 2 + y 2). 921. a;y = a(ar + y), y 3 2 2 922. + y 8 = 189. y + a:y = 180, 923. 9 -f 8 y -f 7 ay/ = 0, 7 + 4 y -f 6 ary = 0. 2 924. + ary = a*, y 2 + xy = b 2 925. xy + x= 15, ary y = 8. 2 * 2 ~ g.V + y 2 = + xy + y a = (a? - y)^ 03 926 12 +y +y 927. 2 + y = 2 a 4- 6, ^ 2 + 2 a:y = a a
912.
a:
2
3
a:
a:
a;
.r
a;
a:
a;
or
or
a:
a?
a:
.
a:
*
a:
a:
ar
928. y
a:
929. yz
=
24,
zx
xy 12,
a:y
=
8.
.
y
2
=
0.
ELEMENTS OF ALGEBRA
300
(* + y)(y +*)= 50, (y + *) = - 102, *(* + #) =24. 152, y(x + y + 2) = 133, z(* + y + 2) = 76. + z) =108, (3 + *)(ar + y + z) = 96,
(*+s)(* + y)=10,
930.
+ z)=18, = ar(a? -f y + 2)
931. s(y 932.
933. (y
+ a)(* + y
y(
934. The difference of two numbers cubes is 513. Find the numbers.
The
935.
their cubes
936. 2240.
is
two numbers Find the numbers.
difference of
270.
The sum of two numbers Find the numbers.
is 20,
3
is
;
is 3,
the difference of their
and the
difference of
and the sum of their cubes
is
A
certain rectangle contains 300 square feet; a second rec8 feet shorter, and 10 feet broader, and also contains 300 square feet. Find the length and breadth of the first rectangle.
937.
tangle
is
The sum
938.
sum
of the perimeters of
two squares is 23 feet, and the Find the sides of the
of the areas of the squares is 16^f feet.
squares.
34
939. The perimeter of a rectangle is 92 Find the area of the rectangle. feet.
feet,
and
its
diagonal
is
940. A plantation in rows consists of 10,000 trees. Tf there had been 20 less rows, there would have been 25 more trees in a row. How many rows are there?
The sum
941. the
sum
of the perimeters of
two squares equals 140 feet; Find the side of each
of their areas equals 617 square feet.
square.
942. The
sum
and the sum of
of the circumferences of
two
their areas 78$- square inches.
circles is
Assuming
IT
44 inches,
= -y,
find
the radii of the two circles.
943. The diagonal of a rectangle equals 17 feet. If each side was increased by 2 feet, the area of the new rectangle would equal 170 square feet. Find the sides of the rectangle. 944. A and B run a race round a two-mile course. In the first heat B reaches the winning post 2 minutes before A. In the second heat
A
increases his speed 2 miles per hour, and B diminishes his as arrives at the winning post 2 minutes before B.
much and A then ;
Find at what
rate each
man
ran in the
first
heat.
REVIEW EXERCISE
301
945. The area of a certain rectangle is 2400 square feet; if its length is decreased 10 feet and its breadth increased 10 feet, its area will be increased 100 square feet. Find its length and breadth. 946. The area of a certain rectangle is equal to the area of a square side is 3 inches longer than one of the sides of the rectangle. If the breadth of the rectangle be decreased by 1 inch and its
whose
length increased by 2 inches, the area lengths of the sides of the rectangle.
947. The diagonal of a rectangular is 476 yards. What is its area?
is
field is
Find the
unaltered.
182 yards, and
its
perim-
eter
948. A certain number exceeds the product of its two digits by 52 and exceeds twice the sum of its digits by 53. Find the number.
949. Find two numbers each of which
is
the square of the other.
950. A number consists of three digits whose sum is 14; the square of the middle digit is equal to the product of the extreme digits, and if 594 be added to the number, the digits are reversed.
Find the number. 951. Two men can perform a piece of work in a certain time one takes 4 days longer, and the other 9 days longer to perform the work than if both worked together. Find in what time both will do it. ;
952. The square described on the hypotenuse of a right triangle is 180 square inches, the difference in the lengths of the legs of the Find the legs of the triangle. triangle is 6. 953. The sum of the contents of two cubic blocks
sum
of the heights of the blocks is 11 feet. each block.
the
954.
Two
travelers,
the same time
and
A
B
starts
;
A
from
B
A and
407 cubic feet;
B, set out from two places,
from
P and
Q, at
P
with the design to pass through Q, and travels in the same direction as A. When
starts
Q
is
Find an edge of
was found that they had together traveled 80 had passed through Q 4 hours before, and that B, at Find the his rate of traveling, was 9 hours' journey distant from P. distance between P and Q. overtook
miles, that
it
A
955. A rectangular lawn whose length is 30 yards and breadth 20 yards is surrounded by a path of uniform width. Find the width of the path if its area is 216 square yards.
ELEMENTS OF ALGEBRA
302 956.
Sum
to 32 terras, 4,
957. Sura to 24 terms, 958.
Sum
959. Find an A.
....
-
2.
,
^ V-
5,
,
such that the sum of the
P.
sum
\
3
to 20 terms,
fourth of the
J,
',
terms
first five
is
one
term being
1
of the following five terms, the first
unity.
Find the sums of the
series
:
1G
+
24
4-
32
4
,
961. 16
-f
21
-f
36
4-
.-.,
962. 36
+
24
1G
4
...,to infinity.
960.
+ -
963.
(iiven a
964.
How many
d
10,
=
terms
to 7
;
to 7 terms;
-
4, s
Find
88.
terms of the series
n.
1
+
3
+
5
+
amount
to
123,454,321?
965.
966.
Sum Sum
to n terms, 1 to n terms,
sum
967. Find the
+
3
5
of
j
4- 1 4-
f
968.
Sum
to infinity,
I
969.
Sum
to infinity,
-^-1 +
971. 972.
Sum Sum Sum
+
f
-
V-j
970.
+
7
12434+
-.
to infinity.
,
*" -j$V
1
1 2
- 4-142
....
- V2
+ O 2 4 y 2 ) + O 8 + y*) + y) + x-(x 2 4 y 2} 4- x*(x 3 -f 8) + y) + (2x + f) + (3 x + y 8 ) 4-
to 10 terms, (x 4-
.
;>/)
to n terms, x(x to 8 terms, (x
.v
973. Evaluate (a) .141414.-.; 974. Find n
^
-f
1-
,
n
(ft)
.3151515....
to n terms, the
terms being in A. P.
975. Find the difference between the sums of the series
5 n
?
and
" 4-
The 10th and
term and the
977.
sum
4.
n
n+l(n + l)
976. first
+ !Lni
The
!Ll^ +
...
n
+ T (
+
(to 2 n terms),
+-
(to V
J' infinity).
!)
18th terms of an A. P. are 29 and 53.
common
9th and llth terms of an A. P. are 1 and
of 20 terms.
Find the
difference. 5.
Find the
REVIEW EXERCISE
303
978. Insert 22 arithmetic means between 8 and 54.
979. Insert 8 arithmetic means between
980.
How many
981. The sum
sum
982. The
terms of 18
+
of n terms of 7
+
17
+
10
and
1
+
-,
+ 11+
9
of n terms of an A. P.
0.
amount ,
is
+ lY "(L V;3 '
is
term.
to 105?
Find
40,
n.
Find the 8th
983. The 21st term of an A. P. is 225, and the sum of the first nine terms is equal to the square of the sum of the first two. Find the first
common
term, and the
difference.
984. Find four numbers in A. P. such that the product of the and fourth may be 55, and of the second and third 03. 985. Find the value of the infinite product 4
986. all
2 of
A
perfect number
by which
integers + 2 1 + 2'2
divisible.
is
2 n is prime, then this
the series
the
If
sum
sum multiplied by
a perfect number.
is
v^5
....
a number which equals the sum
is
it
v7-!
v'i
first
(Euclid.)
of
of
the series
the last term
Find four perfect
numbers. 987. The Arabian Araphad reports that chess was invented by amusement of an Indian rajah, named Sheran, who rewarded the inventor by promising to place 1 grain of wheat on Sessa for the
the 1st square of a chess-board, 2 grains on the 2d, 4 grains on the 3d, and so on, doubling the number for each successive square on the board.
Find the number of grains which Sessa should have received. Find the sum of the series
11 --- - -
988.
V2 989. 5
+
v/2
1.1
992.
What 2 a
-
4
1
+
+
.2
+
.04
2.01
+
3.001
+
1
+
990.
+
:
9
+
-
-
..-,
+
+
.-.,
to oo.
3>/2 to oo
4.001
.
+
.,
to
n terms.
value must a have so that the
+
av/2
+
a
+
+ V2
,
sum
to infinity
of
may
be 8?
ELEMENTS OF ALGEBRA
304
993. Insert 3 geometric means between 2 and 162.
994. Insert 4 geometric means between 243 and 32. 995. The
fifth
term of a G. P.
is 4,
and the
fifth
term
is
8 times
the second ; find the series.
512
996. The sum and product of three numbers in G. P. are 28 and find the numbers. ;
997.
The sum and sum
45 and 765
;
find the
of squares of four
numbers
in
G. P. are
numbers.
998. If a, ft, c, are unequal, prove that they cannot be in A. P. and G. P. at the same time.
999. In a circle whose radius is 1 a square is inscribed, in this square a circle, in this circle a square, and so forth to infinity. Find (a) the sum of all circumferences, (I) the sum of the perimeters of all
squares.
1000. The side of an equilateral triangle equals 2. The sides of a second equilateral triangle equal the altitudes of the first, the sides of a third triangle equal the altitudes of the second, and so forth to Find (a) the sum of all perimeters, (6) the sum of the infinity. areas of all triangles.
1001. Each stroke of the piston of an air of air
is
pump removes
J
of the
What
fractions of the original amount contained in the receiver, (a) after 5 strokes, (6) after n
air contained in the receiver.
strokes?
1002. Under the conditions of the preceding example, after how strokes would the density of the air be xJn ^ ^ ne original
many
density ?
ABC
a circle is inscribed. 1003. In an equilateral triangle second circle touches the first circle and the sides AB and AC. third circle touches the second circle and the to infinity. inches.
What
is
the
sum
same
sides,
and
of the areas of all circles,
if
A A
so forth
AB =
n
1004. Two travelers start on the same road. One of them travels uniformly 10 miles a day. The other travels 8 miles the first day and After how increases this pace by \ mile a day each succeeding day. many days will the latter overtake the former?
REVIEW EXEHCISE 1005. Write down the (a
-
first
three and the last three terms of
*)".
-
1006. Write down the expansion of (3 1007. Expand
(1-2 #)
1008. Write down (x
+
305
2
7
2
5.
2
a;
)
.
the
first
terms
four
in
the
expansion
#).
1009. Find the 9th term of (2 al 1010. Find the middle term of
(
- o/) 14 - l) w
.
.
1011. Find the middle term of (a$
bfy.
-f
1012. Find the two middle terms of
(
- ft) 19
1013. Find the two middle terms of
(
+
1014. Find the
fifth
term of (1
-
x)
.
18
.
9 .
a:)
+
1015. Find the middle term of (a
lQ
b)
.
1016. Find the two middle terms of (a
x)
9 .
(1 *2
1018. Find the
coefficient
a:
\88 1
in
1019. Find the middle term of 1020. Write down the
X ----
1
5a
coefficient of
-
.
-7
x 9 in (5 a 8
1021. Find the eleventh term of /4 x >>
6
|V
- -i-V 2i/
5 .
a:
8 7.
)
of
INDEX [NUMBERS REFER TO PAGES.] 148
Abscissa Absolute term
"
178
.
4
value
....
Addition
Aggregation, signs of Algebraic expression
"
15, 19, 97, .
.
.
9,
....
sum
progression
.
.
.Base of a power Binomial " theorem
Consistent equations
....
130
Constant
155
10
Coordinates
148
18
Cross product
41
Difference
.
246
Discriminant
.
20
Discussion of problems
10
Dividend
45
54
Division
45
Divisor
45
-10 255
Bracket
9
.
.
.
Checks
.
232
23 193 .
.
.241
.
Clearing equations of fractions
'
**
.193 .
difference
....
multiple, lowest
.
.
Completing the square
.
.
...
8
.
fractional
.
.
.
graphic
representa-
....
160
linear
54, 129
literal
54, 112
"
91
" '*
"
"
123
Evolution
807
.130 .108
'
246
105
...
.
graphic solution, 158, 160 in quadratic form 191 .
.181
fraction
.
11
108
251
ratio
63 consistent
tion of
8
Coefficient
130
Equations
20, 37, 49
Composition
.
Elimination
" Character of roots
.
8
9
Complex
120
Consequent
27
Brace
*
53
.210
Degree of an equation
Axiom
**
.
120
mean
Arrangement of expressions Average
Common
t
249
Antecedent *'
.
123
Alternation
Arithmetic
210
....
Conditional equations Conjugate surds . . .
.... .... numerical .... quadratic
.
simple simultaneous
54
.178
.
.
.
129, 232
54
........
169
INDEX
808
8
Exponent Exponents, law of Extraneous roots
34, 45, 195
.
.
65, 184 .
70
...... 227
81)
Factoring
70,
Fourth proportional Fractional equations
u
17
Linear equation
222 Literal equations . 120 Lowest common multiple
Extreme Factor " theorem " II. C
Like terms
227
Mathematical induction
.
.
.
.
.
exponent
Fractions.
.
.
.
proportional Mean, arithmetic
"
Multiplication
.
91
346 120 338 341
geometric
120 Member, first and second .108 Minuend .105 Monomials 03 Multiple, L.C
Geometric progression .251 Graphic solution of simultane.
.
.
Mean
.... .
54, 112
.
.
.31,
.
.
53
23 10
91 102, 212
.
ous equations
100
Negative exponents 11
numbers
.
.
195
.
4
Graphic solution of simple equa158
tions
Graph of a function Grouping terms Highest
common
.
.
.
.154
.
.
.
.
factor
.
53
Identities
Imaginary numbers
Inconsistent equations
Independent equations Index
.
P
130
Power
253
Insertion of parentheses
.
. . Integral expression Interpretation of solutions
.
.
.
.
.
19
42 7
Prime factors
76
63, 114, 143, 180, 189, 243
28
Problem, Product
70
Progressions, arithmetic
241
27
83 10
Polynomial
102 9
.
9,
......
Polynomials, addition of " square of
109
.
.
148 148
89 235 Parenthesis Perfect square
.
13
205
86 Ordinate Origin
Homogeneous equations
Infinite, G.
Order of operations " of surds
1,
'*
7
geometric
.
.
.
.
.
.
246 251 121
Inversion
123
Proportion
Involution
105
Proportional, directly, inversely 122
Irrational
numbers
Known numbers
Law
of exponents
Laws of
signs
.
.
.
.
.
205
.
.
.
1
84, 45, 195 33, 45
Quadratic equations Quotient Radical equations Radicals
....
178
45 221 205
INDEX Ratio national
76,
Rationalizing denominators
Reciprocal
Roots of an equation " character of "
....
171
Substitution
133
104
Sum, algebraic
228
Surds
22
......
.
.
54 193
Theorem, binomial
.
.
193
Third proportional
9
of
33, 45
Term "
23 18
205
27
Rule of signs
10
absolute
178
....
255 120
Transposition
54
Trinomial
10
240
Series
...
Signs of aggregation Similar and dissimilar terms
9, .
Simple equations Simultaneous equations Square of binomial of
27
Unknown numbers
.
129, 232
.... polynomial ....
....
1
17
205 Value, absolute 54 Variable
Similar surds
"
Square root
205
...
.
sum and product
120
215 Subtraction 169 Subtrahend
.
Real numbers
Remainder theorem Removal of parenthesis Root
309
Vinculum
4 155 9
40 42
Zero exponent
Printed in the United States of America.
197
ANSWERS TO
SCHULTZE'S ELEMENTS OF ALGEBRA
COMPILED BY THE AUTHOR WITH THE ASSISTANCK OP
WILLIAM
P.
STrtn
MANGUSE
gork
THE MACMILLAN COMPANY 1918 All rights reserved
COPYRIGHT,
BY
1910,
THE MACMILLAN COMPANY.
Set up and electrotypcd.
Published September, 1910.
Reprinted April, 1913; December, 1916; August, 1917.
NorfoooS
Berwick
J. 8. Gushing Co.
ANSWERS Page
2.
1.
$160.
2.
32,8.
America 46,000,000, Australia 8 in.
9.
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1.
3.
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11.
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4.
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13.
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16.
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4. ft.,
expenditures.
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below
sign.
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7.
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Seattle 12
5.
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in.
16f
in.,
7.
-
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.$9400,
8. 16 in., 16 in., 7. 48 ft., 8 ft. 6. A $90, B $ 128, C $ 16. 10. 150,000,000 negroes, 15,000,000 Indians. 18, 18, 144.
ft.
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ANSWERS
ii
11.
Page 237.
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14.
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04.
31.
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35.
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sq.
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31.
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Page 21. 3 w"
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20.
27. 6
-2. -15. 17. -31. 5.
11.
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17. 14
20
4. 2
a'
2
-
aft
2
4 x.
12.
8
16.
.
+
11.
'.
25.
.
-G.
3.
2 12. Page 48. 7a 2 ftc 4 -4c4-2a.
-
19.
14 r 2
8. 3.
3
4.
12.
10.
^r
2
-13.
3.
2 ?/'
2
22.
!>.
16.
a-.
24.
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-
2 ?/ 4- 3 ry. 5 aft 4 ft 2 4
3
15,
- 5 mp.
18.
23.
4.
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2.
5.
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8.
12.
:r
19. 2
//.
- 25.
2
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r//- 8
4- 2 1
//
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2
2
i
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2
20.
7.
4 pq.
10.
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17.
7.
9.
- 34. 14. - 2 2 - + 77 15.11 _ 5x _ _ o 18. 4- 27 x 2 4- 15 21. -4xy + 13
13.
10.
8 ?/
,
ll'a^-S
ft.
8 ?/
,
2
*3
4-
ft
3
-y
15.
,
4 ,
a2 x 8 4-
ft
8 .
1/*.
a 8 4- 10 xy*,
ANSWERS 56.
Page
1.
10.
2.
9.
2.
6.
5.
3.
!).
11.
1.
4.
12.
11.
vil
8.
7.
5.
6.
5.
13.
4.
6.
2.
7.
14.
2.
8.
4.
15.
22.
2.
22.
10.
%
16.
00.
23.
5.
17.
l.
38.
1&.
-
6.
25.
7.
26.
3.
-
12.
33.
}f.
32.
39.
58.
Page
40.
I.
-
a
1.
10
7.
19.
20.
I.
a -10.
10,
17.
7x 2
-
]
42.
.
2
ri
x.
a
3.
2b- -.
x
59.
x
+
18.
9.
b,a-b
2b -
a.
22.
y
y yr. 10 yr.
26.
y 100 a
29.
xy
2y
+
+
10 b
60.
-f
8 n
(I,
-
5.
-
6
,100-. 3x2.
16.
+
-
b
-
^
34.
12 sq.
ct.
sq.
ft.
a-//
#
41.
10
46.
+
'^
35.
"
39.
lir.
48.
Page 61. c.
y
3
-f
6)(o
-
/>)
2.
+ 1=a.
-
6.
2
+ 10 = c. 4-^ = 100.
2z
3
90
= all-. )
9.
2^ -
4(a
=a-
20
lOx
50.
3.
7.
fix.
y
m -+~m
49.
100
2=10.
1.
x
b.
44.
-f
x .
25
r
43.
x
47.
a.
+
ct.
.r
"mi.
38.
?
42.
20yr.
2x
31.
ft.
ct.
12 yr.,
10 x
x
rn mi.
37.
+
y
ct.,
28.
ct.
3y
;r-1,rr-2.
+
100 d
25.
4x
ct.
/
23.
4
iL*.
(a
2\.
20.
n M.
d.
20.
2.
(>.
2
33.
+
!,
100 a
30. 'nj
r>?imi.
36.
tx mi.
p=
-
44.
11.
-
10 yr.
-f
27.
60
8.
?>i
sq. ft.
(>
+
19.
c ct.
10 act.
32.
Page
4.
15.
s.
X
45.
20.
m+
4.
lO.p+7.
+
d
14.
6 yr., y
$
24.
+ 3x + 2y +
ft,
40.
37.
(
>-_&.
13.
4.
-f-
30.
0.
43.
0.
-13.
1$.
36.
<>.
.
Page 21.
.
-
29.
6.
35.
5.
41.
2.
28.
4.
34.
1.
10.
8.
27.
-Jj12.
21
7.
2.
57.
Page 31.
7.
18.
1.
24.
10
-f
c
=
x
=
-ft)
=
+
w.
2 x. 9.
eZ
7.
7
Page 13.
62.
10.
a
-
100= -^- x700.
9
14.
=
17
50=
100 (6)
-a.
L
= -5 -
m= --
15.
100
=
+
+
=
+
m.
16. (a)'
2x=2(3x~10), v
10)
3 x -f (4 x - 10) 100. 17. (2 a; 10) (6) 2 z-f 20 3^-740, (c) (2z-f 600) (3 (d) 2 x
12.
=4, (c) 2x- 6 = *, (d) 2a + 10 = n, (c) 2a? + 3 (/) (2fl5-8)-h(8ar-ia)=60, (
rrax-lO, (A)
x 460.
?i
100
2x- (3x-
x
11.
(3
sc
+
700)
=
(x
-f
1200)
-
200,
(a) 2 x
- (3x -
(e)
3x -
700)
=
5,
1700) = 12,000, 800 = x + 1300.
x- 200) -f(^ +
ANSWERS
vili
5
J^. = _?_(2ar + 1), ~=90, (6)' --(6 a -30)^ =20, (c) v ; MOO HXT 100 100 -^-~ -(5z-30) =900, v(e) -i* + -A- v(5z - 30) + (2s + 1)'
18.
(a) V
;
^
'
k
'
_
ft
(d)J v
100
?
w (/)
=^8000,
Page
V
100 '
-
2.
30 yr. 14. 30 mi.
7.
250.
15.
Page
67.
1.
55,11.
Page
68.
4.
12,2.
480
Ib.
12.
4pt., 5pt.
9.
Page
70.
Page
71. in.,
3.
12,8,24.
11 in.
Page
14.
Page
74.
1.
4.
600,
1200.
8.
4.
9.
Page
75.
05,5.
80 A.
18.
9.
6.
7.
1250.
12.
19.
24J.
150,000.
30,0.
2. 4.
45
14.
15
in.,
15.
in.
7 hr.
6, 12, 14. 5.
1,3,5.
3,0,16.
6.
20,21,22.
1,000,000 Phil., 2,000,000 Berlin, 4,000,000 N. Y. 10. 21. 11. 20 yr., 10 yr., 25 yr. 12. 6, 7, 8. 8.
90,000,000 gold, 72.
3.
5.
11.
2$.
$40.
17.
42yr., 28yr.
5, 10, 25.
30, 50, 100.
13.
9
300.
2.
18.
4.
10.
6. 52,13. 8. 160 lb., 7. 8,10. 78,79. 10. 40 yr., 10 yr. 11. 29,000 ft., 20,000 ft.
13.
9.
in.,
25.
3.
90 mi.
16.
1.
8
100
5.
13,7.
7.
15.
9.
8.
Pace 65.
100
100
.
10 13.
1.
20 yr. 13. 85 ft.
= SJL+J-
100
64.
'
'
2
480,000,000 pig iron.
180,000,000 copper,
5 Col., 10 Cal., 10 Mass.
15 yd., 20 yd.
200, 3 hr., 15 mi. 10.
12 mi.
11.
by 12 yd. 70^,210^.
10 yd.
2.
1200.
5.
6.
12.
5$ hr.
7.
3.
200.
5 lb.,
1 lb.
82 mi.
3. 6 aty (3 + 4 6) 2. 3x (3r.-2). Oaj(o6-2cd). 2 2 2 5. 11 w(w' + wi - 1). 6. z?/(4^ + 5xy - 6). 7a*fe(2a & -l). 17z8 (l-3z + 2x-'). 8. 8(a6 2 +6c2 -c2 a2 ). 9.
Page 4. 7.
10. 13. 15.
17.
78.
a
1.
a a (a 8 -a+l).
.
10aVy(2a 2 -ay4-3y 2 ).
11.
12.
14. 17 7>c(2 a'^c2 - 3aftc + 4). (2a6-3?2_4 a /^) 16. 13 a 8 4 * 5 (5-3 xyz + x y'W). 11 pV (2 p8 - 5p + 7 g ). 8 2 19. 3 (a +&)(*- y"). 18. (m + n)(a + 6). ?(g -? -g+ 1).
6rt 2
'2
>
2
8
2
?/
21. 13-13. 23. 2 3 6 7. 22. 2.3.4-11. (p + 7)(3a-5&). Page 79. 1. (a -4) (a- 3). 2. (a + 4)(a + 8). 3. (ro-3)(w--2). 5. 6. 4. (z-5)(z-2). (a-5)(a-4). (a + 6) (a + 3). 20.
.
PageSO. 10.
(y
13.
(y
7.
(*-4)( +
+ 8)(y-2). + 7)(y-3).
11. 14.
8. ( + 4)(*-2). (y-ll)(y-4).
9.
2).
(a
+
5)(a
+ 6).
12.
15.
-
(y-8)(y + 2). (y-7)(y + 2).
ANSWERS
ix
18. (az + 9)(ox-2). + 8)(g-3). - 11 6) (a 4- 2 ft). 21. (a2 + 10) (a2 -2). 23. (w + 20)(w + 5). 24. (y + 4)(y-l). 22. 26. (n2 + 12)(n 2 + 5). 25. (a -6 6) (a 4- 4 6). 27. (a 3 + 10)(a- 3). 29. x (z + 2)(x + 3). 30. 100(x- 3)(z-2). 28. (a 4 -10) (a 4 + 3). 32. y(x- 7) (a; + 3). 33. a 2 (w-7)(w + 3). 31. Oa 2 (a-2)(a-l). 35. 200 (x + l)(x + 1). 36. 4 (a - 11 ft)(a-6). 34. 10x2 (y-9)(y + 2). Page 82. 1. (2x-l)(x + f>). 2. (4a-l)(a-2). 3. (3*-2)(.r-2). - 1). 5. 6. 4. (3 n + 4) (2 (3x+l)(x + 4). (5w-l)(m-5). 8. 9.* (2 y + 3)(y- 1). 7. 3(x + 2)(z-l). (4y-3)(3y + 2). 10. (2 *+!)(* -9). 11. (5 a -2) (2 a -3). 12. (9y-4)(y + 4). 15. (4 13. (2w+l)(ro + 3). 14. (5x - 7)(2z -f 1). -3)(3a; - 2). 18. (7 a + 4) (2 a - 1). 16. (6n + l)(+2). 17. (2y-l)(y + 9).
(p-8)0> + l). -7 6) (a -10 6). (ay-8)(ay-3).
16. 19.
17.
(<7
20. (a
(a
a:
19.
(3#-y)(+4y).
22.
(:5-2y)(2a!-3y).
20.
28.
a(2u; + 3)(-c4-4). 100 (a; -y) 2 29.
31.
10(3
25.
26.
-5 6) 2
.
(4a;-5y)(3a; + 2y).
x (5 a;
+ 4) (a;
-f 2).
a*(5a -f l)(flr - 2). 2 -y' (2y-3)(2y-l).
.
32.
(5a-4ft)(2 a~3 ft). 24. 2(2s + 3)(a: + 2).
21.
(15z-2y)(x-5y). 23.
a; -y) (a;- 2 y). 10 y2 (\) x + l)(x~ 3). 33. 10 a 2 (4 - w*)(l -2 n 2 ).
27. 10(2
30.
2(9a:-8y)(8a:-0y). 35. (2 a? 4- 3 y 2 )(2 a: 2 -f y'2 )2 3 Yes, (g - 6) 2 1. 2 No 4. No, Page 83. Yes, (m + w) 2 5. Yes, (a- 2 by2 6. No. 7. Yes, (m-7n) (a; -8) (a; -2).
34.
-
-
-
-
.
.
No.
8.
No.
9.
(3a-26).
13.
Page 84. (15a-y-2)
10.
Yes,
.
2
2
21. 9ft w(?-3) 140 w 2 27. 9. 1. (* + y)(z-y). 4. (2o + l)(2-l). 7. (10a + ft)(10a-ft). .
28.
25. 2. 5.
2
Yes, (6 a; Yes, x\x 24. 24 9.
19.
.
23.
29.
(a
14.
Yes, No.
16.
.
10.
22.
.
.
26.
(y-8) 2
11.
.
.
a-
3 by2 Yes, (4 18. Yes, 10(a - 6) 2
15.
2
.
+ 3?i) 2 (5x-2y) 2
Yes, (w*
9.
30.
40
+
5)
.
a.
- y) 2
.
Yes,
17.
Yes,
20.
25.
aft.
12.
Yes, 216 aft.
x.
+ 8)(a-3).
3.
(l+7a)(l-7a).
(0
6.
(0
+ 6)(6-6). + 0(9-0-
23.
9. (3a;+4 y)(3x-4 y). 8. (ft + ll)(aft-ll). - 8). 11. + 9^)(oxy - 9*). (7 ay + 8) (7 ay 2 2 13. (10 aft + c 2 (10 aft - c 2 ). (5a +l)(5a -l). 2 2 15. (15a + 46*)(16a-46). (13a +10)(13a -10). 2 17. (a*& + 9) (aft + 3) (aft -3). + 2 )(a + ft)(a-ft). ( 2 4 19. 10(a + ft)(a-ft). (l + x )(l + x )(l + x)(l-x). 22. (x + y4 )(x - y4 ). 21. x(x +y)(x -y). 13x(a + ft)(a-ft). 24. 2 y(ll x 2 + 1)(11 x2 - 1). 3a;y (6x + 4)(5x-4).
25.
B 2 (12+ y 2 )(12-y 2 ).
Page 85. 12.
(5xy
)
14.
3
16.
ft
18. 20.
8
26.
(m + n +p)(m + w-p). (m + n + 4p)(w + - 4p).
1.
3.
10.
13x7.
27. 2.
.4.
103x97.
(w ~ n (x
+
y
+
ANSWERS
X
7.
-r)(4x (4x 4- y (m + 2 u + (\p)(m +
9.
(2
5.
-\-
11.
~
a
b 4- 5
f>
-
<:
14.
9
b) (r
Or 4-1) (^4-2).
2 2 (3a -4// )(x4->/).
a
-^)(^-
w
(3
7.
3.
w
-
(a
7.
2
(a
4-
/>; (->
-
5 & 4-#
Page 1(V/
88. 2
m
n
-
2 y) (a 1.
10.
(
-
x(x
14.
10(2
16.
(2x-7)(x 2 -2).
18.
?i(w 4-y)
21.
.r(3x'
-
5
fid).
lj
fc)(
2
8.
(w'
?>
4.
T>).
6.
(
+ 2 //). - 4). K + l) a (a -
2.
ah}.
(m -
3
4-
8.
31.
(
+
1
4-
4-
(2
)
,)
4- >*-)(:> 4-
2
I)'
2
(
-{-
&).
y )(.-?/). 4- 2).
8
-
15 ?>)(a
15
-
(
,))(x
(f> a/>
-
(3
-
x
1).
-2) (m 4-1). a 2 (a-9).
ft)
7>)(3
w 2 )(l
-3
(a
+
-
a
?>).
l
l.'J)('
2
23.
25.
(Ox
fo).
(a
-
10(8x' 4-l) a
4- l)(
4- 1).
x-
7 ?/)(7
(5
y/).
+ .'})(c - 4).
30. 13(
n 4-3*).
- 0+ 12).
^OC 1
3
(<
4).
4- 8).
- 7s) (2
2(5 n
+
4-
20.
1).
27.
32.
-
).
-(7rt-3)(7a~3).
4 (14- w )(l
1).
?>).
3
y).
7.
.
2(5 a
(3 4- a
15.
3)
4-
?/
(^
(7/1
2
11.
29. (5 al)
4- 1) 3.
13>
17.
m - 9).
x
(x-f!/)'
- y).
r)(x-
a-
w)(m- 3- n).
4-
5).
19-
(14.^4-6)
c)((>
4- .'/)('< 4- &
.^
4-2).
6.
.
2.
q).
{I
(x- 4-
8(w
-&)
a 2_rt4-l)(a -rt-
28. (5
2 6).
- y)(fi a - 36).
r4-y-3
(<>
4- ^ 4-
5 />-z
-f y)(jr
y)-
(
-
2
22. 3(4-7>4-4)(^4-'> -22/)((3-x). 24. a(a 2 + !)(+ !)( - !) (x4-2)(x-2). ( rt
(w *
H. (c- 7)(^
2
26.
c).
(5a+l)(9-a).
(2
-
4- ?>-)
O + ?/4-
1.
9.
2 2 4-l)(a' &
62
o-
(wi 4- 4)(?
4-& 2 )(tt4-/>)('e 2
13.
+
a (a
10.
.
2.
(a
+ o) (ff
/>
k
12.
6 -f
r)(5a
fo
y).
(r4-20(4 10.
2
2
+ 9 iZ)
c
x(x4-ti<0.
(.i-4-l)(x4-l)(x~l).
5 (6a 4-l)(a +)2( 2x-2/)(x-2?/).
8.
5
5 b
+
(f>
(m3n + a + b)(m 3n-ab).
?>).
Exercise 47. 4.
-
Exercise 46.
Page 87. (!__/>). 5.
8.
4-
4.
6.
(
Gp).
j).
9.
12.
(2 a
Page 86. 1. (a + (2a-3fc)0*+ tf)-
3.
-
n
'2
12.
y(2x-?/). (5^-4- //)(5y-
6.
2).
.y
33.
&4- 8)
4-
(a
2
35. 2 (a 4- 8) ( (16- 4- 2 y). -8). 36. (x//,?-50)(xt/z- 1). 37. 17(x4-3//)(x-2y). 38. (* _ 2 )(a 4- &). 2 2 3 39. fi(c4-26). 40. 3p (^-9)(j) -4). 41. 3(.e 4- 3)(x 4- 2)(x 2). 1. 2 a 2 13x 3 y. 4. 12. 2 5 a 2 6c 2 3. 5. 450. Page 89.
(^
34.
?>_8).
(16
4-
^
6.
a;
.
.
7.
Page 90. (w4-w) 2
12.
1.
a
7.
13.
6.
2 y)
4 a s &8
+ a
7.
.
b.
4- 3.
Page 92. 42a 3 x.
WIM.
13 x 8
8.
2(m4-l)'
2. 5 x8 3.5 m2 x-2. 9. x4-3. 14. y-6. 15. 2 1.
a8
7.
40
.
2. r
x 2 */3
aV>*>c >d\
4
9.
-
14.
.
.
8.
^
2
13.
-
4.
.
10.
x
3 x4
3. 8.
8x.
5.
!)&(<*
,
a 4.
4- &).
a x
11. 16.
M.
4 a8
11.
.
8.
.
- 4.
a 4-1.
.
15
10.
19.
3x(x-?/)
4-
3
- b. + 3.
6.
12.
x
4- y.
a -4.
6.
24x sy s 9.
.
12
5.
m 2 (m
80a6 4-
n)
2 .
ANSWERS 10.
13.
x
11. 6a2&(rt-6). (a-2y2 (a-3) 2 (a-4) 2 14. (a-2)(a + 2)2. + &)(-&) ( .
12.
15.
+ y) (a: -y). + &) 2 ( - 6).
30(3 2 (a
+ l).
18.
2(2a-l)
-
x 22.
?-_!&. a
1.
23.
+8b
+5
a
-4
a
x 24.
^. -
3 /
//
25.
!+*?. b
w 2 !
^
(!L 5
991
20
+2 3 i
3^-1 rr
Pace
1).
21.
m+1
10
1).
+
63
ANSWERS
Xll
ab-
21
^iie^+JoJ^^ilOa
8.
92
9
.
+ qc + ab
bc
10
2
11
'
238 -
.
aft
-
]
ft
1>*
-
5 x2 y
+
121
+ 84 _y~ z
**/*
12
-8 a 2
2
196 a2
y;2g
:j
ftc
46
12
?t
~ 30 y - 50
1/2
- 80 MP 2
30
?/
r
+
t
S
^
3a 2
2ft
~
'
i
15.
"'
'
180 wv
t+3) 5x
+
13
19
+ *2)(x + 3)'
(x
rtv+Ji:'.
'
a
2
ft
a 9
Page 100.
26.
(wi-8)(w 3ffl
go
2
+
21 1
ft'
7
fi
2)
4m m2
'
26
,,
2 7.
_*^p5_^^_. '
28.
29.
2na
^i
-^-~
30.
94 4
9x -T *
22
9 ^
33
^
37 (a
~
2
b)'
+
(a
a
ft)
42.
41.
+
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;
(c) 2 hr.
;
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;
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ANSWERS 22.
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Page
Page 128.
1.
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mi., 174+
2. 3. 13J, 31J. 19.8 oz. copper, 2.2 oz. tin.
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Page 22.
2.4,3.
1. 3, 7.
25.
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2 horses, 6 cows, 10 sheep. 24.
5, 4, 3.
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12;
Page 152. 2. (a) Apr. Feb. 1 (d) Apr. 16, Nov. 6.
(ft)
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Nov. 15;
25.
On 11.
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21.
20, 40, 30.
100, 00, 20. 4 mi./hr.
a parallel to the x axis. parallel to the x axis
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;
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Page 153.
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Page 166.
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XXii
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ANSWERS
__
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ANSWERS
XXIV 2V3.
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ANSWERS
XXX11
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1
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ANSWERS Page 284.
- 2.
509.
-
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ANSWERS
XX XIV 606.
3
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+
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a-b.
623. 7003.
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ANSWMHti Page 292.
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709.
713.
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ANSWERS
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6+V7.
800.
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813.
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2 (x 4-*4- 1)(V
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3,
1,
;
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879.
886. ;
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2,
891.
2,
4
;
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895.
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881. 4, 4
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ANSWERS 6
oJ---
901.
V 41
Page 299. 904. T3.
902.
2, 1
329; 333, 111. 908.
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909.
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906.
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Page 301. 948.
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952.
Page 302. 960.
280.
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2, 8.
2;
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7 or 30.
985.
8.
18,446,744,073,709,651,615.
ANSWERS
xxxviii
^f (2-f-3V2).
988. 992.
993.
2(2 -v/2).
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-
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.
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is distinguished from a large number of American in that its main theme is the development of text-books history the nation. The author's aim is to keep constantly before the
This book
pupil's mind the general movements in American history and their relative value in the development of our nation. All
smaller movements and single events are clearly grouped under these general movements.
An exhaustive system of marginal references, which have been selected with great care and can be found in the average high school library, supply the student with plenty of historical narrative on which to base the general statements and other classifications made in the text. Topics, Studies and Questions at the end of each chapter take the place of the individual teacher's lesson plans. This book is up-to-date not only in its matter and method,
but in being fully illustrated with
many excellent maps,
diagrams,
etc.
photographs, " This volume
is
an excellent example of the newer type of
school histories, which put the main stress upon national development rather than upon military campaigns. Maps, diagrams, and a full index are provided. The book deserves the attention of history teachers/'
Journal of Pedagogy.
THE MACMILLAN COMPANY 64-66 Fifth Avenue,
BOSTON
CHICAGO
New York
ATLANTA
SAN FRANCISCO
