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VIETA
DESCARTES
NEWTON
GAUSS
DURELL'S
INTRODUCTORY ALGEBRA
BT
FLETCHER DURELL, Ph.D.
FEAD OF THE MATHEMATICAL DEPARTMENT IN THE
LAWRENCEVILLE SCHOOL
NEW YORK
CHARLES E. MERRILL COMPANY
7->
n^
DURELL^S MATHEMATICAL SERIES
ARITHMETIC
Two Book Series
Elementary Arithmetic
Teachers' Edition
Advanced Arithmetic
Three Book Series
Book One
Book Two
Book Three
ALGEBRA
Two Book Course
Book One
Book Two
Introductory Algebra
School Algebra
GEOMETRY
Plane Geometry
Solid Geometry
Plane and Soled Geometry
TRIGONOMETRY
Plane Trigonometry and Tables
Plane and Spherical Trigonometry
AND Tables
Plane and Spherical Trigonometry
WITH Surveying and Tables
Logarithmic and Trigonometric
Tables
[6]
Copyright, 1911, 1912,
By CHARLES E. MERRILL CO.
PREFACE
The main object in preparing this new Algebra
has been to simplify principles and give them interest,
by showing more plainly, if possible, than has been
done heretofore, the practical or common-sense reason
for each step or process. For instance, at the outset it
is shown that new symbols are introduced into algebra
not arbitrarily, but because of definite advantages
in representing numbers. Each successive process is
taken up for the sake of the economy or new power
which it gives as compared with previous processes.
This treatment should not only make each prin-
ciple clearer to the pupil, but should give increased
unity to the subject as a whole. We believe also that
this treatment of algebra is better adapted to the
practical American spirit, and gives the study of the
subject a larger educational value.
Among the special features of this Introductory
Algebra, the following may be mentioned:
A large number of written problems are given in the
early part of the book, and these are grouped in types
which correspond in a measure to the groups used in
treating original exercises in the author's Geometry.
Many informational facts are used in the written
problems. The central and permanent numerical
facts in various departments of knowledge have been
collected and tabulated on pages 280-286 for use in
making problems. Similarly the most important
3
4 PREFACE
formulas in arithmetic, geometry, physics, and engi-
neering have been tabulated for use by teacher and
pupil (pp. 278, 279).
The self-activity of the pupil is aroused by examples
which require the pupil to invent and solve problems
of a specified kind, material for such examples being
made available in the tables of formulas and numerical
facts.
Many of the examples in the book require a frequent
review of the principles of arithmetic, as of decimal
fractions and percentage.
Numerous and thorough reviews of the portion of
the Algebra already studied are also called for. A
unique feature is the series of spiral reviews of the
preceding part of the book by means of examples at the
end of Exercises. Oral work is called for in like man-
ner and is also emphasized in special important Exer-
cises.
The utilities in symholism in general, apart from
technical algebra, are brought out in a special Exercise
(pp. 249, 250) and thus the direct practical value of the
study of algebra is much broadened.
The history of algebra is discussed in Chapter XIV,
and questions on this chapter are inserted in appropri-
ate places in the text.
The author wishes to express his indebtedness to
Professor William Betz of the East High School,
Rochester, New York, and to Dr. Henry A. Converse
of the Polytechnic Institute, Baltimore, Maryland,
for important aid in preparing the book. He is in-
debted also to School Science and Mathematics and
the Mathematics Teacher for a few of the problems.
CONTENTS
CHAPTER PAGE
I. Algebraic Symbols 7
II. Negative Numbers 28
III. Addition and Subtraction; the Equation 39
IV. Multiplication 58
V. Division 76
VI. Equations (Continued) 93
VII. Abbreviated Multiplication and Division 109
VIII. Factoring 134
IX. Highest Common Factor and Lowest
. Common Multiple 160
X. Fractions 166
XI. Fractional and Literal Equations ... 196
XII. Simultaneous Equations 223
XIII. Graphs 251
XIV. History of Elementary Algebra .... 266
Material for Examples 278
INTRODUCTORY ALGEBRA
CHAPTER I
ALGEBRAIC SYMBOLS
1. The Use of Letters.
Ex. Walter and Harold made $27 by gardening one sum-
mer. Walter, who was older and stronger, received a
double share of the profits. How much did each receive?
SOLUTION WITHOUT THE AID OF X
1 share = Harold's part of the profits
2 shares = Walter's part of the profits
1 share + 2 shares = $27
3 shares = $27
1 share = $9, Harold's part
2 shares = $18, Walter's part
SOLUTION BY AID OF X
Let X — Harold's part of the profits
Then 2x = Walter's part of the profits
Hence a; + 2x = $27
3a; = $27
X = $9, Harold's part
2x = $18, Walter's part
We see that by use of the letter x the solution is much
shortened.
2. Algebra is that branch of mathematics which treats
of number by the extended use of symbols.
Later algebra comes to have a wider meaning.
Algebra may also be briefly described as generalized arithmetic.
7
8 SCHOOL ALGEBRA
3. Utility of Algebra. A more extended use of symbols
than is practiced in arithmetic (1) shortens the work of solv-
ing problems; (2) enables us to solve problems which we
could not otherwise solve; and (3) gives other advantages
which will become evident as we proceed (see Art. 143 and
Exercise 76, p. 249).
EXERCISE 1
(Problems of Type I, i. e. of the form x + ax = b.)
1. Two boys together catch 84 fish. If the boy who
owns the boat which they use, receives twice as many fish
as the other boy, how many fish does each boy receive?
2. A man left $12,000 to his son and daughter. To his
daughter, who had taken care of him in his old age, he left a
double share. What did each receive?
3. A man and boy by working a garden one summer made
$128.80. If the man received a share of the profits three
times as large as the share received by the boy, how much
did each receive?
4. - Two boys together gathered 1 bu. 4 qt. of hickory nuts.
If the boy who climbed the trees received a double share,
how many quarts did each receive?
5. Make up and work a similar example concerning two
boys who gathered chestnuts.
6. Two girls made $18.60 by sewing. The girl who sup-
plied the thread and machine received twice as much as the
other girl. How much did each make?
7. Make up and work a similar example concerning two
girls who kept a refreshment stand.
ALGEBRAIC SYMBOLS 9
8. Solve Ex. 1 without the use of x (see Art. 1). How
much of the labor of writing out the solution is saved by the
use of xf Is there any other advantage in the use of x in
solving a problem?
9. The total cotton crop of the world in a certain year
was 15,000,000 bales, and the United States in that year
produced three times as much as all the rest of the
world. How many bales of cotton did the United States
produce?
10. A farm is worked on shares. As the tenant supplied
the tools and fertilizers, he received twice as large a share
of the profits as the owner of the farm. If the profits for one
year are $6000, how much does the tenant receive? The
owner?
11. If the sum of the areas of New York and Massachu-
setts is 57,400 sq. mi. approximately j and New York is 6
times as large as Massachusetts, what is the area of each
state?
12. One number is 5 times as large as another and the
sum of the numbers is 240. Find the numbers.
13. One number is twice as large as another and the sum
of the numbers is 7.26. Find the numbers.
14. One fraction is three times as large as another and their
sum is \.' Find the fractions.
15. One number is 4 times as large as another and their
sum is .0045. Find the numbers.
16. Separate $120 into two parts such that one part is
three times as large as the other.
SuG. Let X = the smaller part.
10 SCHOOL ALGEBRA
17. Separate 5^^ into two parts such that one part is 7
times as large as the other.
18. Make up and work an example similar to Ex. 11.
Also one similar to Ex. 15. To Ex. 16.
Material for examples may be obtained from the lists of
Important Numerical Facts given on pp. 514-520.
19. To look well, the middle part of a steeple should be
twice as high as the lowest part, and the top part 8 times as
high as the lowest part. If a steeple is to be 132 ft. high,
how high should each part be?
20. A man wants to save $6000 in three years. If he is
to save twice as much the second year as the first, and three
times as much the third year as the first, how much must
he save each year?
21. A girl has $42 to spend for a hat, coat, and suit. She
wants to spend twice as much for her coat as for her hat,
and three times as much for her suit as for her hat. How
much does she spend for each?
22. A man bequeathed $84,000 to his niece, daughter, and
wife. If the daughter received twice as much as the niece,
and the wife four times as much as the niece, how much did
each receive?
23. A certain kind of concrete contains twice as much sand
as cement and 5 times as much gravel as cement. How many
cubic feet of each of these materials are there in 1000 cu. yd.
of concrete?
24. Make up and work a similar example for yourself
where the materials in the concrete are as 1, 2, 4.
25. In a certain kind of fertilizer the weight of the nitrate
of soda equals that of the ground bone, and the weight of
I
ALGEBRAIC SYMBOLS 11
the potash is twice as great as that of the ground bone. How
many pounds of each of the materials are there in a ton of
fertiUzer?
26. If the amount of potash in a given kind of glass is 5
times as great as the amount of lime, and the amount of
sand 3 times as great as the amount of potash, how many
pounds of each will there be in 4000 lb. of glass?
27. The railroad fare for two adults and a boy traveling
for half fare was $49.50. What was the fare for each person?
SuG. Let X = the smallest of the fares.
28. Separate 120 into three parts, such that the second part
is twice as large as the first, and the third part three times
as large as the first.
29. Separate 120 into three parts which shall be as 1, 2, 3.
30. Separate .0372 into three parts in like manner. Also y g .
31. Separate 240 into four parts which shall be as 1, 1,
2,4.
32. Separate $1800 into three parts, such that the second
is three times as large as the first, and the third 5 times as
large as the second.
33. In one kind of concrete the parts of cement, sand, and
gravel are as 1, 2, and 4; in another kind three parts are as
1, 2, and 5. How many more pounds of cement are needed
in a ton of one than of the other?
34. How many of the examples in this Exercise can you
work at sight?
To get the greatest possible benefit from the use of letters
to represent numbers, we now make further definitions and
rules.
12 SCHOOL ALGEBRA
4. Three Classes of Symbols. Three principal kinds of
symbols are used in algebra: (1) Symbols of quantity, (2)
Symbols of operation, and (3) Symbols of relation,
5. Symbols for Known Quantities. Known quantities are
represented in arithmetic by figures; as 2, 3, 27. They
are represented in the same way in algebra, but also in
another more general way, viz.: by letters; as by a, b, c.
The advantages in the use of letters to represent known num-
bers are: (1) letters are brief to write; and (2) a letter may stand
for any known number, and thus by the use of letters we obtain
results which are true for all numbers. See Exs. 34-40, p. 97.
6. Symbols for Unknown Quantities. Unknown quan-
tities in algebra are usually denoted by the last letters of
the alphabet; as x, y, z, u, v, etc.
The advantages in the use of distinct symbols for unknown
quantities are numerous and will be gradually realized as we
proceed. Some of these advantages are stated in Art. 3. See
also Art. 143.
7. The Signs +, — , X, -=-, and = are used in algebra, as in
arithmetic, to denote addition, subtraction, multiplication,
division, and equality respectively.
In algebra, multiplication is also denoted by a dot placed
between the two quantities multiplied, or by placing the
quantities side by side without any intervening symbol.
Thus, instead oi a X h, we may write a & or ab.
8. Signs of Aggregation. The parenthesis sign, ( ), is used,
as in arithmetic, to indicate that all the quantities inclosed by
it are to be treated as a single quantity; that is, subjected to
the same operation.
Thus, 5(2fl — 6 + c) means that the quantities inside the paren-
thesis, viz. 2a, — b, and + c, are each to be multiplied by 5.
If ALGEBRAIC SYMBOLS 13
kgain, (a + 26) (a + 2b + c) means that the sum of the quanti-
s in the first parenthesis is to be multiplied by the sum of those
in the second parenthesis.
Instead of the parenthesis, to prevent confusion, the fol-
lowing signs are sometimes used: the brackets [ ], the braces
II, and the vinculum .
9. The Sign of Continuation is ... . This sign is read
"and so on" or "and so on to."
Thus, 1, 3, 5, 7, ... . is read ''1, 3, 5, T and so on."
But 1, 3, 5, 7, ... . 19is read " 1, 3, 5, 7 and so on to 19."
10. The Sign of Deduction is .'. and it is read "therefore"
or "hence."
This sign is used to show the relation between succeeding
propositions.
EXJilRCISE 2
Express in words :
1. 5 + a.
7.
56 — a.
13.
a + 6 ^ 3.
2. d — a.
8.
2a + 3c.
14.
4 + 5(a + 6).
3. a ^ b.
9.
cd — ah.
15.
(a + b){x - y)
4. ad.
10.
7{a + 6).
16.
2a + 36 - 5c.
5. 2a + 36.
11.
7{a - 6).
17.
a ^ {x -\- y).
c d
'-a-b
12.
5a + 6
X + y
18.
a + 6 c
5 d
19. If a = 1, 6 = 2, c = 3, c? = 4, find the value of
the combinations of symbols in Exs. 1-10.
20. Make and read an example similar to Ex. 5. To
Ex. 10. Ex. 14.
14 SCHOOL ALGEBRA
Express in symbols:
21. X plus 3. The sum of x and 3. The number which
exceeds a: by 3.
22. X diminished by 3. The number 3 less than x,
23. Two times a plus three times h,
24. The sum of 4 and of 5 times x,
25. One third of the sum of a and h.
Answer the following in algebraic language:
26. If a boy has a cents and earns 10 cents, how many
cents will he then have?
27. How many, if he has a cents and earns 6 cents? How
many, if he then spends c cents?
28. Walter has x marbles and his brother has 10 more
than Walter. How many marbles has his brother?
29. Walter has h marbles and his brother has 5 more than
twice Walter's marbles. How many has his brother?
30. If Mary is a years old now, how old will she be in 3
years? In 5 years? In x years?
31. What is the next larger number than 5? Than x? n?
a: + 1? a: + 2? n - V. x - 21
32. What is the next larger even number than 6? Than
22/? 2a:? 2n + 2?
33. Taking x as the smallest number, write two consecu-
tive numbers. Three consecutive numbers. Four. Five.
(The following problems are mainly of Type II, i. e. of
the form x + x + a = b.)
1 ALGEBRAIC SYMBOLS 15
. If there are 214 pupils in our school, and the number
of girls exceeds the number of boys by 8, how many boys and
how many girls are there?
I Let X = the number of boys
Then a; + 8 = the number of girls
Hence a; + a: + 8 = 214
Or 2x + 8 = 214
Subtracting 8 from the - 8 - 8
» equals gives 2x = 206
X = 103, the number of boys
X +8 = 111 f the number of girls
35. Walter and his brother together had 60 marbles, and
his brother had 10 more than Walter. How many marbles
had each boy?
36. Make up and work an example similar to Ex. 35.
37. At New York on Dec. 21, the night is 5 hr. 32 min.
longer than the day. Find the length of the day.
38. Separate 28^^ into two parts such that out. shall exceed
the other by 2f .
39. A baseball nine hag played 62 games and won 8 more
games than it has lost. How many games has it won?
40. In a certain election 12,784 votes were cast. If the
successful candidate had a majority of 1732, how many votes
did he receive?
41. Make up and work an example similar to Ex. 40.
42. The sum of two consecutive numbers is 15. Find the
numbers.
43. The sum of three consecutive numbers is 33. Find
the numbers.
16 SCHOOL ALGEBRA
44. If 112,216 sq. mi. are added to 24 times the area of
the British Isles, the result will be 3,025,600 sq. mi. (the
area of the United States). Find the area of the British
Isles.
45. Twice the height of Mt. Washington with 1567 ft.
added equals the height of Pike's Peak, or 14,147 ft. Find
the height of Mt. Washington.
46. How many of the examples in this Exercise can you
work at sight?
47. Which of the symbols mentioned in Arts. 6-10 are
symbols of quantity? Of operation? Of relation?
48. Make up and work an example similar to Ex. 44. To
Ex. 45.
Definitions and Principles
11. The term Factors has the same meaning in algebra
as in arithmetic; that is, the factors of a number are the
numbers which, multiplied together, produce the given
number.
For example, the factors of 14 are 7 and 2; the factors of abc are
a, b, and c.
12. Coefficients. A numerical factor, if it occurs in a
product, is written first and is called a coefficient. Hence,
A coefficient is a number prefixed to a quantity to show
how many times the given quantity is taken.
For example, in 5xy, 5 is the coefficient.
When the coefficient is 1, the 1 is not written, but is
understood.
Thus, xy means Ixy,
II DEFINITIONS AND PRINCIPLES 17
The following enlarged definition of coefficient is often
used. In the product of several factor's, the coefficient of
any factor, or factors, is the product of the remaining factors.
Thus, in habxy, the coefficient of y is 5a6x; of xy, is 5a6; of ob is
hxy. What is the coefficient of 6? Of a? xl 5a? 5?
A numerical coefficient is a coefficient composed only of
figures; as 15 in Ibah.
A literal coefficient is a coefficient composed only of letters;
as ah in ahx.
What, then, is a mixed coefficientf Give an example of one.
" 13. Power and Exponent are used in the same sense in alge-
bra as in arithmetic.
Hi A power is the product of equal factors.
^Ha. power is expressed briefly by the use of an exponent.
^B/\.n exponent is a small figure or letter written above and
to the right of a quantity to indicate how many times the
quantity is taken as a factor.
Thus, for xxxx, or four x's multiplied together, we write x"^, the
exponent in this case being 4. The expression is read "a; to the
fourth power."
When the exponent is unity, it is omitted. Thus, x is
used instead of x^, and means x to the first power.
A power is composed of two parts: (1) the base (i. e. one
of the equal factors) ; and (2) the exponent.
Thus, in the power a', the base is a and the exponent is 3.
14. Root and Radical Sign have the same meaning in
algebra as in arithmetic.
A root of a number is one of the equal factors which, when
multipHed together, produce the given number.
18 SCHOOL ALGEBRA
The square root of a number is one of two equal factors
which, multipHed together, produce the given number.
What is the cube root of a number? The fifth root?
Thus, 4 is the cube root of 64, and a of a^.
The radical sign is ^/, and means that the root of the
quantity following it is to be found. The degree of the root
is indicated by a small figure placed above the radical sign.
The number denoting the degree of a root is the index of
the root. For the square root, the figure or index of the root
is omitted.
Thus, VO means "square root of 9."
Va means "cube root of a."
15. Aids in Solving Problems ; Axioms. In solving prob-
lems like those given in Exercise 1 and the latter part of
Exercise 2, certain principles are often important aids in
discovering the relations used and simplifying them.
The most important of these principles are as follows:
1. The whole is equal to the sum of its parts.
2. Things equal to the same things, or equal things, are equal
to each other.
3. // equals are added to equals, the results are eqvxiL
4. // equals are subtracted from equals, the results are equal
5. If equals are multiplied by equals, the results are equal.
6. // equals are divided by equals, the results are equal.
7. Like powers, or like roots, of equals are equal.
These principles are sometimes called axioms.
ALGEBRAIC SYMBOLS 19
EXERCISE 3
Write in words:
1. 561 a^ + 6' 12. Va + \^b.
*• '^ • 10. a + (6 + c)2.
6. 2a2 4- 363. ^^- c a * 5 4 *
16. If a = 1, 6 = 2, and c = 3, find the value of the
combinations of symbols in Exs. 1-8.
■ 17. If a = 4, 6 = 8, and c = 3, find the value of the
expressions in Exs. 9-12.
Write in symbols :
18. The square of the sum of a and b. Of 2a minus 36.
19. The cube root of the sum of a and 6.
20. X plus X increased by 4 equals 14.
21. X plus twice X plus x increased by 3 equals 108.
22. Make up and work an example similar to Ex. 18. To
Ex. 20.
23. Reduce to its simplest form 6 + 6 + 6 + 6 + 6. Also
6X6X6X6X6.
If 6 = 2, what is the value of each of these results?
24. Make up and work an example similar to Ex. 23.
25. Reduce Saaa + 76666 — Bcccccc to its simplest form.
How many more symbols are used in the long form than in
the short form?
20 SCHOOL ALGEBRA
26. Find the value of 2" when n = 1. Also when n = 2,
3. 5. 7.
27. Find the value of a", when a = 3 and ?i = 4. Also
when a = 5 and ti = 3.
28. Express the number of your great-grandparents as a
power of 2.
(The following are miscellaneous problems of Types I
and II.)
29. A man and boy together spade up a garden containing
6000 sq. ft. If the man spades four times as much ground as
the boy, how much does the boy spade?
30. Two boys earn $38 by taking passengers on a motor
boat. If the boy who owns the boat receives $10 more than
the other boy, how much does each receive?
31. A certain macadam road cost $1800, of which the
county paid twice as much as the state, and the township the
same amount as the county. How much did each pay?
32. The top of the Statue of Liberty in New York Harbor
is 306 ft. above the surface of the water. If the altitude of
the pedestal is 4 ft. greater than the height of the statue,
how high is each?
33. In a certain kind of gunpowder the weight of the char-
coal equals that of the sulphur, and the amount of niter equals
the charcoal and sulphur combined. How many pounds of
each substance are needed to make a ton of gunpowder?
34. In a certain year in the United States 200,000,000
bushels plus three times the number of bushels in the wheat
crop equaled the corn crop, or 2,600,000,000 bushels. How
many bushels were in the wheat crop?
ALGEBRAIC EXPRESSIONS 21
to Type I. Also those which belong to Type II.
K 36. Make up and work an example similar to Ex. 29. To
'"ex. 31.
37. How many of the examples in this Exercise can you
work at sight?
I
Algebraic Expressions
16. An Algebraic Expression is an algebraic symbol or
combination of symbols representing some quantity; as
bx^y — 6ab + l\ax.
17. A Term is a part of an algebraic expression which
does not contain a plus or minus sign. (Signs occurring
inside a parenthesis are not considered in fixing the terms.)
Ex. 1. bx^y - Qab + 7\/ax.
This algebraic expression contains three terms: viz. bofy, — 6ab,
and 7^ ax.
Ex. 2. 5x + a -^b -h c.
This expression also contains three terms: 6x, a -i- b, and c.
Ex. 3. 7ax2 + 5(cf _|. j) _ ^3.
Since the parenthesis, (a + &), is treated as a single quantity,
three terms occur in this expression: 7ax'^, 5{a +6), and — c'.
18. A Monomial is an algebraic expression of only one
term; as 5x^y or c.
19. A Polynomial is an algebraic expression containing
more than one term; as 3ab — c + 2a: + 5?/^.
A monomial is sometimes called a simple expression, and a
polynomial a compound expression.
20. A Binomial is an algebraic expression of two terms;
as 2a — 36.
22 SCHOOL ALGEBRA
A Trinomial is an algebraic expression of three terms; as
2a - 36 + 5c.
Evaluation of Algebraic Expressions
21. The Order of Operation in obtaining numerical values
is the same in algebra as in arithmetic.
I. In a series of operations involving addition, subtrac-
tion, multiplication, division, and root extraction, the multi-
plications, divisions, and root extractions are to be performed
before any of the additions and subtractions,
Ex. 1. Find the value of 4 + 12 X 3.
4 -K12 X 3 = 4 + 36 = 40 Ans.
(hence 4 +12 X 3 does not equal 16 X 3, etc.)
Ex. 2. What is the value of 60 - (8 -^ 2 + l3 X 7?
60 -84-2+3x7= 60 -4+ 21 =77 Ans,
II. If a given expression contains one or more parentheses
(or other signs of aggregation), each parenthesis is to be re-
duced to a single number before the operations of the expression
as a whole are to be performed.
Ex. 1. 5 +4(6 -2) =5+4x4=5 + 16 =21 Ans.
(hence 5 + 4(6 - 2) does not equal 9(6 - 2) or 9 X 4, etc.)
Note that in an expression like V16 + 9 the bar above
the 16 + 9 is a vinculum, or sign of aggregation.
Ex. 2. Vl6 +9 = V25 = 5 Ans.
(hence a/16 + 9 does not equal a/16 + V9, etc.)
22. The Numerical Value of an Algebraic Expression is
obtained thus:
Substitute for each letter in the expression the number which
the letter stands for;
Perform the operations indicated.
I
ALGEBRAIC EXPRESSIONS 23
I
Thus, if a = 1, 6 = 2, c = 3:
Ex. 1. Find the numerical value of 7ab — c\
7a6 - c2 = 7 X 1 X 2 - 32
= 14-9
= 5 Ans.
qh
Ex. 2. Find numerical value of 5ah^ + 7(a' + 2b)^ + Sc^.
c
iThe given expression
9 X2
- „ -5X1X22+ 7(P + 2 X 2)2 + 3 X 32
= 3X2-5X4+ 7(1 +4)2 +3X9
= 6-20 + 175+27
= 188 Ans.
EXERCISE 4
In each of the following examples, state the order of
operations before working the example. Wherever possi-
ble, use cancellation. When a = 5, 6 = 3, c = 1, and
X = 6, find the numerical value of
1. 2 + 3a. 13. a^ — bx^.
2. X - 2c. 14. 2(2a - c).
3. 46 — 2x, 15. x(a — 6).
4. a + 3a:. 16. 4(a — 3c)^.
5. 5a - Sx. 17. 2a:(2a - 36)^.
6. 3(a + c). 18. 3 + 2(a; - a).
7. a + 3c - a;. 19. 5a: - 3(26 + c).
8. 5a: — 26 + a. 20. 2(x^ — a-) + 3ac.
9. a + a: -i- 6 — c. 21. 3a:(a: — 3)^ — 9a:.
10. 5 x-7- b — c, 22. (a: — 1) (a: — 3) + a: (x—a),
11. 36 - x. 23. 3(2a: - 5c) - a(262 - 3a:).
12. 2a: — 46c. 24. (56 + a:) (a: — 6 + a — Sc^).
24 SCHOOL ALGEBRA
25.
a 4- 7c
X
26.
3x'-b
a-f-2c
29
5a2 . 3c
X — 1 b
(x - 1) (6 + 1) (5c ~ b)
2o. ■
abx
f3a^b\ ( 2b \ Uc - l\
' \ 5x J\x-lJ\ Sb J
31.
by. 33. a'^x'^.
38.
6(10?/ - 36).
39.
3a:(4a + 36).
40.
ax + 5a!(36 - y).
41.
3a + a(3x - lOy)
42.
5x - 3{by - ab).
If a = §, 6 = f , a: = 2, 2/ = f , find the value of
30. 6a. 32. abx. 34. 3aW. 36. 2x + by,
35. X — 26. 37. 6a6 - Wy,
43. 5a: (6?/ — a^) — 6a:.
44. 6(a + 6)2 + 10(2/ - a)2.
45. a: + VSa.
46. VSa + V36.
47. 5?/ — VOaa:.
48. Does a:2 + a: = 12, if a: = 2? If a: = 3? 4? 5? 1?
49. Does 3^2 - 4a: = 4, if a; = 1? If a: = 2? 3? f ? 0?
50. Doesa:2-5a:4-6 = 0,ifa: = 1? If a: = 2? 3? 4? 5?
51. Doesa:2-|a:-2 = 0,ifa: = 1? If a: = 2? 3? 4?^?
52. Show that (a - 26)^ = a^ - 4ab + 46^, when a = 3
and 6 = 1.
^3 _ 53
53. That 7- = a^ + a6 + 6^, when a = 2 and 6 = 1.
a — 0
54. Find the value of 20:^ when a: = 1. When
a: = 2. 5. i 1.5.
SuG. The results may be conveniently arranged as in
the following tabulation:
Find the value of each of the following and
tabulate results:
55. 2a: + 1, when a: = 1. When a: = 2. 3. 5.
X
2a:2
1
2
2
8
5
50
1
1
2
2
.5
4.5
1
4
. 1.5
ALGEBRAIC EXPRESSIONS 25
16. x^ + 2, when a; = 1. When a; = 2. 3. J. 1. 5.
M. x{x + 1), when x = I. When x = 2. 3. .2. i J.
►8. In Exs. 4-10 state which of the expressions used are
monomials. Also which are binomials. Trinomials. State
the same for Exs. 35-40.
EXERCISE 5
1. If ^ = Iw, find the value of A when ^ = 12 and w = 5j.
Also when / = 10.4 and w = 5.8.
Do you know what use is made of the formula A = Iw m. arith-
metic in finding areas?
2. If F = Iwh, find V when I = 12, w = b, and A = 3.
Also when I = 10.4, w = 5.8, and h = 3.05.
Do you know what use is made of the formula V = Iwh in arith-
metic in finding volumes?
^ 3. If p = br, find p when b = 350 and r = 1.07. Also
when b = 7.68 and r = .045. Also when b = 84,000 and
r = mh
What does the formula p = hr mean in arithmetic in connec-
tion with the subject of percentage?
'4. If i = prt, find i when p = $300, r = .05, and t = 2j.
Also when p = $9327.50, r = .06, and ^ = 3f .
What is the meaning in arithmetic of the formula i = prt?
5. If ^ = irR^, find the value of A when tt = 3.1416 and
i^ = 10.
Do you know of any use that is made of the formula A = ttR^
in arithmetic?
6. If ^ = Va^ + 6^, find the value of h when a = 8 and
6 = 6.
Do you know of any use that is made of the formula
h = Va^ + 62 iti arithmetic?
26 SCHOOL ALGEBRA
7. If 5 = igfy find s when g = 32.16 and < = 4. Also
when g = 32.16 and t = 2|.
Can you find out the meaning of the formula s = J gf^?
8. A stone dropped from the top of a precipice reaches its
base in 5 seconds. How high is the precipice?
9. If C = UF - 32), find C when F = 95°. Also when
F = 100°.
Do you know the meaning of the formula used in this example?
10. If iron melts at a temperature of 2700° F., at what
temperature does it melt on the centigrade scale?
11. If ^ = itB? - iTr\ TT = 3.1416, R = 13, and r = 12,
find A in the shortest way.
12. If 1 orange costs 3 cents, how many oranges can be
bought for 12 cents? For x cents? For x -\- y cents?
13. If 1 orange costs a cents, how many oranges can be
bought for 25 cents? For x cents? For x -\- y cents?
14. If 1 acre of land costs x dollars, what will one half an
acre cost? f of an acre? J of an acre?
(The following problems are variations of Type I.)
15. If a 12-year-old boy and a 16-year-old boy together
earn $48 in mowing lawns, and the younger boy receives only
half as much as the other, how much does each boy receive?
Let X = no. dollars received by 16-year-old boy
Then ^x = no. dollars received by 12-year-old boy
Hence x -\- ^x = $48
or |x = $48
Multiplying these equal numbers by 2 (Art. 15, 5)
Sx = $96
Dividing equals by 3 (Art. 15, 6)
X = $32, share of older hoy
\x = $16, share of younger boy
K ALGEBRAIC EXPRESSIONS 27
6. A man left .$24,000 to his son and daughter. As his
daughter had cared for him in his old age, he left his son only
f as much as he left his daughter. How much did each
receive?
17. A man and boy together made $124.80 by working a
garden one summer. If the boy received J as much as the
man, how much did he receive?
18. A farm is worked on shares. As the owner of the farm
supplies the tools and fertilizers, the tenant receives only f
as large a share of the profits as the owner. If the profits for
one year are $4410, how much does each receive?
19. Two men manage a store, and as one of them owns
the building, the other receives only f as large a share of the
profits as the owner of the store. If the profits for one year
are $6600, what does each receive?
20. Separate 126 into two parts such that one of them is
i as large as the other, f as large.
21. Separate .028 in the same manner as in Ex. 20.
22. A macadam road cost $18,000. The county paid J as
much of the cost as the township, and the state paid J as
much as the township. How much did each pay?
23. A certain kind of concrete contained J as much sand
as gravel and J as much cement as sand. How many pounds
of each material were there in If tons of concrete?
24. Make up and work an example similar to Ex. 16. To
Ex. 20.
25. How many of the examples in this Exercise can you
work at sight?
CHAPTER II
NEGATIVE NUMBERS
23. Positive and Negative Quantity. Negative quantity is
quantity exactly opposite in quality or condition to quantity
taken as 'positive.
If distance east of a certain point is taken as positive, distance
west of that point is called negative.
If north latitude is positive, south latitude is negative.
If temperature above zero is taken as positive, temperature
below zero is negative.
If in business matters a man's assets are his positive possessions,
his debts are negative quantity.
Positive and negative quantity are distinguished by the signs +
and — placed before them.
Thus, $50 assets are denoted by + $50, and $30 debts by - $30.
We denote 12° above zero by + 12°, and 10° below zero by - 10°.
The use of the signs + and — for this purpose, as well as to indi-
cate the operations of addition and subtraction, will be explained
in Art. 26.
24. Algebraic Numbers is a general name for both positive
and negative numbers.
The absolute value of a number is the value of the number
considered without regard to its sign.
Thus, if one man travels 5 miles east and another man travels 5
miles west, the absolute distance traveled by the two men is the
same, viz.: 5 miles. The two distances traveled, however, are dif-
ferent algebraic numbers, one distance being + 5 miles and the other
distance being — 5 miles.
In general the absolute value of both + 5 and — 5 is 5; and of
both + a and — a is a.
28
I
NEGATIVE NUMBERS 29
The Utility of Negative Number lies in the fact that
the use of negative number enables us to use two opposite
or contrasted kinds of quantity in working a given problem.
Also by the use of negative quantity we are often able to
choose an advantageous starting point in solving a problem.
" The full meaning of these utilities and other advantages in the
use of negative quantity will appear as we advance in the study of
algebra.
EXERCISE 6
"l. What is meant by a temperature of — 8°? By a latitude
of - 23°? By the date - 776? (Dates after the birth of
Christ are taken as positive.)
2. If the temperature was 17° at noon and — 8° at mid-
night, how many degrees did it fall?
3. If in a given time the temperature should fall from
— 5° to — 12°, how many degrees would it fall?
4. If the temperature were 15° at a given time, what would
it become after a fall of 10°? Of 28°? - 15°?
5. If the temperature were — 8° at a given time, what
would it become after a rise of 4°? Of 15°? - 8°?
6. Make up and work an example similar to Ex. 3. Also
to Ex. 5.
7. If a traveler is in latitude — 4° and travels north 7°,
what does his latitude become? What does it become if
instead he travels south 7°? ^
8. If a man's property is — $7000 and he saves $2000 a
year for 8 years, what does his property become?
9. If a vessel; at latitude 3°, sails south 345 miles, what
does her latitude become if 60 miles equal 1°?
30 SCHOOL ALGEBRA
10. If a man bought a horse for $L50 and sold it for $200*
what was his gain? What would his gain have been if he
had sold it for $125? For $100?
11. What is meant by saving — $10? By a distance — 10
miles north?
12. What is the absolute value of — 4 miles? Of + 4
miles? - 5 inches? - 3°? - $4200?
13. Make up an example for yourself showing the meaning
of absolute value.
(The following problems are variations of Type II, or are
of Type III, viz.: x + ax + b = c.)
14. Walter and his brother together had 90 marbles, and
his brother had 10 less than Walter. How many marbles hac
each boy?
Let X = no. of marbles Walter had
Then a; — 10 = no. of marbles his brother had
X +x -10 = ,90
2a; - 10 = 90
Adding 10 to each of these equals (Art. 15, 3)
2x = 100
X = 50, no. of marbles Walter had
x — 10 = 40, no. of marbles his brother had
15. A basket ball team has played 27 games and has lost
3 less than it has won. How many games has it won?
16. In a certain election 12,420 votes were cast, and the
defeated candidate had 210 less votes than the winning can-
didate. How many votes had each candidate?
17. Make up and work a similar example for yourself.
18. Walter and his brother together have 83 marbles. If
his brother has 7 less than twice the number Walter has,
how many has each boy?
NEGATIVE NUMBERS 31
19. One number exceeds 4 times another number by 5,
and the sum of the numbers is 100. Find the numbers.
20. One number exceeds 3 times another number by .12,
and the sum of the numbers is 4.4. Find the numbers.
21. One fraction exceeds 5 times another fraction by ^,
and the sum of the fractions is V • Find the fractions.
22. The distance from New York to Chicago is 912 miles.
If this is 24 miles less than ^ times the distance from New
York to Boston, what is the latter distance?
23. The Eiffel Tower is 984 ft. high. If this is 126 ft. less
than twice the height of the Washington Monument, what
is the height of the Washington Monument?
24. How many of the examples in this Exercise can you
work at sight?
25. Which of Exs. 14-23 are of type x -{- x — a = b, and
which are of type x -\- ax ^ h = c'l
26. Make up and work an example similar to Ex. 19. To
Ex. 23.
26. Double Use of + and — Signs. The signs + and —
are employed for two purposes (see Arts. 7 and 23) : first,
to indicate the operations of addition and subtraction; and
second, to express positive and negative quantity. We are
able to make this double use of these signs because, in each
use, the signs are governed by the same laws.
--7 -6 -5 -4 -3 -2 -1 . +1 +2 +3 +4 +5 +6 +7 +8
w I I I I I I I I I I ^
K B O A F -^
A person walks from 0 toward E a distance of 5 miles (to F) and
then walks back toward W a distance of 3 miles (to A). If the dis-
32 SCHOOL ALGEBRA
tance to the right of 0 is regarded as positive, and therefore the dis-
tance to the left of 0 is negative, the distance from the starting
point to the destination may be expressed as the sum of a positive
quantity and a negative quantity; that is,
(positive distance OF) + (negative distance FA)^
or, +5 + (-3) = 5 -3 = 2.
The position arrived at may be determined in another waj?" — viz.
by deducting 3 miles from 5 miles. We obtain
5 - (+3) = 5 -3 = 2.
From this example we see that adding negative quantity is
the same in effect as subtracting 'positive quantity.
Therefore, in the expression 5 — 3, the minus sign may
be considered either a sign of the quality of 3, or as a sign of
operation to be performed on 3. Hence, we are able to use
the signs + and — to cover two meanings.
27. Laws for the Use of + and — Signs. Whichever of
the two meanings of + and — named in Art. 26 is assigned,
we see that + (- 3) = - 3; also, - (-h 3) = - 3.
The signs + and — applied in succession to a quantity are
equivalent to the single sign — .
Or in symbols,
4- (— a) = — a; and _ (+ a) = — a.
Ex. Find the value of 8 + 4-11 + 3- 6. On squared
paper show the meaning of the numbers involved.
8+4- 11 +3-6 = 15 -17= -2 Ans.
Taking the distances to the right of OP as positive, we have the
diagram on p. 33 showing the meaning of the numbers involved.
Note that the above process holds true whether a number pre-
ceded by a minus sign is regarded as the subtraction of a positive
number or the addition of a negative number.
If in the illustration on p. 31 a person walks in the nega-
tive direction from 0 (i. e. toward W) a distance of 4 miles
I
NEGATIVE NUMBERS
33
to K, and then reverses his direction and goes 2 miles, he
will be at B. Or stated in another way, diminishing the
n
+
8
+
4
-11
+ 3
-
2
—^
_—
^^
'6
F
distance traveled west by 2 miles, brings him to the same
place as walking the full direction west and then walking 2
miles east.
It may be well to study another illustration of this principle.
If a man owes two notes of $500 and $100 respectively, removing
the note for $100 is the same in effect as annexing $100 in money to
the debts as they are. That is,
- $500 - $100 - (- $100) = - $500 ^ $100 + $100 = - $500
Hence:
The sign — applied twice to a given positive quantity gives
a + result.
Or in symbols, — (— a) = -f a.
These laws enable us to u^e
negative quantity with as great
freedom as we u^e positive quan-
tity, and hence are an important
source of power, as will become
more evident later.
Ex. On squared paper show
the meaning of — 5 — (— 3). Also of —5 + 3. Hence,
show that - 5 - (- 3) = - 5 + 3.
4
~5
n^
D
1
r
i-3
\
\
34 SCHOOL ALGEBRA
On the lower diagram on p. 33 — 5 — ( — 3) means OA — DA,
.or OD. Also -5+3 means OA + BC, or OD.
Hence, — 5 — (— 3) and —5+3 give the same result; or we
may say — 5 — (— 3) = — 5+3.
28. The Algebraic Sum of two or more algebraic numbers
is the result of combining the given algebraic numbers into
a single number.
Thus, the algebraic sum of 4 and — 7 is — 3.
EXERCISE 7
Find the value of each of the following and verify the
result on squared paper:
1. 5 - 2. 6. 0 - 4. 11. 5 - (- 8).
2. 6 - 8. 7. 8 - 6 —4. . 12. - 7 + (- 2).
3. 5 - 5. 8. 7 - 5 + 4. 13. 0 - (- 5).
4.-4 + 2. 9. 3 + 1 - 5. 14. 0 + (- 5).
5.-4-2. 10. - 4 - (- 3). 15. - 4 - (- 1.5).
16. 4 + 5 - 12 + 3 - 5.
17. -3 + 8-6-2 + 2-1.
18. At 6 A. M. a thermometer read 57°. It then made
successive changes as follows: + 7°, — 2°, + 5°, — 3°, — 2°.
What was the final reading?
19. In a certain football game, taking a distance toward
the north goal as positive, during the first seven plays the ball
started at the middle of the field and shifted its position
in yards as follows: +50-10-15-5 + 10-5-20.
Find the final position of the ball with reference to the middle
of the fitild. On squared paper show the changes in the posi-
tion of the ball, letting 5 yd. equal one space on the papePr ;
NEGATIVE NUMBERS 35
20. State in the language of debts and credits the mean-
ing of
- $700 - $200 - ( - $200) = - $700 - $200 + $200
!Hi SuG. If a man has debts of $700 and $200, the removal of the
$200 debt is the same as leaving his debts unchanged and adding
i_S200 to his possessions. He becomes worth — $700 in either case.
^^ 21. State in the language of distance traveled east and
west the meaning of
- 10 mi. - 2 mi. - ( - 2 mi.) = - 10 mi. - 2 mi. + 2 mi.
(The following are miscellaneous problems of Type II and
Type III.)
22. A man and a boy together catch 320 fish, and the man
receives three times as many fish as the boy. How many fish
does each have?
23. A man has $3220 in two banks and the amount in one
bank exceeds that in the other by $540. How much has he
in each bank?
, 24. Two girls make $24.60 by sewing, and the younger
girl receives only one half as much as the older. How much
does each receive?
25. Separate $12.68 into two parts one of which shall be
smaller than the other by $5.
26. A given piece of bronze weighs 4600 lb. It contains
twice as much tin as zinc, and 8J times as much copper as
zinc. How many pounds of each metal does the bronze
contain?
27. The distance from the mouth of the Mississippi River
to the source of the Missouri River is 4500 miles. The dis-
tance between the mouth of the Mississippi and the mouth of
36 SCHOOL ALGEBRA
the Missouri is 1700 miles less than the length of the Missouri.
What is the length of the Missouri?
28. A farmer obtained 2720 pounds of cream in one month
by the use of a separator. This is ^ more than he would
have obtained if his milk had been skimmed by hand. How
much would he have obtained by the latter process?
29. The cost of a macadam road was $24,000. The county
paid twice as much as the state, and the township three times
as much as the state. How much did each pay?
30. Three partners divided $14,000, the second partner
receiving $2000 more than the first, and the third partner re-
ceiving twice as much as the first. How much did each
receive?
31. Mt. Washington is 6290 ft. high. This is 170 ft. more
than 10 times the height of the Singer Building (N. Y.).
How high is the latter?
32. Make up and work an example similar to Ex. 18.
Ex.21. Ex.24. Ex.29.
33. How many of the examples in this Exercise can you
work at sight?
34. Which of Exs. 22-33 of this Exercise are of Type I?
Of Type n? TypeHI?
29. Graphs. A set of numerical facts may often be com-
bined as a geometrical picture called a graph. The meaning
and use of negative numbers are often well illustrated on a
graph.
Ex. On a given day the following were the temperatures
at a given place:
NEGATIVE NUMBERS
37
Midnight - 15° 9 A. M.
3 A. M. - 20° Noon
6 A. M. - 10° 3 P. M.
2°
10°
15°
6 P. M. 10°
9 P. M. 0°
Midnight - 10°
Temperatures
20'
Hours
Graph these facts.
We draw a horizon-
tal line and on it mark
oflf spaces to represent
hours, as in the dia-
gram. Perpendicular to
this we draw a line and
on it mark off spaces to
represent temperatures.
Above or below each
point which represents
an hour, a point is lo-
cated which represents
the temperature at that
hour. Through the points thus located a continuous line ABCD
is drawn. This is the required graph.
EXERCISE 8
Graph each of the following sets of temperatures:
1.
Mid-
night
3A.M.
6A.M.
9 A.M.
12 M.
3 P.M.
6 P.M.
9 P.M.
-20°
-30°
-20°
-10°
0°
10°
10°
0°
2.
-10°
-20°
-10°
0°
10°
20°
10°
6°
3.
-10°
-15°
- 5°
10°
15°
25°
15°
-5°
4.
0°
-10°
- 5°
15°
25°
30°
15°
5°
5. Make up and work a similar example for yourself.
Graph each of the following sets of temperatures:
6.
7.
8.
Jan. 1
Feb. 1
Mar. 1
Apr. 1.
May 1
June 1
New York
New York
London
31°
-1°C.
37°
31°
-1°C.
38°
35°
1°C.
40°
42°
6°C.
45°
54°
12° C.
50°
64°
18° C.
57°
38
SCHOOL ALGEBRA
July 1
Aug. 1
Sept. 1
Oct. 1
Nov. 1
Dec. 1
71°
22° C.
62°
73°
23° C.
62°
69°
21° C.
59°
61°
16° C.
54°
49°
9°C.
46°
39°
4°C.
41°
9. Convert the temperatures given for London in Ex. 8
to temperatures on the Centigrade scale and graph them
(see Ex. 9, p. 26).
10. Collect and graph sets of numerical facts similar to
those given in the preceding examples.
H ADDITION AND SUBTRACTION; THE EQUATION
^K Addition
30. The Utility of Addition in Algebra.
Ex. Find the value of 3a6^ + Saft^ + 2a}? y when a = 2
f
nd 6 = 3.
PROCESS WITHOUT ALGEBRAIC ADDITION
If we substitute directly in the given expression, we obtain
3a62 + 5a62 + 2a62 =3x2x32+5x2x32+2x2x32
= 54+90+36
= 180 Ans.
PROCESS AIDED BY ALGEBRAIC ADDITION
3a62 + 5a62 + 2ah'' = lOafe^
= 10 X 2 X 32
= 180 Am,
In solving the above example, algebraic addition enables
us to save more than half the work. Algebraic addition has
other uses which will appear later.
Why do we now make definitions and rules?
31. Addition, in algebra, is the combination of several
algebraic expressions into a single equivalent expression.
Addition is sometimes described as collecting terms in an expression.
32. Similar Terms (or like terms) are terms which contain
the same literal factors and the same radical signs over the
same factors.
• 39
40 SCHOOL ALGEBRA
Thus, lab"^ and — bab"^ are similar terms. Also 5a V 2 and — 6a V2
are similar terms.
Dissimilar terms (or unlike terms) are terms which are
unlike either in their literal factors or in the radical sign
over the same factor.
Thus, Sa^ft and bab"^ are dissimilar terms. Also 3v 5 and 3a/5
are dissimilar terms.
The addition of dissimilar terms can only be indicated.
Thus, h added to a gives a + h; also o? — Sa% added to Sa^ — ¥
gives a3 - 3a% + Sa^ - bK
33. Method for Addition. The most convenient general
method for addition is shown in the following examples:
Ex. 1. Add 4a:2 + 3a! + 2, 3x^ - 4x - 3, - 2x^ - x - 5.
Arranging similar terms in the same column, and adding each
column separately, we obtain
CHECK
4x2+ 3a; +2= 4+3+2= 9
3x^ -4x -3 = 3 -4 -3 = -4
-2x2- X - 5 = -2-1-5^ 8
Sum 5x2 -2x -6= 5-2-6=-3
To check the accuracy of the work, we let x = any convenient
number, as ,1; find the numerical value of each row; and compare
the sum of these results with the numerical value of the algebraic
expression obtained as the sum.
Ex. 2. Add 2a^ - 5a^h + Aa¥ + aW, ^o?h + 2o^ - a¥ - 3ah\
a^h -a' + 2ah\
Proceeding as in Ex. 1,
CHECK
2a3 - 5a26 + 4a62 + a^h^ =2-5+4 + 1 =2
2a3 + 4a26 - ZalP- -aft^ = 2+4-3-1 =2
- a^ + a26 + 2a62 = -1+1+2 =2
Sum 3a3 + 3a62 + ^253 _ fj^^ = 3+0+3 + I-I =6
In the second column the algebraic sum of the coefficients is
— 5+4 + 1, which = 0; and as zero times a number is zero, the
I
ADDITION 41
sum of the second column is zero, which need not be set down in
the result.
The work is checked by letting a and h each = 1.
Hence, the process for addition may be stated as follows:
^jk^rraiige the terms to be added in columns, placing similar
terirns in the same column;
Find the algebraic sum of the numerical coefficients jof each
column and prefix this result to the literal factors common to the
terms in the column.
Sometimes the algebraic sum of the coefficients of each group of
similar terms is found without arranging the terms in columns.
I
EXERCISE 9
dd and check each result:
1.
2.
3.
4.
5.
- 11
4
Sx
— x
-7x
6
- 10
- 6x
- 3x
12x
6.
7.
8.
9.
10.
2a
-x"
Ixy
a'b
7xV
ba
3x2
- 10x1/
Ba'b
- lOxV
- 12a
5x2
2xy
-Sa^b
xY
11. 3ax, — 2ax, 5ax, ax, — 3ax.
12. 5x2, i2a:2, - lOx^, x^, - IGx^, Sx^, - x\
13. 7aW, - \2aW, - aW, - ^aW, baW, Mb\
14.
15.
16.
17.
3(a + b)
- 6(x - y)
b^a-\-x
47rr2
5(a + b)
^{x - y)
- eVa 4- X
-27rr2
- 4(a + b)
- 5(x - y)
2Va + x
i-r^
42 SCHOOL ALGEBRA
18.
19.
20.
3a; -22/
5^2+ 7
a^ — ax -\- Ax^
- 2x + 31/
x^ - 10
3a2 -f 2aa; - Sar^
X- y
-7a;2+ 1
— 0? — ax — x^
21. a — 26, 3a + 46, a + 56, — 5a — 6, a — 56.
22. ,3a:2 + 2/2, 2a:2 - T?/^, - 4ir2 _ 5^2^ ^^ _|_ 3^2^ _ 3^2^
23. 3aa:^ — 56?/^, 2aa:^ + 46i/^, 26?/^ — 4aa:^, 6^/^ — aa:^.
Reduce each of the following to its simplest form:
24. x" - xy + Sy^ + 2^2 + 2xy-2y^ + x^ + y^ + 3x^ -xy.
25. mn — 3ri2-j- 772,2 -|- m^ + 2^2 — Smn + m2 — 7i2 -^ jjifi _ 2^2.
26. a:2 + 2/2 - 2^2 + 3^2 - 2/2 + 222 + 2;2 - 2a:2 + a:2 _ ^2.
27. 2a:2 _ a:^/ + 3xy - 5y^ + 32^2 _ 3^^ _^ ^.-2 + 2y^ - 2xy.
28. 7x + y -{- 5z — lOxy -\- 2y — 3z + ISxy — Axz + 52
— 6x — 4:xz -\-2xy — ^y -{- 9z -{- 7x — xz -{- 2\xz — I62 -\-x — 5xy.
29. 0^ + ^x'^y + 3a:2/2 _[_ ^^ _|- ^^3 _ 3^2^ _j_ 3^^2 — y^ -\- 2x^y
— 2xy'^ -\-y^ + x^ — y^ + x'^y — 4:0^ — xy^ — i^ + y^ + a^ —
x^y + xy"^.
Collect similar terms in the following and check each result:
30. 2a; - 3?/ - 5a: + 42 + 4i/ + z - 2?/ - a: - 32 +2.T-3y.
31. Zxy — 5ax + 3y^ — 2xy — 3a;2 + 4aa; — 2y^ + 3aa; — 2xy.
32. a; - 3i/ + 22 + 2^/ - 2a; - 2 - 3a; - 42-2a; + 2 + 2a;.
33. 2a; - 1 + 5?/ - 2 + 3a; + 2 + 32/ - 3 - 2.T + 1 - a; - 32/.
34. 3d26 - 2a2c + 3a2 - 5a26 - a2 - 3a2c + a26 + 6a2c - 2a2.
35. 5a^ -Sx + 4:-2x^ -63^ + Ax-7 - x'^ + a^ -i- 3x^
— a; -I- 5 + 3a;2 - 6a; - a;2 + 4a; - 2a;2 + 2a;.
36. 2a;" - 5a:"* + 3a;2 - a;" - 7a; + 3a;2 - 3 + 2x"' - bx^
+ 5 + 3a;"*.
I
ADDITION 43
7. Reduce Sxxxyy + Sxxxyy — 5xxxyy — 2xxxyy to its
simplest form. About how much briefer is the form you
obtain than the given form?
38. Make up and work an example similar to Ex. 37.
39. Make up and work an example showing the use of
algebraic addition (see Ex. of Art. 30, p. 39).
40. State in general language the use of algebraic addition.
|m(The following are mixed problems of Types I, II, III.)
41. Three partners in a retail business made $18,000 in
one year. The second partner owned the building and re-
ceived twice as large a share as the first partner. The third
partner supplied most of the capital and received three times
as large a share as the first. How much did each receive?
42. Make up and work a similar example for yourself.
43. Find three consecutive numbers whose sum is 36.
44. Find four consecutive numbers whose sum is 106.
45. Make up and work an example concerning five con-
secutive numbers.
46. The area of the United States and its outlying posses-
sions is 3,742,155 sq. mi. The area of the United States
exceeds that of its outlying possessions by 2,309,045 sq. mi.
What is the area of the outlying possessions?
47. How many of the examples in this Exercise can you
work at sight?
48. Name the type to which each of the above problems
belongs (Exs. 41-46).
44 SCHOOL ALGEBRA
Subtraction
34. The Utility of Subtraction in Algebra.
Ex. Find the numerical value of 17aW — 15aW, wheq
a = 3 and 6 = 2.
PROCESS WITHOUT ALGEBRAIC SUBTRACTION
17a263 _ I5a%^ = 17 X 3^ x 2^ - 15 X 3^ x 2'
= 17X9 X8 -15X9X8
= 1224 - 1080 = 144 Am.
PROCESS AIDED BY ALGEBRAIC SUBTRACTION
= 2 X 32 X 23
= 144 Ans.
In solving the above example, algebraic subtraction
enables us to save more than half of the work. Algebraic
subtraction has other advantages which will appear later.
Why do we now proceed to make definitions and rules?
35. Subtraction, in algebra, is the process of finding a
quantity which, added to a given quantity (the subtrahend),
will produce another given quantity (the minuend).
Thus, if we subtract Sab from lOab, we obtain 7ab, for 7ab
added to Sab (subtrahend) gives lOab (minuend).
36. Signs in Subtraction. From Art. 26 it follows that
Subtracting a positive quantity is the same as adding a nega-
tive quantity of the same absolute magnitude; and
Subtracting a negative quantity is the same as adding a
positive quantity of the same absolute magnitude.
37. Method for Subtraction. The most convenient general
method in subtraction is to
I
SUBTRACTION 45
CHECK
5x' -2x^ + X -S =5
-2+1-3=1
- 2x3 a 3a,2 _ 3. 4. 2 =2
-3-1+2=0
Write the terms of the subtrahend under the terms of the
minuend, placing similar terms in the same column;
Change the signs of the terms in the subtrahend mentally, and
proceed as in addition.
Ex. 1. From 5a^-2x^ -j- x - 3 subtract 2a^ - 3x'^ - x + 2.
Cheek the work by letting x = 1.
I_..„.............
|^P?he coefficient of a;^ is 5 - 2, or 3, of a:^ is - 2 + 3, or 1, etc.
Ex. 2. Subtract 2a' - 3a'b - 6a''b^ - 2a¥ + 2b' from
a' + 5a^6 — Qa^b^ — 3a¥. Check the work by letting a = 1
and 6 = 1.
CHECK
^4 + 5^35 _ Qa%^ _ ^a¥ = 1+5-6-3 =-3
- 2a^ + Sa% - Qa%^ - 2ab^ ^2¥ = 2-3-6-2+2= -7
- a^ + 8a^b -a¥-2b^=-l-\-S-i-0-l-2= 4
The coefficient of a'^¥ is - 6 + 6, or 0. The coefficient of b^ is
0 - 2, or - 2.
EXERCISE 10
Subtract and check each result :
1.
2. 3.
4. 5.
6.
7ab
5ic X
5a: - 3a:2
- 7a:2/
Sab
9a; 2a:
- 3a: - 4a:2
Sxy
7.
8.
7(a: + y)
9.
10.
5(a + b)
- 2Va + X -
- 4V6 _ 2/
3(a + 6)
- 3(a: + y)
- sVa + a:
2Vb-y
11.
12.
13.
14.
3x^ - 4x
3a: -9
2x? -b
5a:2 + 4a: - 3
2x^+ X
5a: + 1
- x?^2
ar^ - 3a: + 5
46 SCHOOL ALGEBRA
15. From 3a + 2b - 3c - d take 2a - 2b + c - 2d,
16. From 7 - 3a: + 2x'^ take 15 - 4x - 5x^.
17. From ic^ - 2/2 - 2;2 + 8 take 2x'' + y^ - 2z^ + 10.
18. From bxy — 3xz + ^yz + x^ take ^xz — 20*2/ — ^r^.
19. From 2 - x + x^ -{-x"^ take 3 + x-a:2-a:3_ 2x\
20. Subtract lOa:^?/ + 3x'^y'^ — \3xy'^ from x^?/ — xy"^ + 2a;2y2^
21. Subtract 3 — 2ab + 3ac — Acd from 5 — ac + Scd — 5a^.
22. Subtract 1 + x — x'^ + o^ — x^ from 2 — a: — a:^ — a:^.^^^
23. Subtract a -\- 2b — 3c -{- 4d from m + 26 + c? — a: +a.
24. Subtract 3a;^ - 2a:2 _^ 5^ _ 7 ^j.^^^ Sa:^ ^ 2a:2 - a: - 7.
25. Subtract - a:^ - 2a;^ + a:^ + 5 from a:^ - a:^ + a:^ - 2a: + 5.
26. Subtract 3a:'" — 3x" + a: — 3 from a:"» + a:" — a:^ + a: — 1.
27. From the sum of 2a: and 3y subtract their difference.
28. From 0 subtract — 3a:. From 0 subtract x — y.
From zero subtract 3a2 — 2ab + 6^.
29. Reduce laaabb + baaahb — 3aaabb to its simplest
form.
30. Make up and work an example similar to Ex. 29.
If ^ = a:^ - 3a:2 + 1, 5 = 2a:2 - 5a: - 3, C = 3a:3 + a:2 + 3^^
find the value of
31. A + B + C 33. A + B - C
32. B - A-\- C 3i. A - B + C
(The following problems are variations of Types I and II.)
35. Find the value of a:, if 3a: — 2 in. = 7 in.
36. Separate $24.80 into two parts such that one part is
smaller than the other by $4.60.
I
USE OF THE PARENTHESIS 47
37. Separate $24.80 into two pai-ts such that the smaller
!part equals | of the larger part.
I 38. Separate $5000 into three parts such that the second
part shall exceed the first by $300, and the third shall exceed
the first by $800.
39. Separate $5000 into three parts such that the second
part shall exceed the first by $300, and the third shall exceed
the second by $800.
40. Separate $6000 into three parts such that the second
part equals J of the first, and the third part equals ^ of the
first.
41. Separate $6000 into three parts such that the second
part is double the first, and the third part is double the second.
42. Make up and work an example similar to Ex. 36.
To Ex. 40.
43. Name the type of which each of the above problems
(Exs. 36-43) is a variation.
44. How many of the examples in this Exercise can you
work at sight?
45. How many of the examples in Exercise 1 can you now
work at sight?
Use of the Parenthesis
38. Utility of the Parenthesis. The parenthesis is useful
in indicating an addition or a subtraction in a brief way.
Thus, 2a + 36 - 5c - (3a - 26 + 3c) indicates that 3a - 26 + 3c
is to be subtracted from 2a + 36 — 5c.
The parenthesis will also be found useful in indicating
multiplication and division in a brief manner, and other uses
of the parenthesis will become evident as we proceed
48 SCHOOL ALGEBRA
39. Removal of a Parenthesis. From the processes of addi-
tion and subtraction it follows that
When a parenthesis preceded by a -\- sign is removed, the
signs of the terms inclosed by the parenthesis remain unchanged.
But
When a parenthesis preceded by a minus sign is removed, the
signs of the terms inclosed by the parenthesis are changed, the
+ signs to —, and the — signs to -\-.
Ex. Simplify 2a + 36 - 5c - (3a - 26 + 3c).
2a + 36 - 5c - (3a - 26 + 3c) = 2a + 36 - 5c - 3a + 26 - 3c
= — a + 56 — 8c Ans.
Let the pupil check the work by letting a = l,6 = l,c = l.
40. Parenthesis within Parenthesis. Using the parenthesis
as a general name for all the signs of aggregation, it is evident
that several parentheses may occur one within another in
the same algebraic expression. The best general method of
removing several parentheses occurring thus is as follows :
Remove the parentheses one at a time, beginning with the
{innermost;
Collect the terms of the result.
It is also possible to remove the parentheses in reverse order,
that is, by removing the outside parenthesis first, etc. Working an
example in this way often forms a convenient check on the first
process.
I
Ex. Simplify bx
- y -
[4a: - 6?/ + { -
3a: + 2/ -f-
&x - z)}].
5x - y -
-[ix-
-62/ + 1
- 3x + ?/ + 22 -
(2a: -2)11
= 5x - y -
-[4x-
-6y + {
-Zx +y +2z -
2x + 2}]
= 5x - y -
-[4a:-
-62/
-Zx +y ■\-2z -
2a: +2]
= 5x - y -
- 4:X
+ 6?/
+ 3x - 2/ - 22 4
- 2x -z
= 6x +^y -
- Sz Ans.
USE OF THE PARENTHESIS 49
The work may be checked by removing the parentheses in reverse
order, or by the method of substitution as follows:
Letting x = 1, y = 2^ z = S, we have
5x -y -l^x -Qy + { - Sx -\- y +2z - {2x - z)}]
= 5-2- [4- 12 { -3+2+6- (2- 3)}]
= 3-[-8 + {5-(- 1)|] = 3 - [ - 8 + {5 + 1}]
= 3-(-8+6) =3-(-2) =3+2=5
Also 6a; +42/ -32 =6+8-9=5
EXERCISE 11
Remove parentheses and collect similar terms. Check
each result either by substitution of numerical values, or by
reversing the order in which the parentheses are removed.
. 1. 3a + (2a - 6). 7. x - [2x -\- (x - 1)].
2. 2x- (x- 1). 8. 5a: + (1 - [2 - 40!]).
3. a: + (1 - 2a:). 9. 2 - 11 - (3 - a) - a}.
4. 3a; - (1 + 3a;). lo. 2a: - [- a: - (a; - 1)].
5. X- (-X- 1). 11. 2y+ {-X- (2y - x)}.
6. X + 2y - {2x — y). 12. a - {- a — (— a — 1)^.
13. [a:^ - (x^y - z^) - z^] + (x'^y - x^).
14. 1 - {1 - [1 + (1 - a:) - 1] - 1} -a:.
15. a: -[-{-(- a: - 1) - a:} - 1] - 1.
16. 1 - 12 + [- 3 - (- 4 - 5-6) - 7]^
17. a- \a-h[b - (a+ 6 + c-a + 6 + cZ)- c]}.
18. X - {2x^ + (3a^ - 3a: - [a: + x^]) + [2a: - (x^ + a^)]},
19. x' - [4a:3 _ [3^.2 _ ^2x + 2)] + 3a:] - [x' + (Sa^ -\-2x^
- 3a: - 1)].
20. - [- 2a: - {-. (- 2a: - 1) - 2a:i - 1] - 2a:.
21. X -[x + (a: - 2/) - {^ + (2/ - ^) - 22/} + 2/] - 2/ + ^.
1
50 SCHOOL ALGEBRA
22. 25a: - [12 + ^3a: - 7 - (- 12a: - 5 + 15a:) - (3 -f
2a:)}] + 7 - (3a: + 5) + (2a: - 3) + a: + 8.
23. In 3a: — (5a — 26 + c), what is the sign of 5a as the
example stands?
24. In Ex. 12, Exercise 10, indicate the subtraction by
use of a parenthesis. Do the same in Ex. 13.
Remove the parentheses and find the value of x in each
of the following:
25. X+ {x + 2) = 7. 27. 5a: - (2a: - 3) = 12.
26. 3a: - (a: + 2) = 8. 28. 4a: - (a: - |) = 2j.
29. Make up and work an example similar to Ex. 16. To
Ex. 26.
30. How many of the examples in this Exercise can you
work at sight?
31. How many of the examples in Exercise 3 (p. 19) can
you work at sight?
41. Insertion of a Parenthesis. It is clear that the process
of removing a parenthesis may be reversed; that is, that
terms may be inclosed in a parenthesis.
Inverting the statements of Art. 39, we have
Terms may he inclosed in a parenthesis preceded by the plus
sign, provided the signs of the terms remain unchanged;
Terms may be inclosed in a parenthesis preceded by the minus
sign, provided the signs of the terms are changed.
Ex. a— b-\-c-\-d— e = a — b + (c -{- d — e),
or, =a— b — {— c— d-\-e) Ans.
USE OF THE PARENTHESIS 51
EXERCISE 12
In each of the following insert a parenthesis inclosing the
last three terms, each parenthesis to be preceded by a minus
sign. Check the work either by removing the parenthesis in
the answer, or by numerical substitution.
1. 0^3 - 3a:2 _|_ 3^ _ 1^ 5 ^4 _|_ 4^ _ ^2 _ ^
2. a-b + c-\-d. 6. a262 - 2cd - c^ - (P.
3. 1 + 2a - a2 - 1. 7. ^x^ - 9a:2 + 12xy - Ay\
4. 1 - a2 - 2ab - h\ q. a:^ - 4a^ + 42:^ + 4a: - 4 - r^.
It is often useful to collect the coefficients of a letter into
a single coefficient.
Ex. Collect the coefficients of x, y, and z in the expression,
3x — 4y -^ 5z — ax — by — cz — hx -\- ay -i- az.
The complete coefficient of a: is (3 —a — 6) ; of 2/, ( — 4 — b + a)
or — (4 H- 6 — a) ; of 2;, (5 — c -}- a).
Hence, the expression may be written,
(3 — a — b)x — (4 + 6 — a)y + (5 — c + a)z Ans.
In like manner collect the coefficients of x, y, and z:
9. mx — ny -{- 3z + 2x -{- nz — 4cy.
10. X — y — 2z — ax -\- by — az — bx — ay -\- cz.
11. - 7x + 12y - lOz - 2ax + 36^ - cy -{- 2bx - 6dy .
12. ^y — Sacx — 5cdz — 4abx — 3cdy + 2cx — 4:Z — bax.
Collect the coefficients of :x?, x^, and x\
13. ?>d? -\- X — 2x^ — ao:^ — 5 + aa:^ — 2ax — c:i? — cx^ —ex,
14. — x^ — x — ax^ -\- :>? — ax -\-bx^ — ax? — 3bx — 2ba^-\- 3a,
15. a^x^ — Qx — a — Wx? — 26V + Zbx — aV — cx^ -{-
3cx — c.
52 SCHOOL ALGEBRA
Equations and Transposition
42. An Equation is a statement of the equality of two
algebraic expressions.
An equation, therefore, consists of the sign of equality and
an algebraic expression on each side of it; as 3a: — 1 = 2a; + 5.
The solution of an equation is the process of finding the
value of the unknown number (as of x) in the given equation.
43. Members of an Equation. The algebraic expression
to the left of the sign of equality is called the first member of
the equation; the expression to the right of the sign of equality
is called the second member.
Thus, in the equation 3x — 1 = 2a; + 3,
the first member is 3a; — 1; the second member is 2x + 3.
The members of an equation are sometimes called sides of
the equation.
The members of an equation are similar to the pans of a set of
weighing scales which must be kept balanced. (See Art. 15, p. 18.)
44. Utility of Equations. An equation expresses the re-
lation of at least one unknown quantity to certain known
quantities. By means of an equation, we are often able to
determine the value of the unknown quantity.
See the problems solved in Exercises 1, 2, etc., by the aid of
equations.
I 45. The Transposition of a Term is moving the term from
one member of an equation to the other member. We shall
see that when a term is transposed, the sign of the term must
be changed.
Ex. 1. Find the value of a; in x— 5 = 7.
I
i
r
EQUATIONS
i
PROCESS
WITHOUT TRANSPOSITION
We have
given
x-5 =
5 =
7
5
53
I
Adding 5 to each of the equals, x = 7 + 5
or a; = 12 Ans.
(Art. 15, 3)
PROCESS WITH TRANSPOSITION
We have given x—5= 7
Transferring 5 to the right-hand^ x = 7+5
member of the equation and^ a; = 12 Ans.
changing its sign, J
Hence transposition is a short way of adding equal num-
bers to the two members of an equation. The labor saved
by means of transposition is more evident when several terms
are to be transposed at the same time.
For the present, however, in order to fix firmly in mind the na-
ture of the process, we shall not transpose terms, but shall add
equals to the members of an equation when we wish to transfer
terms from one member to the other.
Ex. 2. Solve 5x-(x + 2) = 3x - {2x - 7). //-^ V
Removing parentheses, 5x—x — 2 = 3a;— 2a; + 7
Adding - 3x + 2x + 2 to j _ 3;^ + 2x + 2 = - 3x + 2x + 2
each member, )
5a; - a; - 3a; + 2a; = 2 + 7
3a; = 9
a; = 3 Ans.
46. Checking the Solution of an Equation. The result
obtained by solving an equation may be checked by substi-
tuting in each member of the original equation the value of
X obtained by the solution. If the two members reduce to
the same number, the value found for x is correct.
Thus, in Ex. 2, putting 3 in the place of x,
The left member, 5a; - (a; + 2) = 15 - (3 + 2) = 15 - 5 = 10
Also the right member, 3a; - (2a; - 7) = 9 - (6 - 7) =9 + 1 =10
54 SCHOOL ALGEBRA
EXERCISE 13
Solve the following equations without transposition o{
terms. Verify each result obtained.
1.
o: + 2x - 3 = 6.
6. 3a: - 2 - 2a: + .74.
2.
3a: = a: + 10.
7. 5a: - (2a: - 3) = 6.
3.
5a: - 1 = 14.
8. 7a: - {5x + 4) = -2.
4.
4a: - 3 = 12 - a:.
9. 9a: = 10 - (a: + 5).
5.
5a: - 1 = 3a: + 7.
10. 8a: + (3a: - 4) = 25.
11. 10a: - (x ■
- 5) = 4 - (a: + 2).
12. 10 - (3a:
- 5) = 8 - (7:c + 2).
13. Solve Exs. 1-12 by aid of transposition of terms.
Solve the following problems and cheek each result:
14. If 5 times x equals 9 diminished by twice a:, find x,
15. If f of a: equals 12 less Ja;, find x.
16. If 12 is added to a given number, the result equals
three times the given number. Find the number.
17. One number exceeds another by 5 and the sum of the
numbers is 12. Find the numbers.
18. The difference of two numbers is 5 and the sum of the
numbers is 13. Find the numbers.
19. Separate 12 into two parts such that one part exceeds
the other by 5.
20. One number exceeds another by 1.4 and the sum of
the numbers is 16.4. Find the numbers.
21. The difference of two numbers is 1.4 and the sum of
the numbers is 16.4. Find the numbers.
EQUATIONS 55
22. Separate 16.4 into two parts such that one part ex-
ceeds the other by 1.4.
23. Make up and work three examples similar to Exs.
14-16. 'Also to Exs. 17-20.
24. Find three consecutive odd numbers whose sum is 45.
Also five consecutive odd numbers whose sum is 45.
25. Find three consecutive even numbers whose sum is 60.
Also five consecutive even numbers whose sum is 60.
26. Make up and work an example similar to Ex. 24.
27. Make up and work an example similar to Ex. 25.
28. To what type does each of the above problems belong,
or of what type is each a variation?
EXERCISE 14
Review
1. Find the value of a + 3(6 — a;), when a = 5, 6 = 2, and
x= I.
2. Find the value of Sx - {x — 2)2 + 2(a; + 1) (4 - a:) -
V^x + 1, when x = S.
3. li s = vt + igt^, find the value of s when v = 10, g = 32.16,
and t = 4.
4. lix = 3, find the value of ix'^. Also of {^xY.
Simplify:
5. )2x* -^5a^ - 3a;2 + 2x - 5 -\- 2oi^ - Sx* -2x + 2x^ - 2x + 2x^
-6 +,3.t2 + *4 -^x^ _[. 7_X_f- 2 + 3a:' + 2a;^ - 4x -2x^.
6. 3a/2~- 5Vs + 8 +6^- 2v^ - 7 + 3V3"- 4VT- 2.
Subtract:
7. 3x3 _2x^ -^5x -^ from 8x^ - x^ - 1.
8. 5x3 - 3^.2^ + yz from Sx^ + 7xy^ - y\
56 SCHOOL ALGEBRA
Simplify and collect:
9. Sx - { - 2x + [- 4x - {x - 2) - x] - x} - 1.
10. 9x-{-8x-[7x + i-ex + l) -dx]- 4a;} -(3a: + 1) - 2x.
Bracket coefficients of like powers of x:
11. x^ -x^+2-Sx*- ax^ + ax^ -cx^- 2ax^ + 3cx^ - 2cx^-5x\
12. 1 - X - x"^ - x^ +2a - 2ax + 2ax^ - 2ax^ - 3bx + Sbx"^
+ Sbx^ + ex.
Solve and give the reason for each step :
• 13. 3a; - 5 = X + 7. 15. 4a; + (a; - 1) = 3a; - (x +2).
14. 5a; - (a; - 4) = 16. 16. 3 - (a; - 2) = 7 - 5a;.
17. Subtract 5a;2 - 3aa; - 2a^ from - 3a;2 + 2ax^ - a\
18. Find the value of 5x^ — 3 (a — 2a;) + Sa^, when a = 4 and
a; = L
19. Add 5a;2 — 3aa; + 4a2, 5aa; — 3a;2 + a^, and 3aa; — x^ — 2ax.
20. Simphfy a;^ - [5aa; + {a^ - 2x^ - ax) - 3x^\ - 5a2. Test the
accuracy of your work by letting a = 1 and a; = 2.
21. Solve 5-a;=4-(7+ 3a;).
22. The land surface of the world is 51,240,000 square miles. If
the land area of the rest of the world is seven times that of North
America, find the area of North America.
23. Add ia;2 - |a;+ i, \x^ + ia; - J, and \x^ -Ja; + f.
24. Subtract \x^ — \x-\-\ from \x'^- \x - f .
25. Add .5a2-.15a +2.5, 1.2a2 +.3a-1.5, and-.75a2 + .3a-.7.
26. Subtract .27a2 - .12a - 2.3 from l.Sa^ + 2a - 1.7.
27. Add 2(a; + 2/) - 3(a;+ z) + 2fo + z), 4(a; + 2) - 3(a; + y)
- 5(2/ + z), and 4(a; + ?/) - (a; + 2) + 4 (?/ + z).
28. From the sum of a^ - 7ah + 36^ and 2a2 - 662 + 7a'^b^, take
the sum of 4:a%^ - 3a^ + 2a2 - b^ and 3a6 - 2¥ + a^.
29. What must be added to x" — a; + 1 that the sum may be a;^?
That the sum may be 3a;? 15? 0?
30. What must be subtracted from 2a;2 — 3a; + 1 that the re-
mainder may be a;^? x^ + 10? 7? a - a; + 1?
EQUATIONS 57
^P B =4x^ - x^y - xy^ - Zy\ D = x^ - 2xy^ + y\
find the value of
31. A - 5 + C - D 33. A - (5 + C) + Z)
32. A -[B -{D -{■ C)] 34. B + \A -[C - D]]
35. By a diagram show that — 7 — ( — 3) and —7+3 have
the same value.
36. In an election for two candidates, 32,544 votes were cast.
The successful candidate had a majority of 2416 votes. How many
votes did each candidate receive?
37. The Panama Canal is 49 miles long and the part of it through
the lowlands is 4 miles more than 8 times the part through the hills
(called the Culebra Cut). How long is each part?
38. How many examples in Exercise 2 (p. 13) can you now work
at sight?
CHAPTER IV
MULTIPLICATION
47. Multiplication, at the outset, may be regarded as the
process of finding the result (called the product) of taking one
quantity (the multiplicand) as many times as there are units
in another quantity (the multiplier).
The term multiplication has acquired a much broader
meaning than this, which is sometimes expressed as follows:
Multiplication is the process of finding a number (the
product) which is obtained from a given number (the multi-
plicand) in the same way that another number (the multiplier)
is obtained from unity.
Multiplication is useful as a means of shortening addition
or subtraction. Later many other uses (often indirect) of
multiplication will become evident.
Multiplication of Monomials
48. Multiplication of CoeflBicients. To multiply 4a by 36,
we evidently take the product of all the factors of the
multiplier and the multiplicand, and thus get 4 X a X 3 X 6.
Rearranging factors, we obtain as the product,
4X3XaX6or 12ab.
Hence, in multiplying two monomials.
Multiply the coefficients to prodvxie the coefficient of the
product,
58
MULTIPLICATION OF MONOMIALS 59
49. Multiplication of Literal Factors or Law of Exponents.
Ex. Multiply a^ by a\
Since a^ = a X a X a
and a^ = a X a
: . a? X a^ = a X a X a X a X a '= a\
This may be expressed in the form
a^ Xa? = a^'''^ = a^,
or, in general, a"* X a" = a"* "*" ",
;vhere m and n are positive whole numbers.
Hence, in multiplying the literal factors of a monomial,
Add the exponents of each letter that occurs in both multiplier
ind multiplicand.
Ex. 4a26c3 X U^h^x = l2aW(^x.
50. Law of Signs. The law of signs in multiplication
"ollows directly from the general law of signs as stated in
irt. 31.
(1) + $100 taken 5 times gives + $500,
)r, in general, a + quantity taken a + number of times gives
a + result.
(2) $100 of debts, that is, - $100, taken 5 times gives - $500,
)r, in general, a — quantity taken a + number of times, gives
a — quantity as a result.
(3) $100 deducted 5 times, or $100 X - 5, gives as the total
amount of deduction — $500,
)r, in general, a + quantity taken a — number of times, gives
a — quantity as a result.
(4) Deducting $100 of debts 5 times from a man's possessions
is the same as adding $500 to his assets; that is,
- $100 X - 5 = + $500,
ir, in general, a — quantity taken a — number of times gives
a + quantity as a result.
60 SCHOOL ALGEBRA
We see from (1) and (4) that
either + X +, or — X — , gives +,
and from (2) and (3), that
either — X +, or + X — , gives — .
In brief, in multipHcation
lAhe signs give plus; unlike signs give minu^.
51. Multiplication of Monomials. Combining the results
of Arts. 48, 49, and 50, we may express the method of multi-
plying one monomial by another as follows :
Multiply the coefficients together for a new coefficient;
Annex the literal factors, giving each factor an exponent
equal to the sum of its exponents in the terms multiplied
together;
Determine the sign of the result by the rule that like signs
give +, and unlike signs give — .
Ex. 1. Multiply 5a^b3^ by - 6a¥y\
The product is - 30 a^¥x^y'^.
Ex. 2. Multiply 5a"+3 by 2a"-^
Since n + 3 and n — 1, added, give 2n + 2,
the product is 10a2"+2.
EXERCISE
15
1.
2.
3.
4.
5.
6.
Multiply
-5
-3a
Zah
30a:y
4a:
-5a:
by
4
-2
-5
- 1
-2x
-3a:
7.
8.
9.
10.
11.
12.
Multiply
?>ax
-^xf
lax
- 5a26
6c2^ -
- 2x^yz
by
— 4:ax
-Ixf
-Zay
-4c^2
-3c^2 _
- 8a:2/V
I
MULTIPLICATION OF MONOMIALS 61
i^
13.
14. 15.
16.
17.
18.
Multiply
4x^
^aa^ .5x
2.12/3
2|x3
ix^
by
.2x3
ia'a^ mx
.052/2
io:
.5x
19.
20.
21.
22.
23.
Multiply
2n-l
2n-l
2n-l
a;"~^
a;»-2
by
22 ' 23
Exs. 19-21 when n =
2
4. Also
x^
0^3
Verify
Exs. 22
and 23
when n =
4 and x
= 3.
24.
25.
26.
27.
28.
Multiply
a^x^-^
a^x--'
a2a:"-3
- a'^x--^
^2n
by 0^3 _ (j^^4 _ ^3^ aa:"+^ a:3n
29. 30. 31. 32.
5(a + 6)3 3(a-j-by - 6(a + 6) . 7(a + by-^
2{a + 6)2 - (a + 6) - 2(a + 6)3 3(a + bf
33. Multiply 2 "-2 by 2 and verify the result when n = 4:,
■ 34. Write out all the factors of 7a\ Of (7a)\
35. (aby is how many times as large as ab'^ when a = 3
and 6 = 2?
36. How many ic's are there in the product of 5aa:3 j^y
6a2x^? How many a's?
37. How much money do five empty pocket-books'contain?
5X0=?
38. Find the value of 7 times 0. Of 5a X 0. Of 0 x'^x^.
Of 3(x + 2/) X 0.
If a = 4, 6 = f , c = 0, .T = 1, and y = 9, find the value of
39. abc. ,.. 41 5cx2/2. 43. 4c2 + a.
40. a^c. 42. a2 + Scy, 44. (3c + a;)3.
62 SCHOOL ALGEBRA
ac + y 2a + c{x + y)
45. -3-. 47. 1 .
^^ 5ac^ + l ^g 5(a; - 1) + 8
a: -f y * * 2a
49. How many of the examples in this Exercise can you
work at sight?
50. How many examples in Exercise 3 (p. 19) can you
now work at sight?
Multiplication of a Polynomial by a Monomial
52. Utility of the Distributive Law in Multiplication;
Rule. In arithmetic we have become familiar with the fact
that, for instance, 5X67 = 5(60 + 7)= 5X60 + 5X7; and
that this principle enables us to perform all multiplications
by committing to memory only the products up to 9 X 9.
Similarly, in algebra, a(6 + c) = ah +^c. This is called
the Distributive Law of Multiplication. By use of this law,
all multiplications in algebra can be performed as a multi-
plication of pairs of monomials.
Hence, to multiply any polynomial by a monomial.
Multiply each term of the multiplicand by the multiplier,
and set down the results as a new polynomial.
Ex. Multiply 2a^ - 5a^b + 3a¥ by - ^ab\
2a3 - 5a^b + Sab^ =4
-Sab^ = - 12
Product - 6a*b^ + I5a^b^ - Qa'b* = - 48
The check is obtained by letting a = 1, and 6=2.
EXERCISE 16
1. 2. 3. 4.
Multiply 2a -\- 3x 3x - 2y 4x^y - xy^ 7 ax - 4by
by Sax — 5xy 2xy — 3abxy
MULTIPLICATION OF A POLYNOMIAL 63
Multiply:
5. Sac^ — 3m^n by 5an. 9. Sx""*"^ + 7a:" by — 4x.
6. m — m^ — 3m^ by — 1m?n. 10. 3a:"-^ + Sa;""^ by 7?,
7. 8x^2/ — 5a:?/2 ^ ^\^y 3^2/. 11. 3a:^« + ^7?" by a:*".
2a:" - 3a:"~^ by 7?, 12. 2a^» - Ta^" by - 2a2».
13.
14.
Multiply 2.5a:2 - 3.7a: + .51
i^ - ^^ - i
by Ax
i^
15.
16.
la,? - \ax ~ |a
Ax-^3?-l3?+ .5a;*
-^ax
-.25x'
17. What is the value of 7a: — by times zero?
Multiply:
18. 5(a + hf - 3(a + 6) - 5 by 2(a + 6).
19. 7(a: - yf + 2(a: - i/) - 6 by 3(a: - y)\
20. 2(3a + 26)2 _ 5(3^ _|. 26) + 4 by 4(3a + 26).
21. Reduce {7aaabb — 5aaabb) X 6aa66 to its simplest
form. Compare the size of the result with that of the original
expression.
(The following problems are mixed variations of Types I,
n, and III.)
22. What number diminished by 19 equals 37?
23. What number increased by 19 equals 37?
24. What number diminished by 1.067 equals 4.5?
25. What number increased by twice itself and then by
24 equals 144? '
64 SCHOOL ALGEBRA
26. What number increased by twice itself and then
diminished by 24 equals 144?
27. What number increased by J of itself and then by 20
equals 60?
28. What number diminished by J of itself and then
increased by 30 equals 90?
29. What number of dollars diminished by i of itself and
then by $30 equals $160.60?
30. If a number is multiplied by 3 and then diminished
by 40, the result is 140. Find the number.
31. If 5 times a certain number is increased by 20.5, the
result is 870. Find the number.
32. If five times a certain number is increased by 20.5, the
result is equal to three times the number increased by 160.
Find the number.
33. A man who died left $16,000 to his son and daughter.
The share of his daughter, who had taken care of him in his
illness, was $500 less than twice the share of the son.
How much did each receive?
34. A cubic foot of iron and a cubic foot of aluminum
together weigh 618 lb. If the weight of the iron is 14 lb. less
than three times the weight of the aluminum, find the weight
of each.
35. A baseball nine has played 54 games, and the number
of games it has won is 3 less than twice the number it has
lost. How many has it lost?
36. Of which type is each of the above problems (Exs.
22-35) an instance or a variation?
I
MULTIPLICATION OF A POLYNOMIAL 65
Multiply each member of the following equalities by — 1
and solve:
i^
37. — 2x — 5=— x-\-4:. 38. — 4a: — a; = — 6 — 9.
9. Make up and work an example similar to Ex. 10. To
Ex. 19.
40. Make up and work an example similar to Ex. 30. To
Ex. 35.
41. How many of the examples in this Exercise can you
work at sight?
Multiplication of a Polynomial by a Polynomial
53. Arranging the Terms of a Polynomial. The multi-
plication of polynomials is greatly facilitated by arranging
the terms in each polynomial according to the powers of some
letter, in either the ascending or descending order.
Thus, 5x^ +S — X + X* - 7x^, arranged according to the as-
ccDding powers of x, becomes
S -X +5x'^ -7x^ + x\
Also, a'* + 6'* — \.a^\P- — 5a^b, arranged according to the descend-
ing powers of a, becomes
a* - 5a'b - ^a%^ + ¥.
54. Multiplication of Polynomials. By a double use of
the Distributive Law:
(a + b)(c + d) = a(c -i- d) + b{c + d)
= ac -\- ad -{- be -\- bd
We see that a similar result is obtained, no matter how
many terms occur in each polynomial.
Therefore, to multiply two polynomials, ,u
66 SCHOOL ALGEBRA
Arrange the terms of the multiplier and the multiplicand
according to the ascending or descending powers of the same
letter;
Multiply each term of the multiplicand by each term of the
multiplier;
Add the partial products thus obtained.
Ex. 1. Multiply 2x - 3y by 3a: + 5y.
The terms as given are arranged in order.
The most convenient way of adding partial products is to set
own similar terms in columns, thus:
2x - Sy = - 1
Sx + 5y = 8
1 6x^ Qxy
Partial products J _^ ^^^^ _ ^^^^
Product 6^2 + xy - 15y^ = - 8
The check is obtained by letting x = 1 and y = l(ora: = y = 1).
Note that this method checks only the signs and coefficients, not
the letters or their exponents. Mistakes in letters and exponents,
however, are rare in comparison with mistakes in signs and coeffi-
cients. A convenient check for all elements in the process is ob-
tained by letting x = y = 2. A useful check on the letters and
exponents in many examples is given in Art. 56.
Ex. 2. Multiply 2a: - o:^ + 1 - Sa:^ by 2a; + 3 - x\
Arrange the terms in both polynomials according to the ascending
powers of x. (Why is the ascending order chosen rather than the
descending?)
1 +2x -Sx^ -
S -\-2x - x^
x^
-1
4
S -\-6x -9x^ -
+ 2a; + 4a;2 -
- a;2 -
Sx^
6x3 _ 2x^
2x' + Sx' + x^
Product 3 + 8a; - 6a;2 - lla;^ + x* -^ x^ = - 4
Now multiply the two polynomials together with their terms in
the order as first given. This will show you the advantage of ar-
ranging the terms in order before multiplying.
MULTIPLICATION OF A POLYNOMIAL 67
Ex. 3. Multiply a^-\-b^-\-c^ + 2ab-ac-bchy a-\-b-\- c.
Arranging the terms according to powers of a,
a'
a
+ 2ab
+ b
+
ac
c
+
62
- bc-\- &
= 3
= 3
a'
+ 2a26
+ a'b
+
aH
G?C
+ a62
+ 2a62
- abc + ac^
- abc
-h 2abc - ac^
+ ¥
-¥c
+ b^c
+ 6c2
-bc^ +(^
a' + Sa^b + 3a62 _j_ 53 + c^ = 9
55. Degree of a Term; Homogeneous Expressions. The
degree of a term is determined by the number of literal fac-
tors which the term contains. Hence, the degree of a term is
equal to the sum of the exponents of the literal factors in the
term.
Thus, 7a^bc'^ is a term of the 6th degree, since the sum of the
exponents in it is 3 + 1 + 2, or 6.
The degree of an algebraic expression is the same as the
degree of that term in the expression which has the highest
degree.
Thus, 7x^ + Sx^y"^ + ?/ is of the 4th degree.
A homogeneous polynomial is a polynomial of which all the
terms are of the same degree.
Thus, 5a^b — ¥ + ab^ is a homogeneous polynomial, since each
of its terms is of the 3d degree.
56. Multiplication of Homogeneous Polynomials. If two
monomials are multiplied together, the degree of the product
must equal the sum of the degrees of the multiplier and the
multiplicand.
For instance, in Ex. 3, above, the multiplicand is of the 2d
degree and the multiplier is of the 1st degree, and are both homo-
geneous. Their product is seen to be homogeneous and of the 3(X
4egree.
68 DURELL'S ALGEBRA: BOOK ONE
The fact that the product of two homogeneous expressions
mn^t also he homogeneous affords a partial test of the accuracy
of the work.
If, for instance, in Ex. 3, p. 67, a term of the 5th degree, such as
5a%^, had been obtained in the product, it would have been at once
evident that a mistake had been made in the work.
57. Detached Coefficients; Symmetrical Expressions. The
process of multiplying algebraic expressions may often be
further abbreviated by using only the signs and coefficients
of terms, omitting the letters and their exponents and thus
saving much of the work of ordinary multiplication.
EXERCISE 17
Multiply and check each result:
1. a: - 4 by 2a; + 1. 5. 7x^ - 5y^ by Ax^ + B?/^
2. a; - 3 by 3a: + 2. 6. 5xy + 6 by Qxy - 7.
3. 2a; + 5 by a; - 7. 7. ^a^ - b^c by Sa^c + 2a6V.
4. 3a; - 42/ by 4a; - 3y. 8. ll^y - 7xf by Sa;^ + 2y\
9. a^ — ab -{- ¥hy a + b.
10. ^ -{-x^y -[■ xy'^ -\- fhy x - y.
11. 4a;3 _ 3^2 _^ 2a; - 1 by 2a; + 1.
12. 2a;2 - 3xy + 2/ by 3a; - 5y.
13. a;^ - 3a;2 + 2a; - 1 by 2ar^ + a; - 3/
14. Sx^y - 4a;2/2 - i/^ by a;^ - 2xy - y\
15. a;^ — ^x^y + Sxy^ — fhy x^ — 2xy + y\
16. 4a;3 _ 33^ ^ 5^ _ 2 by a;2 + 3a; - 3.
17. a;^ - 3x2 + 5 by a;2 - a; - 4.
18. 3? -dxy -\- fhy a^ - Sxy - f.
I MULTIPLICATION OF A POLYNOMIAL 69
9. a" - ab + ¥ by 0" + ah + b\
0. 4a:2 + 91/2 - Qxy by ^x^ + V + 6a:y.
21. x' - rx'y^ + 6a;!/3 _ ^/^ by a:^ _ 2x1/2 _|_ ^^^
22. 3^ - 6aa:2 + 12a2a: - 8a^ by - a:^ - 4aa: - Aa\
23. a2 + 62 -j- ic2 + 2a6 - ax - 6x by a + & + a;.
24. ab + cd -\- ac -\- bd by a6 + cc? — ac - 6c?.
|H|5. ia + 46 by Ja - |6.
26. !a:2-4a: + ibyfa: + |.
27. .5a - .46 by .2a - .36.
28. 1.8a:2 - 3.2a; + .48 by 2.5a: + .5.
29. x^ + 2x»-i + 3a;"-2 - 2 by a; - 2.
30. a;"+^ - 3a;" + 4a;"-^ - Sa;"-^ by a;" + 2a;»-^
31. a;"-^ - 2a;"-^ + 3a;"-2 - 4a;"-^ + 5a;" by 2a,'2 + 3a; + 1.
32. Multiply 3a; - 5 + 4a;2 + a;3 by 2a; - 3 + a;2 without
changing the order of the terms. Now arrange the terms in
each expression in descending order and multiply. About
how much easier is the second process than the first?
33. Make up and work an example similar to Ex. 32.
Arrange the terms of the following in descending order of
some letter, and multiply:
34. 4.T - 3a;2 - 5 + 2a;3 by a; + 4.
35. 3a;2 — 5 — a; by a; + 4a;2 — 2.
36. 231^ -\- y^ — 4iXy^ + Sa^y by y^ + Soi^ — 2xy.
37. Which of the polynomials in Exs. 16-24 are homo-
geneous?
70 SCHOOL ALGEBRA
38. A number increased by 3 times itself and then by 40
equals 180. Find the number.
39. Separate 180 into two parts such that one part exceeds
three times the other by 40.
SuG. Let X = the second part.
40. A number increased by | of itself and then by 20
equals 95. Find the number.
41. Separate 95 into two parts such that one part exceeds
§ the other part by 20.
42. A number increased by f of itself and then diminished
by 30 equals 70. Find the number.
43. Separate 70 into two parts such that one part exceeds
f of the other part by 30.
44. A number din: inished by | of itself and then increased
by 30 equals 66. Find the number.
45. A number increased by .06 of itself and then by $100
equals $312. Find the number.
46. Separate 400 into two parts such that one part exceeds
3 times the other part by 60.
47. Separate $1000 into two parts such that one part is
smaller than 4 times the other part by $100.
48. Of which type is each of the above problems (Exs.
38-47) an instance or a variation?
58. Multiplication Indicated by the Parenthesis; Simpli-
fications. The parenthesis is useful in indicating multipK-
cations or combinations of multiplications.
Thus, (a - b + 2c)2 means that a - 6 + 2c is to be multiplied
by itself.
(a — b + 2c)' means that a — 6 + 2c is to be taken as a factor
three times and multiplied.
1
MULTIPLICATION OF A POLYNOMIAL il
o perform the multiplication expressed by a power is to
expand the power.
IBpAgain, (a — b) {a — 26) {a -\-h — c) means that the three factors,
a — b, a — 2b, and a -\- b — c, are all to be multiplied together.
Also, (a — 2xY — (a + 2x) (a — 2x) means that a + 2x is to be
multiplied by a — 2x, and the product is to be subtracted from the
m-oduct of a — 2a; by itself.
BPWe simplify an expression in which multiplications are
indicated by parentheses and exponents by performing the
operations indicated and collecting terms.
Simplify 3(0- - 2y) {x + 2y) - 2{x - 2y)\
\
Z{x - 2y) {x + 2y) - 2{x - 2yY
= 3(a;2 - 42/2) - 2{x'^ - 4xy + 4y^)
= 3x2 _ 12^2 _ (2a:2 - 8xy + Sy^)
= 3^2 - 12?/2 - 2a;2 + 8xy - 8y^
= a;2 + 8xy - 20y^ Ans.
Check this result by letting x = 1 and y = 2.
EXERCISE 18
Find the product of
1. (- a) (- a) (- «) (- «) (- G^).
2. (- 1) (- 1) (- 1) (- 1) (- 1) (- 1).
3. {x - y) {x - y) {x -\- y) {x - y) {x + y) in parenthe-
sis form.
4. Find the value of (- 2)\ Of a" when a = - 1 and
71 = 7.
Simplify by removing parentheses and collecting terms:
t-5. x-2{x-^ 1). Ue. (2.r + 3) {^x - 4).
6. {x - 2) {x + 1). 9. la - 3(4a - 8).
^ 7. 2t + 3(5a: - 4). lo. 9a + 5(3a + 4).
72 SCHOOL ALGEBRA
11. 3z(x - 2) - 2a:(a; - 3).
^ 12. (2a:2 - 3.T + 1)^
A 13. (2a - 36 + 5)2 - (2a + 36 - 5)2.
14. {x - 5)2 - (a; + 5)2.
15. 3a: - 2(3a:2 - 5x + 2).
16. X - 2(a: - 1) {x + 3).
17. (a: -2){x- 1) (a; + 3).
^18. 3a2 - (a - 26) (3a + 46).
19. {x — y — zY - x(x — 2y + 2z).
20. 2^2 - 3(x - 1)2 + {x - 2)2.
21. 3^2 - x{l - x)(2 + x) + a^.
22. 2 - 3(.r - 2)2 - 2(3 - 2x) (1 + x),
23. a2 — [x{a — x) — a{x ~ a)] — 3?.
24. {x -\){x-2)-{x- 2) (a; - 3) + (a: - 3) (a; - 4).
25. 3(a: - yf - 2\{x -\- yY - (x - y) {x + y)} + 2f,
26. x(x - y - z) - y{z - X - y) - z{z - y - x) - if.
27. 3[(a + 26)x + 2my] - 5[(m - c)y + 6a:] - 4[(a: - a)
a + c?/].
28. 26a6 - (9a - 86) (5a + 26) - (46 - 3a) (15a + 46).
29. Multiply the sum of (a — 2a:)2 and (2a — a;)2 by
3a - 2(a - x),
30. Subtract (x — 2yy from a:^ — 8?/^ and divide the re-
mainder by X — 2y.
31. Find the value of 3X0 + 4. Of 8-7X0.
Of 6 X 0 X 5 + 7.
I
MULTIPLICATION OF A POLYNOMIAL 73
If a = 3, 6 = 0, X = — 2, and ?/ = — 5, find the values of
32. 2ax, 36. by^ + 3x{x - y),
33. 6a:^2/. 37. 4a;2 — a6a:(4x — 2/).
34. Sx^ + a6i/. 38. 3a: - 3(2x + 3).
35. bxy — ax^. 39. 2{x^ + y) — ciby + aic^.
40. 2(1 - 2xy + (a: + 2/) (c^' + ^).
41. (a: - 1)2 - 3(0: + 1) (a: + 2) - xix" - 2) (y - 2x).
42. Sa(a - 2x) - {a - {a - 1) (x + 1) - (a + x)^ + 5ax.
Find the vahie of
43. (x -\- ay — (x — ay when x = 2a.
44. 5(a; + py — (x -\- p) {x — 2p) when x = 3p.
45. 3a:2 _j_ 4^ _ 5(^;j. _ ly when a: = ab.
46. If a: = 2 and 2/ = 1> find the value of (x + yY. Also
of a:^ + ?/^.
47. From the sum of 2a + 56 and 36 — 5a, subtract three
times a — 76, and verify the result when a = 2 and 6 = 5.
Also when a = 3 and 6 = — 1.
48. If a certain number is diminished by 24 and the result
multiplied by 3, the final result will be 78. Find the number.
49. If a certain sum of money is increased by $150 and
the result multiplied by 4, the final result will be $1000.
What is the original sum of money?
50. Separate $1000 into two parts such that one part
equals four times the sum of $150 and the other part.
51. Separate ..001 5 into two parts such that one part equals
3 times the sum of .0001 and the other part.
74 SCHOOL ALGEBRA
52. The sum of two fractions is If, and the larger is three
times the sum of the smaller and ^. Find the fractions.
53. Separate SI 00 into two parts such that the sum of one
part and $10 equals the other part.
54. Separate $100 into three parts such that 3 times the
sum of $5 and one of the parts equals each of the other parts.
55. Separate $100 into four parts such that twice the sum
of one part and $1 equals each of the other parts.
56. A man walked 15 miles, rode a certain distance, and
then took a boat for twice as far as he had previously trav-
eled. Altogether he went 120 miles. How far did he go by
boat?
57. The sum of three numbers is 50. The first number is
twice the second, and the third is 16 less than three times the
second. Find the numbers.
58. Find five consecutive numbers whose sum is 3 less than
6 times the least of the numbers.
59. The difference between two numbers is 6, and if 3 is
added to the larger, the sum will be double the less. Find the
numbers.
60. Divide $4500 among two sons and a daughter so that
each son gets $100 less than twice the daughter's share.
61. Find two numbers, whose difference is 14, such that
the greater exceeds twice the less by 3.
62. The difference of the squares of two consecutive num-
bers is 43. Find the numbers.
63. Three boys together earned $98. If the second earned
$11 more than the first, and the third $28 less than the other
two together, how much did each earn?
MULTIPLICATION OF A POLYNOMIAL
75
64. Which of Exs. 48-63 are instances or variations of
Type I? Of II? III?
65. Make up and work an example similar to Ex. 48. To
Ex.49.
66. Make up and work an example similar to Ex. 58. To
Ex. 62.
67. How many examples in Exercise 6 (p. 29) can you
m work at sight?
. CHAPTER V
DIVISION
59. Division is the process of finding one factor when the
product and the other factor are given.
The dividend is the product of the two factors, and hence
it is the quantity to be divided by the given factor.
The divisor is the given factor.
The quotient is the required factor.
Thus, to divide lOxy by 5x, we must find a quantity which,
multiplied by 5a:, will produce lOxy. The factor 5x is the divisor,
lOxy is the dividend, and the other factor, or required quotient, is
evidently 2y.
The division of a by 6 may be indicated in each of the
following ways:
h)a, a -^ b, 7, or a/b
60. General Principle. Division being the inverse of mul-
tiplication, the methods of division are obtained by inverting
thie processes used in multiplication.
Division of Monomials
61. Index Law for Division. If a^ is to be divided by a^,
we have
a^ aX a X aX-GrX-et-
Or, in general, — = a"*-",
(Ji
where m and n are positive whole numbers.
76
DIVISION OF MONOMIALS 77
62. The Law of Signs in Division is obtained by inverting
the processes of multiplication.
Thus, in multiplication, if a and h stand for any positive
quantities (see Art. 50, p. 59),
■\- aX -\-h = -{- ah]
-i-aX— b=— ab
^aX +b = - ab
- a X ~ b = -{- ab^
^+ab ^ -\-b = -}-a...(l)
- ab -^ -b = + a...{2)
- ab -ir +b = - a...{S)
^-a6-^ - 6 = - a...(4)
Hence, by
definition
of divi-
sion.
From (1) and (2) we see that the division of like signs
gives +. From (3) and (4) we see that the division of unlike
signs gives —. Hence, the law of signs is the same in divi-
sion as in multiplication.
63. Division of Monomials. Combining the results ob-
tained in Arts. 60, 61, and 62, we have the following method
for the division of one monomial by another:
Divide the coefficient of the dividend by the coefficient of the
divisor;
Obtain the exponent of each literal factor in the quotient by
subtracting the exponent of each letter in the divisor from the
exponent of the same letter in the dividend;
Determine the sign of the result by the rule that like signs give
plus, and unlike signs give minus.
Ex. 1. Divide 27a36V by - 9a%7?,
■ = — 3a63 Quotient
since the factor x^ in the divigor cancels x^ in the dividend.
Ex.2. Divide a2"-3 by-a--i.
-~zr = a"*"^ Quotient
Check the work In each of the above examples by multiplying the
quotient bv the divisor.
7S
SCHOOL ALGEBRA
EXERCISE 19
Di\dde and check the result:
1. 15a by — 5a. 10. — m^n by — m'.
2. - 3^:^ by a-. 2.1. - 3x2 by _ 1.
3. Sa^x^ by — 4aa-2. 12. — 8aar by Jar.
4. - 30a:^i/2 by _ Gor^i/. 4*3. 166/ by - f 6y.
5. - 7x:^ by 72:^. 14. 8ma: by .2x,
6. 2Lt?/22 by — 3xz. 15. .4aa:2 by .8ar^.
4.-J7. 186c^(f by — 9c^d. 16. .04aa; by .Sox.
UQ - 33.TY27 by lla-2/3^. 17. 2^3^ by f.x-l
9. 2Sx^y^^ by — Ma:?/^^^. is. — jx^ by .5a;.
19. 47rr2by27r. By r^. By vrr.
20. ig^hygt By ig. By i<.
21. JmiJ^ by fm. By .5v^, By .252?.
22. 20(x + 7/)3 by - 4(a: + y). By - 2(a: + 2/)'-
23. - 1.4(a - by by - 7(a - hf. By - 2(a - by.
^ 24. a^'' by a^". B?/ a^". — a".
25. - 6a"+3by2a
n+l
Bva
t+2
- 3a
n-l
,2n-3
26. a2"+5 by a"+^ By a
2n+3
27. How many 2's are multiplied together in 2^''? In 2*1
In the quotient of 2io -^ 2^?
28. How many a:'s in ic^°? In o:^? In the quotient
a:io ~ a:^
29. Divide 2""^ by 2 and verify your result when n = 5J
Treat 2'^~^ -^ 2^ in the same way.
DIVISION OF A POLYNOMIAL 79
30. If an empty box is divided by partitions into 5 equal
parts, will each compartment of the box be empty?
31. What is the value of 0 ^ 5? Of 0 ^ 7? State the
meaning of the latter in a manner similar to that used in
Ex. 30.
32. Give the value of 0 -^ 10. Of 0 -^ a. Of 0 ^ 2x,
0.0 0 0
Of
7a lah 7ahx la'h^T?
3aa:
33. What is the value of -— when a = 0? When a; = 0?
72/
If a = 2, 6 = 3, c = 0, a: = 1, find the value of each of the
following:
34. *-« 36. ^^?^^)
a 4a
35. '^^ + ^) 37. ^^'
h ' h + x
38. What is a polynomial? A binomial? A monomial?
Give two examples of each. .
Division of a Polynomial by a Monomial
64. Utility in the Distributive Law of Division; Rule. In
arithmetic we have become familiar with the fact that, for
instance,
65 _ 50 + 15 _ 50 , 15 _ .^ , ^ _ ,.
- - _ + _ _ 10 + 3 - Id;
and that this principle enables us to perform all divisions by
committing to memory only the quotients up to 81 -^ 9.
Similarly, in algebra, divisions can be greatly simplified by
the fact that
axf-\- he dc , he , ,
= 1 = a -j- b
c c c
80 SCHOOL ALGEBRA
This is called the Distributive Law of Division.
Hence, to divide a polynomial by a monomial,
Divide each term of the dividend by each term of the divisor,
I and connect the results by the proper signs.
Ex. 1. Divide 12a^x - lOa^y + 6aV by 2a\
i 2a^)12a^x - lOa^y + ea^z^
I 6ax — 5y + Sa^z^ Quotient
Ex. 2. Divide 6a^^+^ - 4a2n+2 _ 2a^n-s by 2a^-\
3a2»+4 _ 2a"+3_fl4n-2 Quotient
Check the work in each of the above examples by multiplying the
quotient by the divisor.
I
EXERCISE 20
Divide and check:
1. a:^ — Sa:^ by — x.
2. 20^2 - 8xy by 4a;.
3. 4a62 - Qa^bc by - 2ab.
4. - Sa:^ + 7a;2 _ a: by - x.
5. 15x^y — lOx^y^ — 5xy^ by 5xy,
6. — m — m^ -\- m^ — m^ by — m.
7. 14::ji?yh — 21a:i/V + xyz by — xyz.
a - 3a:2 - 2a; + 5 by - 1.
9. .^3? - .12a: + 9 by - .3.
10. .02a2 - Mah - .Sb^ by .5.
11. ix'' - ix - I by - |.
12. |a462 - iaW - Wb' by - f a&.
DIVISION OF A POLYNOMIAL 81
13. 9a:^" - 6a;2» + 12x« by - 3x\
14. - 4a:2n+l + lOx^n+2 _ Q^n+2 ^y 2x^\
15. a;»»+3 - 2a:»+2 _|- 3a;n+l _|. ^n ^y ^.n-l^
16. 8a:"»+2 _ i6a;"»+i - 4a;"» - 12a;^-^ by - 4a:"»^^
17. 9x2«-2 - 6a:2»-i ^ i2a:2" - 3a:2"+i by 3a;»-^
18. 10(a + 6)2 - 8(a + 6) by - 2(a + 6).
19. .5(a; - yY — ,lb{x — yf by .5(a: — yY,
20. (a + 6)a; — (a -f 6)?/ by (a + 6).
21. (a — h)x -\- {a — h)y by (a — 6).
22. x(.T + 1) + (a: 4- 1) by {x + 1).
23. \ of a number added to twice the number gives 210.
Find the number.
24. f of a number added to 5 times the number gives 340.
Find the number.
25. f of a number added to J the number gives 140. Find
the number.
26. The difference between f and § of a certain number is
14. Find the number.
27. What number increased by .06 of itself gives 318?
28. What sum of money at simple interest for one year at
6% will amount to $318?
29. What number increased by .15 of itself will amount to
690?
30. What sum of money at simple interest at 5% will
amount to $690 in 3 years?
31. For every nickel which a girl put in her savings bank
her father put in a dime. If her bank contained $18.75 at
82 SCHOOL ALGEBRA
the end of one year, how many nickels did the girl save in
that time?
32. For every dime that a boy spent for books, his father
gave him a quarter to spend for the same purpose. If he
spent $52.50 in all, how much did his father give him?
33. A purse contains $10.50 in dollar bills and quarters,
but there are twice as many quarters as bills. How many
are there of each?
34. How can $2.25 be paid in 5 and 10 cent pieces so that
the same number of each is used?
35. How can $5.95 be paid in dimes and quarters using
the same number of each?
In the following equations divide each member by — 1
and solve, checking each result:
36.^ - 1 - 3a: = - a: - 5 38. - x -{2x - 1) = - 5
37. -5x-8-x=-7x+l 39. - 7a: - 5 = - 3a: + 4
40. How many of Exs. 23-35, pp. 81-82, belong to Type
I? To Type II? III?
41. Make up and work an example similar to Ex. 31. To
Ex. 36.
42. Make up and work an example similar to Ex. 13. To
Ex. 18.
43. How many of the examples in this Exercise can you
work at sight?
Division of a Polynomial by a Polynomial
65. General Method. The method of dividing one poly-
nomial by another is to arrange the polynomials, according
to the ascending or descending powers of some one letter.
I
DIVISION OF A POLYNOMIAL 83
and then, in effect, to separate the dividend into par-
tial dividends, which are divided in succession by the
divisor.
Ex. 1. Divide Qx^ -j- 7a^ - 3x^ + Ux - 6 by 2x^ + 3a:-2.
We divide the first term of the dividend, 6x*, by the first term
of the divisor, 2x^, obtaining the quotient Sa:^. Multiplying this
quotient, Sx^, by the entire divisor, we obtain the first partial divi-
dend. If we subtract this from the entire dividend and divide the
remainder by 2x^y we have a process like the following:
Dividend Divisor
A , ^ ,
Qx^ + 7x^ - Sx^ + llx - Q\2x^ H-3a; -2= 15 -v- 3
Qx* + 9x3 _ 6a;2 3x2- xj-3= 5
- 2x3 ^ 3a;2 4_ 112; _ 6 ^~y~^
~2x3-3x^+ 2x «^^^^^
6x2 ^ Qx -Q
6x2 4, 93. _ 6
A quick check on the parts of the work in which errors are most
likely to be made is obtained by letting x = 1, as is done in the
solution above. A more complete check is obtained by finding the
product of the divisor and quotient and noting whether the result
equals the dividend.
Now state the process of dividing a polynomial by a
polynomial as a general rule.
Ex. 2. Divide Slah^ - 206^ - lOa'b^ + Qa^ - a^b by
3a2 - 562 ^ 4^5^
6a* - a% - 10a%^ -f Slab^ - 206^3^2 + 4a6 -562 =6^-2
6a* + Sa% - 10a262 2a^ - Sab + 462 = 3
- 9a36 + 31a6'
- 9o36 - 12^262 + I5a63
+ 12a262 + 16a63 - 206*
+ 12a262 + 16a63 - 206*
Is the dividend homogeneous? The divisor? The quotient?
84 SCHOOL ALGEBRA Jj
Ex. 3. Divide x^ + 2/^ + s^ + Sx^y + Sxy^ hy x + y + z.
Arranging terms according to the descending powers of x,
x' + Sxhj + Sxy^ + 2/3 + z^x +y -{-z =9-5-3
^ + x^y + xH x^ + 2xy - xz + y^ + z^ -yz -• 3
+ 2xh/ - x^z + Sxy^ + 2/' + 2'
-f 2x^ + 2xy^ + 2xyz
- x-'z + xy^
- xH - xz^
- 2xyz + 7/ + 2'
- xyz
xy^ + xz^
xt
- xyz +2/^+2'
+ y^+ y^z
•Vxz^
- - +xz^
- xyz -y^z +z^
- xyz - yH - 2/22
- xyz - y'^z - yz^
The process of algebraic division may often be abbreviated
by the use of detached coefficients. (See Appendix, p. 466.)
EXERCISE 21
Divide and check each result:
1. 3r^ + 7x + 2bya; + 2.
2. 6a:2 + 7a: + 2by3a; + 2.
3. 12x2 -i-xy - 20?/2 by 3a: -f 4?/.
4. 3a:2 + X - 14 by a; - 2.
5. 6a:2 _ 31^2/ + 352/ by 2a; - 7y,
6. 12a2 - 1 lac - 36c2 by 4a - 9c.
7. - 15a:2 -f 59a; - 56 by 3a; - 7.
8. 44a;2 — xy — 3y^hy llx — Sy.
9. a2 - 462 by a - 26. 12. 9a;2 - 49 by 3a; -f 7.
10. a^ — y^hy X — y. 13. 125 — 64a;^ by 5 — 4a;.
11. 27a;3 + 8 by 3a; + 2. 14. 8aV + / by 2aa; + 2/^.
DIVISION OF A POLYNOMIAL 85
15. 2a:3 __ 9^ ^ H^ _ 3 by 2a: - 3.
16. 353^ + 47a:2 + 13a: + 1 by ox + 1.
17. 6a3 - ITa^a: + Uax"^ - ^^ by 2a - 3a:.
18. iy" - W + 222/2 - 72/ + 5 by 2i/ - 5.
19. (^ + c*x + (^x'^ + c^a:^ _^ ^^4 _^ ^5 ^y ^ _|_ ^.^
20. 11a: - 8a:2 _|_ 5^ _ 20 + 2a:4 by a: + 4.
21. 4a: + 6a:^ + 3a:2 - lla:^ _ 4 ^y 80^2 _ 4.
22. - a:^2/ - lla:2/^ - 2a:y + Ga:* - ^y^ by 2a: - 3y.
23. 42/3 _^ 6^6 _ 13^4^^ by 3a:2 - 2z/.
24. a:^ - 16?/* by a: - 2y. 26. a:^ - / by a: + t/-
25. a:^ + 32?/5 by a: + 2?/. 27. 256a:8 - i/S by 4a:2 _ ^^2^
28. 9a: - ISa:^ + Sa:^ - ISa:^ + 2 by 4a:2 + x - 2.
29. 10 - x^ -21x^ + 12a:4 - 3a: by a: + 4a:2 - 2.
30. 22a:2 _ 13^ _|_ i^^ _ 13^4 _|_ 5^. _ g by a: + Sa:^ - 2.
31. 14a:V - 16^:^2/^ + 6a:^ + 2/^ + 5a:42/ " 6ar2/^ by 3a:2 _^ ^/^
-2a:2/.
32. ba'h - '^aW - aW + 3a^ - 4&^ by a" + 3a6 + 262.
33. a:^ — 2/^ + 2^ — 3:2/2 — 2a:22 + 2yz^ hy x — y — z,
34. c^ + ^3 + n^ — 3ccZn by c + ^ + n. ,•
35. 2/' -2f + l by 2/2 - 22/ + 1.
36. 2x« + 1 - 3x^ by 1 + 2a: + a:2.
37. Qxhf - 62/2^2 - 6a:2s2 - iSxyz^ - bxyh by ^xy + 2yz ;
+ 3a:z.
38. x^ - 39a: + 15 - 2a:3 by Zx^ j^ ^x -^ ;>? -\- 15.
ao 4^6 - ^a:* + 25 - 14a:3 _ ^.2 by 2a:3 _ ^ _ 5 _l_ 3^^^
86 SCHOOL ALGEBRA
40. Ja2 _ 152 by la + 16.
41. ia;2 - 1^2/^ by ^x - \y.
42. \a^ - jj¥ by ia - |6.
43. i:^ - -V-^ + f by ix - 2.
44. A6x^ — .25y^ by Ax + .5y.
45. 2.880:^ + 10.86a; - 19.2 by 1.2^^ + 1.5x-j- 6.4.
46. 6.t2«+i - 13a:2n ^ Q^r^"-! by 3a:"+i _ 2x».
47. 12.T^" + ISo:^" - a;" by 3.T" + 1.
48. 4a;»+3 + 5a:»+2 _ ^n+i _ a;n ^ ^.n-i j^y 3.2 _|_ 2^ + 1.
49. 6a-+i - 5a:" - 6a:"-i ^ i3a.n-2 _ g^n-s ^y 2.r2 - 3a: + 2.
50. In Ex. 20 try to divide without arranging the terms of
the dividend either in ascending or descending order.
51. What is the value of — ^ when a: = 0? When 2/ = 0?
x-\- y
52. Divide a:^ + 2/^ bya: — y to 5 terms and note the
remainder.
53. Divide 1 by 1 — a: to 4 terms and note the remainder.
54. Divide 1 by 1 — aa: to 3 terms.
55. If a boy walks at the rate of 3 miles an hour, how far
will he walk in 5 hours? In a hours? In x hours? In a: + 2
hours?
56. A boy starts at a given time and walks 5 hours. An-
other boy then starts and rides a bicycle x hours until he over-
takes the first boy. How many hours does the second boy
ride? How many if the first boy has a start of a hours? Of
y hours?
DIVISION OF A POLYNOMIAL 87
57. Two men A and B start from places 35 miles apart
and walk toward each other at the rate of 4 miles and 3
miles an hour respectively. How many hours will it be
before they meet?
SuG. In forming an equation, it is an aid to diagram a problem
of this kind : 35^1^
If the two men start j^(^ ^ )^
at the same time and ^^ |
walk toward each other
•intil they meet, they must travel the same number of hours.
Let X = the number of hours each man travels.
Then 4x = number of miles A travels.
3a; = number of miles B travels.
4x -^Sx = 35 (Art. 15, 1)
7x = 35
X = 5, no. hours before they meet.
Check. 4x = 20, distance A travels.
3a; = 15, distance B travels. ;
20 + 15 = 35 '
In working Exs. 58-72, draw a diagram as an aid in each
solution:
58. Two men, A and B, start from places 42 miles apart
and walk toward each other, at the rate of 4 and 3 miles per
hour respectively. How many hours will it be before they
meet?
59. Make up and work an example similar to Ex. 58.
60. Two bicyclists, A and B, start respectively from New
York and Philadelphia, 90 miles apart, and ride toward each
other. A rides 8, and B, 12 miles per hour. How long and
how far will A ride before meeting B?
61. Boston is 234 miles from New York. If two automo-
biles start from the two cities at the same time and travel
88 SCHOOL ALGEBRA
toward each other at the rate of 12 and 14 miles per hour
respectively, how far will each go before they meet?
62. Make up and work a similar example concerning trains
which travel between New York and Chicago, which are 912
miles apart.
63. One boy starts at a certain time from New York on
a bicycle and travels toward Philadelphia at the rate of 8
miles an hour. One hour later another boy starts from
Philadelphia and goes toward New York at the rate of 6
miles an hour. How long before they will meet?
64. New York and Washington are 228 miles apart. A
train starts from New York at a given time and goes at the
rate of 26 miles an hour, and two hours later a train starts
from Washington and proceeds at the rate of 34 miles an
hour. How long before they will meet?
65. Make up and work an example similar to Ex. 64 con-
cerning trains which travel between Cincinnati and New
Orleans, which are 830 miles apart,
66. Two boys start at the same place and travel in oppo-
site directions on bicycles at the rate of 8 miles and 10 miles
an hour. How long before they will be 108 miles apart?
67. If they travel in the same direction, how long before
they will be 16 miles apart?
68. Two boys start from New York and Philadelphia at
the same time and travel toward each other until they meet.
If one goes twice as fast as the other and they meet in 7^
hours, what is the rate per hour of each boy?
69. If in Ex. 68 one boy went 5 miles an hour faster than
the other, and they met in 6 hours, what was the rate of each?
REVIEW 89
70. Make up and work an example similar to Ex. 68
concerning automobiles traveling between New York and
Washington.
71. Make up and work an example similar to Ex. 69 con-
cerning railroad trains traveling between New York and
Buffalo, which are 440 mi. apart.
72. A set out from a town, P, to walk to Q, 45 miles distant,
an hour before B started from Q toward P. A walked at the
rate of 4 miles an hour, but rested 2 hours on the way; B
walked at the rate of 3 miles an hour. How many miles did
each travel before they met?
73. How many examples in Exercise 10 (p. 45) can you
now work at sight?
EXERCISE 22
Review
1. Express the following in as few terms as possible: 3.2x2 —
2.6xy + .16t/2 + 1.5x2 - .8y^ - .S2xy + Ay^ - l.bx^ + Axy.
2. Subtract .ISa^ + .362 _ 2.5a6 from - 7a? - 4ab - 1.5&2.
3. Add 2ip2 - l.5p + I, .75p2 + |p -'.4, | - ^If- - .5p.
4. Simplify 3.2^2 - [.8^2 + (3.5x - J - .2x2) - 1.5 - 3x].
5. Solve .3x - 4 = .2x + .5,
6- What is the root of an equation? How do you check your
solution of an equation? Check Ex. 5.
7. Simplify 5x - 3(x - 2) (x + 7) + 3(x - 2)2.
If a = 0, 6 = 1, c = 4, X = - 2, find the value of
8. a{b +c) - 3x.
9. (c + 2x) (b -a) -3(x + 4) (x + 5).
^Q 3a + 5(2 4- x)
b +c
90 SCHOOL ALGEBRA
11. Multiply a; - 2 + 4a;2 by 2^2 - 1 - 5a;.
12. Divide x* + S - Gx^ + x^ + 8x - llx' by 2x - x^ + 3.
13. Find three consecutive numbers whose sum is 33.
14. In a certain kind of concrete, twice as much sand is used as
cement, and twice as much gravel as sand. How many pounds of
each are used in making 2800 lb. of concrete?
15. The record time for the 100 yd. swim at a certain date was
55f sec. This was 7§ sec. more than 5 times that for the 100-yd.
dash. What was the record time for the latter?
16. Solve Ex. 15 without using x to represent the unknown
number. How much of the labor of writing out the solution is
saved by the use of x? Is there any other advantage in using x to
solve problems?
17. What is the dividend when the quotient is a;^ + 2x^ + 7x
+ 20, the remainder 62a; + 59, and the divisor a;^ — 2a; — 3?
18. What is the divisor if the quotient is x^ + 3a;, the dividend
x^ — 8, and the remainder 9a; — 8?
19. If a; = — J and y = — i, find the value of
(3a; - 2yy {9x^ + ^y^) - 6(y - x) VGxy{x + 2y^ + 4).
20. Add a to h. Also add 3a - 56 to 4c + 7d.
21. Subtract Sx — 2y + z from — 7. From — a. From h.
22. Subtract 2a - 36 from 0. Also 5 from 0.
23. Can 3 + 2a6 be united in a single term? Give a reason.
24. The product of an even number of negative factors has what
sign? Of an odd number of negative factors? Give an example
using not less than five factors.
25. Express the following in a simpler form: 6aaa{x — y) ix — y)
{x -y){x - y).
26. If a boy's mark on each of three recitations is 0, what is his
average on the recitations? Give the value of 0 + 0 + 0. Of 3 X 0.
0,|
27. Find the value of 8 X 0 - 5 + ^.
REVIEW 91
28. Form the power whose base is 5 and exponent 2. Also the
power whose base is 2 and exponent 5. Find the difference in value
between these two powers.
29. Find the value of each of the following products and verify
each result for the values x = 2, a = S, b = 1.
(1) a:«.a;2« (2) x'^ + K x"-^ (S)x^ + \x^-^
30. From the product of Sx^ — 2 and 2x — 5 subtract 7 times
the product of x and x — 2.
31. Show on squared paper that 3x4+5x4=8x4.
Also that 4X5+7X5-2X5=9X5.
32. Multiply 1^2 - ax -h fa^ by fx^ + ^ax + ia\
33. Divide x^" - 2/^" by x"" - ?/".
34. Divide 2 — a; by 1 + re to five terms in the quotient.
35. Divide [{x'' - 2x - 1) {x - 1) -{- 2x' - 2x\ by {{x + 2)
(a; + 1) - {x^ + 2a^ + 3)].
36. Multiply Z.2x^ - 4..bxy + l.Si/^ by 1.5a; - 3.52/.
37. Divide a;^ — 15 by a;^ + a; — 1 to five terms.
38. Divide 36^2 j^ W + \ - ^^V - 6a; + ly by 6x - Jy - i
39. Divide 2.4a;3 - {).l2x'^y + 4.322/^ by 1.5a; + l.St/.
40. Divide a? +h^ -\-(? - Sabc by a + 6 + c.
Solve and verify
41. (2a; + 1) (a; - 3) + 7 = a; - 2(a; - 4) (2 -x).
42. 7a; - 2(a; - 1) (2 - a;) - 17 = a;(3a; + 7) - (a; + l)'.
Simplify:
43. 62 + [40 - {8x - (22 + 4a;) - 22a;! - 7a;] - [7a; + [142 -
(42 -5a;)}]. -
44. a'^ih -c) - h\a - c) + c\a - b) - (a - b) (a - c) (b - c).
45. What is the advantage of regarding a polynomial as made up
of terms? (What- is a polynomial? A term?)
46. What is the name of an expression containing two terms?
Three terms?
47. Write a homogeneous expression containing three terms, and
*Jie letters x and y.
92 SCHOOL ALGEBRA
48. If s = ar"-*, find the value of s when a = 2, r = 3, n = 4.
Also when a = 2, r = 1, and n = 5. When a = 3, r = J, and n = 5.
49. Who first used the letters x, y, and z to represent unknown
numbers in equations in algebra? (See p. 455.) Find out all you
can about this man.
50. Give some of the other symbols that were used to represent
numbers before the use of the three last letters of the alphabet was
suggested.
51. Can you point out any advantages in the use of x, y, and z
instead of the other symbols once used for the same purpose?
52. Find out, if you can, whether any other symbols than the
last letters of the alphabet are now used to represent an unknown
number in an equation? How many dififerent symbols can be used
for this purpose?
53. Who invented the parenthesis sign and when?
54. How many examples in Exercise 13 (p. 54) can you work at
sight?
CHAPTER VI
EQUATIONS (continued)
66. The Equation, members of an equation, and transpo-
sition have already been explained. (See Arts. 42-45, pp.
52-53.)
67. A Root of an equation is a number which, when substi-
tuted for the unknown quantity, satisfies the equation; that
is, reduces the two members of the equation to the same
number.
Ex. If in the equation, 3a; — 1 = 2a: + 3,
we substitute 4 in the place of x in each member,
we obtain 3x - 1 = 12 - 1 = 11
2x +3=8+3 = 11
The equation is satisfied. Hence, 4 is the root of the given equation.
68. The Degree of an Equation having One Unknown
Quantity. If an equation contains only one unknown quan-
tity, the degree of the equation (after the equation has been
reduced to its simplest form) is determined by the highest
exponent of the unknown quantity in the equation.
Thus, if X is the only unknown,
2a; + 1 = 5a; — 8 is an equation of the first degree.
ax = b^ -\- ex is of the first degree.
4x^ — 5x =20 is of the seeond degree.
3x2 _ 3.3 = 6a; + 8 is of the third degree.
A simple equation is an equation of the first degree.
An equation of the first degree is also often termed a linear
equation, for reasons which will be explained later. (See Art. 148.)
93
94 SCHOOL ALGEBRA
69. Identities and Conditional Equations. If we take the
expression (x — 2) (x + 2) = x^ — 4, and substitute x^= 1,
we obtain — 3 = — 3. The two members of the expression
are found to be equal.
Similarly, they are found to be equal if we let x = 2, 3, 4,
etc.; 0,-1,-2, etc.; or any number. An expression hav-
ing this characteristic is termed an identity. I
An identity (or identical equation) is an equality whose
two members are equal for all values of the unknown quantity
(or quantities) contained in it.
A conditional equation is an equation which is true for
only one value (or a limited number of values) of x. For
the sake of brevity, a conditional equation is usually termed
an equation.
The equations studied in Art. 42 (p. 52) and Exercise 13 (p. 54)
are conditional equations.
Hence, the sign = is used in two senses in elementary
algebra, viz.: to indicate sometimes an equation, and some-
times an identity. The context enables us to decide readily
which of these two meanings the sign = has in any given case.
Later it will be found useful to use the mark = to indicate an
identity, and = to indicate a conditional equation, or equation
proper.
70. The Aids in Solving an Equation, given in Art. 15,
p. 18, stated more precisely, are as follows:
The roots of an equation are not changed if
1 . The same quantity is added to both members of the equation.
2. The same quantity is subtracted from both members of the
equation.
3. Both members are multiplied by the same quantity or equal
quantities (provided the multiplier is not zero, or an expression
containing the unknown).
EQUATIONS 95
4. Both members are divided by the same quantity (provided
the divisor is not zero, or an expression containing the
unknown).
Other principles similar to these are used later as aids in solving
equations.
Transposition (see Art. 45, p. 52) is a short way of using Prin-
ciples 1 and 2 of this article.
71. The Method of Solving a Simple Equation may now
be stated as follows:
Clear the equation of parentheses by performing the operations
indicated by them;
Transpose the unknown terms to the left-hand side of the
equation, the known terms to the right-hand side;
Collect terms;
Divide both members by the coefficient of the unknown quantity.
' Ex. Solve x{x - 2) = x(a; + 4) - 3(a: - 3) (1)
Removing parentheses, x^—2x= x^ +4a; —Sx+9
Transposing terms (Art. 70, 1, 2), x'^ -x"^ -2x -4tx-{-Zx =9
Collecting terms, —3a; =9. . . .(2)
Dividing by -3 (Art. 70, 4), a;= -3 Root
Check. x{x -2) = - 3( - 3 - 2) = - 3( - 5) =15
a:(x + 4) - 3(a; - 3) = - 3(- 3 + 4) - 3(- 3 - 3)
= -3 -3(-6) = -3 + 18 = 15
EXERCISE 23
Solve the following; refer to each principle in Art. 70 as
you use it, and check each answer:
1. 2a: = 15 - 3a:. 6. 3.T - 7 = 14 - 4a:.
. 2. 15 + 3a: = 27. • 7. 2a: - 7 = 8 + 5a:.
3. 4a: - 11 = 29. 8. 2a: - (a: - 1) = 5.
4. 16a: + 3 = 15a: + 7. 9. 2 ft. + a: = 12 ft.
5. 14a: - 10 = 12a: - 3a:. lo. 7 in. + a: = 2 ft.
96 SCHOOL ALGEBRA
11. x'-xix-hd) = x-{- 12. 13. 7(2 - 3x) = 2(7" - Sx).
12. 2a; - 3(a: - 3) + 2 = 0. i4. 3 - 2(3a: + 2) == 7.
15. (x - 8) (a; + 12) - (a: + 1) (x - 6) = 0.
16. 5(a; - 3) - 7(6 - a:) + 3 = 24 - 3(8 - x).
17. 3(a; - 1) (a; + 1) = a:(3a; + 4).
18. 4(a: - 3)2 = (2a: + 1)1
19. 8(a: - 3) - (6 - 2a:) = 2(a: + 2) - 5(5 - a:).
20. 5a: - (3a: - 7) - {4 - 2a: - (6a: - 3)} = 10.
21. a: + 2 - [.T - 8 - 2{8 - 3(5 - x) - a-}] = 0.
22. 2a:(a: - 5) - {x^ -\- (3a: - 2) (1 - a:)} = (2a: - 4)^.
23. Sx'' + 13a:-2{a:2-3 [(a:-l) (3 + a:)-2(a: + 2)2]} = 3,
24. .25a: - 2 = .2a: + 3. 26. fa: -f 6 = ^a: + 8.
25. 1.6x - .7 = 1.5a: - .3. 27. gx - | = | - fa:.
28. (.2a: + .2) (.4a: - .3) = (.4a: - .4) (.2a: + .3).
29. What right have we to change the equation 3a: = 15 —2a;
xo the form 3x -f 2a: = 15?
30. If 2x — 3 = 5, what right have we to transpose the
— 3, and to write the equation in the form 2a: = 5 + 3?
31. What is the advantage in being able to add the same
number to both members of an equation? To transpose a
term? To divide both members of an equation by the same
number?
32. Determine which of the following are identities and
which conditional equations, or equations proper:
(1) (x + 3) (a: - 3) = a:2 - 9.
(2) (a: + 3) (x - 3) = (a: + 1) (a: - 2).
(3) (a: + 2) (a: - 1) = a:2 + a: - 2.
(4) (a: + 2) (a: - 1) = x\
EQUATIONS 97
33. Write an identity and an equation of which the first
members are the same.
34. Prove that the sum of any three consecutive numbers
equals three times the middle one of the numbers.
SuG. Let the three numbers be indicated by n, n — 1, and n — 2.
35. Find a similar result for the sum of five consecutive
numbers. Of seven consecutive numbers.
36. Prove that the product of the sum and difference of
any two numbers is equal to the square of the first, minus
the square of the second. Illustrate by a numerical example.
SuG. Denote the two numbers by a and b.
37. Prove that the square of the sum of any two numbers
«^quals the square of the first number, plus twice the product
of the two numbers, plus the square of the second number.
Illustrate by a numerical example.
38. State and prove a similar property for the square of
the difference of two numbers.
39. Prove that if the sum of the cubes of two numbers is
divided by the sum of the numbers, the quotient equals the
square of the first number, minus the product of the first by
the second, plus the square of the second.
40. State and prove a similar property of the difference of
the cubes of two. numbers.
Find the value of the letter in each of the following:
41. 3a - 2 = 7. 44. 24 = 12 - 3p.
42. 5 - 26 = 1. 45. 3(2/ - 4) = 5(2 - y).
43. 5(c - 1) = 12 - c. 46. r - 3(r - 1) = 5.
47. In ^ = Iw, if A = 42 J and I = 8 J, find w,
48. If^ = 48.36 sq. ft. and w = 6.2 ft., find I
98 SCHOOL ALGEBRA
49. Convert each of the two preceding examples into an
example concerning areas.
50. If V = Iwh, and V = 504, I = 12, and ^ = 5, find w.
51. Convert Ex. 50 into an example concerning volumes.
52. In i = prt, if i = $27, r = .05, and p = $240, find t.
53. Convert Ex. 52 into an example concerning interest.
54. So far as we know, who first used an equation to solve
a problem? Give this first problem thus solved, and tell all
you know about the document in which it was found.
(See pp. 266 and 274.)
55. Form an equation whose root is 2 and which contains
four terms.
56. Make up and work an example similar to Ex. 15. To
Ex. 31. Ex. 46.
57. How many of the examples in this Exercise can you
work at sight?
72. Solution of Problems. In solving problems, the stu-
dent will find it necessary to study each problem carefully
by itself, as no rule or method can be found which will cover
all cases. The following general directions will, however, be
found of service:
By study of the problem, determine what are the unknown
quantities whose values are to be obtained;
Let X equal one of these unknown quantities;
State in terms of x all the other unknown quantities which are
either to be determined or to be used in the process of the solution;
Obtain an equation by the use of a principle (such as, the
whole is equal to the sum of its parts, or things equal to the
same things are equal to each other) ;
EQUATIONS 99
Solve the equation, and find the value of each of the unknown
quantities.
In solving problems it is especially important to note that
we let X = a definite number , not a vague quantity.
Thus, in working Ex. 1 of Exercise 24
we do not let x = A's marbles,
nor X = what A has,
but let X = number of marbles A has.
73. Checking the Solution of a Written Problem. The
best way of checking the result obtained by solving a prob-
lem is to observe whether the result obtained satisfies the
conditions as originally stated in the language of the problem.
(This method is better than that used in checking the example
in Art. 46, p. 53.)
Thus, to check Ex. 18, p. 54 : after the answers 9 and 4 have been
obtained, we note that the difference of 9 and 4 is 5, and that the
sum of 9 and 4 is 13. 9 and 4 thus satisfy the original conditions ot
the problem.
What is the advantage in this method of checking the solu-
tion of a problem?
EXERCISE 24.
# i^ Oral
1. A has X marbles, and B has twice as many. How many has B?
How many have both?
2. There are 100 pupils in a school, of which x are boys. How
many are girls?
3. If I have x dollars, and you have three dollars more than
twice as many, how many have you? How many have we together?
4. Two boys together solved a examples. One did x examples.
How many did the other solve?
5. The difference between two numbers is 15, and the less is x.
What is the greater? What is their sum?
100 SCHOOL ALGEBRA
6. If n is a whole number, what is the next larger number? The
next less?
7. Write three consecutive numbers, the least being x. Write
them if the greatest is y.
8. John has x dollars, and James has seven dollars less than
three times as many. How many has James?
9. If I am a: years old now, how old was I ten years ago? a years
ago? How old will I be in c years?
10. A man bought a horse for x dollars, and sold it so as to gain
a dollars. What did he receive for it?
11. A man sold a horse for $200, and lost x doUars. What did
the horse cost?
12. If a yard of cloth cost m dollars, what will x yards cost?
13. A boy rides a miles an hour; how far will he ride in c hours?
14. A bicycUst rides x yards in y seconds. How far will he ride
in one second? In n seconds?
15. In how many hours can a boy walk x miles at a miles an hour?
16. A man has a dollars and h quarters. How many cents has he?
17. How many dimes in x dollars and y half-dollars?
18. I have x dollars in my purse and y dimes in my pocket. If
I give away fifty cents, how much have I remaining?
19. By how much does 30 exceed xt
20. What number is 40 less than a;? What number is x less
than 40?
21. What number exceeds x by a? What number exceeds a by x?
22. By how Imuch does a -\-h exceed x'i
23. How much did a girl have left if she had $5 and spent 15ff?
If she had a dollars and spent h cents?
24. A boy had a dollars, received h cents, and then spent c cents.
How many cents did he have left?
25. What is the interest on a dollars at h per cent for c years?
26. Express algebraically the following statement: a divided by
6 gives c as a quotient and rf as a remainder.
I
EQUATIONS 101
27. A man having x hours at his disposal, rode a hours at the
rate of 8 miles an hour, and walked the rest of his time at the rate
of 3 miles an hour. How far did he ride? How far did he walk?
EXERCISE 25
l^ft 1. Separate $84 into two parts such that one part is three
^mes as large as the other.
2. Separate $84 into two parts such that one part exceeds
the other by $12.
3. Separate $84 into three parts such that the first part
is twice as large as the second, and the second part is twice as
large as the third.
4. A boy has three times as many marbles as his brother,
and together they have 48; how many has each?
5. A and B pay together $100 in taxes; if A pays $22 more
than B, what does each pay?
6. Two boys made $67.50 one summer by taking passengers
on a launch. The boy who owned the launch received twice
as large a share of the profits as the other boy. How much
did each receive?
7. How many grains of gold are there in a gold dollar, if
the gold dollar weighs 25.8 grains and 9 parts of the dollar
are gold and 1 part copper?
8. A ball nine has played 64 games and won 12 more than
it has lost. How many games has it won?
9. A man left $21,000 to his wife and four daughters. If
the wife received three times as much as each daughter, how
much did each receive?
10. If he had left $21,000 so that the wife received $10,000
more than each daughter, how much would each have
received?
102 SCHOOL ALGEBRA
11. A cubic foot of water and a cubic foot of alcohol to-
gether weigh 112.5 lb. The alcohol weighs 5 as much as the
water. What is the weight of a cubic foot of each?
12. Find three consecutive numbers whose sum is 63.
13. In a certain grade of milk the other solids equal three
times the weight of the butter fat, and the liquid part of the
milk weighs 7 times as much as the solids. How many pounds
of butter fat in 4800 lb. of milk?
14. The difference of the squares of two consecutive num-
bers is 43. Find the numbers.
15. At a certain date the record time for the quarter-mile
run was 47 seconds, and 5 times the record time for the 100-
yard dash exceeded the record time for the quarter-mile by
1 second. Find the record time for the 100-yard dash at this
date.
16. The difference of two numbers is 13 and their sum is
35. Find the numbers.
17. John solved a certain number of examples, and William
did 12 less than twice as many. Together they solved 96»
How many did each solve?
18. Three boys earned together $98. If the second earned
$11 more than the first, and the third $28 less than the other
two together, how many dollars did each earn?
19. The sum of two numbers is 92, and the larger is 3 less
than four times the less. Find the numbers.
20. The sum of three numbers is 50. The first is twice
the second, and the third is 16 less than three times the second.
Find the r umbers.
I
EQUATIONS 103
21. A farmer paid $94 for a horse and cow. What did
each cost, if the horse cost $13 more than twice as much as the
cow? X '\ '\ "^ -^ ■■■'^ - ?^/
22. Ex. 1 (p. 95) might be stated as a problem concerning
an unknown number, thus: Twice a certain number equals 15
less three times the number. Find the number.
In like manner, convert Ex. 2 (p. 95) into a problem con-
cerning an unknown number. Also Ex. 3. Ex. 8.
23. In reducing iron ore in a furnace, 7 times as many car-
loads of, coke as of limestone are used, and 8 times as many
carloads of iron ore as of limestone. If 800 carloads in all are
used on a certain day, how many carloads of each is this?
24. One side of a triangle is twice as long as the shortest
side. The third side exceeds the length of the shortest side
by 12 yards. If the perimeter of the triangle is 360 yards, find
each side.
25. A man spent $3.24 for coffee and sugar, buying the
same number of pounds of each. If the sugar cost 5 cents a
pound and the coffee 22 cents, how many pounds of each did
he buy?
26. The distance from New York to Chicago is 912 miles.
If this is 24 miles less than four times the distance from
New York to Boston, find the latter distance.
27. On a certain railroad in a given year the receipts per
mile were $3085. If the receipts per mile for freight exceeded
those for passengers by $265, find the receipts per mile from
each of these sources.
28. A man left $64,000 to his wife, daughter, and niece.
To his daughter he left $4000 more than to his niece, and to
his wife $8000 more than to his daughter and niece together.
How much did he leave to each?
104
SCHOOL ALGEBRA
a; + 5
29. Find the number whose double exceeds 24 by 6.
30. The perimeter of a given rectangle is 26 feet, and the
length of the rectangle exceeds the
width by 5 feet. Find the dimen-
sions of the rectangle.
31. The perimeter of a given
rectangle is 18 yards, and the
length exceeds the width by 3 ft. Find the dimensions.
Make up and work a similar example for yourself.
32". The length of a rectangle exceeds a side of a given
square by 3 inches and the width of the rectangle is 2 inches
less than a side of the
square. If the area of
the rectangle equals the ^
area of the square, find
a side of the square.
SuG. Denote the sides of the square and rectangle as in the
diagram.
Since the areas of the two figures are equal,
x^ = (x + 3) (x - 2), etc.
In working Exs. 33-38, draw a diagram for each example.
33. The length of a rectangle exceeds a side of a given
square by 8 ft. and the width of the rectangle is 4 ft. less than
a side of the square. If the area of the rectangle equals the
area of the square, find a side of the square.
34. If one side of a square is increased by 4 yd., and an
adjacent side by 3 yd., a rectangle is formed whose area ex-
ceeds that of the square by 47 sq. yd. Find a side of the
square.
35. The perimeter of a rectangle is 120 ft., and the rec-
tangle is twice as long as it is wide. Find its dimensions.
ic + S
EQUATIONS 105
36. A certain rectangle is three times as long as it is wide.
If 20 ft. is added to its length and 10 ft. is deducted from
its width, the area is diminished by 400 sq. ft. Find the
i dimensions of the rectangle.
IV 37. A rectangle is 5 ft. longer than it is wide. If its length
is increased by 4 ft., and its width by 3 ft., its area is in-
creased by 76 sq. ft. Find the dimensions of the rectangle.
38. A rectangle is 4 in. longer than it is wide. If its length
is increased by 4 in., and its width diminished by 2 in., its area
remains unchanged. Find the dimensions of the rectangle.
139. Make up and work an example similar to Ex. 38.
40. A tennis court is 42 ft. longer than it is wide. If a
argin of 15 ft. on each end and of 10 ft. on each side is
added, the area of the court is increased by 3240 sq. ft.
Find the dimensions of the court.
41. The length of a football field exceeds its width by 140
ft. If a margin of 20 ft. is added on each side and end of the
field, the area is increased by 20,000 sq. ft. Find the dimen-
sions of the field.
42. A boy is three times as old as his brother. Five years
hence he will be only twice as old. Find the present age of
each.
43. A man is twice as old as his brother. Five years ago
he was three times as old. Find the age of each at the present
time.
44. How many pounds of coffee at 30^ a pound must be
mixed with 12 pounds of coffee at 20f^ a pound to make a
mixture worth 24^ a pound?
45. How many pounds of tea at 60^ a pound must be
mixed with 25 lb. of tea at 40^ a pound, to make a mixture
worth 4i5^ a pound?
106 SCHOOL ALGEBRA
46. Make up and work an example similar to Ex. 45.
47. Find five consecutive numbers whose sum shall be 3
less than six times the least.
48. Find three consecutive odd numbers whose sum is 63.
49. A telegram at a 25-2 rate cost 47 cents. How many
words were in the telegram?
SuG. A 25-2 rate means a cost of 25 cents for the first 10 words
and 2 cents for each additional word.
50. Make up and work an example, similar to Ex. 49, con-
cerning a telegram sent at a 40-3 rate.
51. A talk over a long distance telephone at a 50-7 rate
cost 85f^. How many minutes did the talk last?
SuG. A 50-7 rate over a long distance telephone means a cost
of 50 cents for the first 3 minutes and 7 cents for each additional
minute.
52. A rectangle is 8 ft. longer than it is wide and the pe-
rimeter is 120 ft. Find the dimensions of the rectangle.
53. If 5 is subtracted from a certain number and the difTer-
ence is subtracted from 115, the result equals three times the
given number. Find the number.
54. If J is added to double a certain fraction, the result ii^
the same as if | had been subtracted from three times the
fraction. Find the fraction.
55. What number subtracted from 100 gives a result equal
to the sum of 14 and the number?
56. Find the number which exceeds 12 by as much as
three times the number exceeds 24.
57. Find five consecutive numbers such that the last is
twice the first.
EQUATIONS 107
58. Find two consecutive integers such that the first plus
5 times the second equals 53.
59. A man is 48 years old and his son is 18. How many
years ago was the father four times as old as the son? Also
how many years hence will the father be twice as old as the
son?
60. Find two numbers such that their difference is 20, and
one is four times as large as the other.
61. The length of a single tennis court exceeds the width
by 51 ft. If the width is increased by 9 ft., we have a double
court, the area of which exceeds that of the single court by
702 sq. ft. Find the dimensions of each court.
62. A boy sold a certain number of newspapers on Monday,
twice as many on Tuesday, on Wednesday 5 more than on
]\Ionday, and on Thursday 7 less than on Tuesday. If he
sold 310 newspapers on the four days, how many did he sell
on each of the days?
63. Twenty-five men agreed to pay equal amounts in
raising a certain sum of money. Five of them failed to pay
their subscriptions, and as a result each of the other twenty
had to pay one dollar more. How much did each man sub-
scribe originally?
64. A boy starts from a certain place and walks at the
rate of 3 miles an hour. Three hours later another boy starts
after the first boy and travels on a bicycle at the rate of 6
miles an hour. How many hours will it be before the second
boy overtakes the first? (Draw a diagram.)
65. If the boys had traveled in opposite directions, how
many hours after the second boy started would it have been
before they were 81 miles apart?
108 SCHOOL ALGEBRA
66. A boy was engaged to work 50 days at 75^ per day for
the days he worked, and to forfeit 25^!^ every day he was idle.
On settlement he received $25.50; how many days did he
work?
67. Which of the above problems belong to, or are varia-
tions of. Type I? Of Type II? III?
68. How many examples in Exercise 15 (p. 60) can you
now work at sight?
CHAPTER VII
ABBREVIATED MULTIPLICATION AND DIVISION
p Abbreviated Multiplication
74. Utility of Abbreviated Multiplication. In certain
cases of multiplication, by observing the character of the
expressions to be multiplied, it is possible to write out the
product at once, without the labor of the actual multiplica-
tion. This is true of almost all the multiplication of binomials,
and that of many trinomials, and by the use of the abbre-
viated methods at least three fourths of the labor of multi-
plication in such cases may be saved. The student should
therefore master these short methods as thoroughly as the
multiplication table in arithmetic.
75. I. Square of the Sum of Two Quantities.
Let a + 6 be the sum of any two algebraic quantities.
By actual multiplication, a -{- b
a + h
a^ 4- ab
a^ + 2ab + b^ Product
Or, in brief, (a + bf = a^ + 2ab + ¥,
which, stated in general language, is the rule:
The square of the sum of two quantities equals the square of
the first, plu^ twice the product of the first by the second, plus the
square of the second.
109
110 ' SCHOOL ALGEBRA
Ex. 1. (2x + 3yy = 4a:2 + 12xy + V Prodmjt
Ex. 2. 1042 = (100 + 4)2 = 1002 H- 8 X 100 + 42
= 10,000 + 800 + 16 = 10816 Ans.
76. n. Square of the DiiFerenee of Two Quantities.
By actual multiplication, a — b
a — b
€? — ab
-ab -\-b'
G? - 2ab -f- 62 Product
Or, in brief, (a - bf = a" - 2ab + b\
which, stated in general language, is the rule:
The square of the difference of two quantities equals the square
of the first, minu^ twice the product of the first by the second,
plus the square of the second.
Ex. 1. (2x- 3yf = 4a:2 - 12xy + 9y^ Product
Ex. 2. [{x + 2y)-3f = (x + 2yy- 10(a: + 2y) + 25
= a:2 + 4.T2/ + 4?/2 - 10.r - 2O2/ + 25
Product
To check the work of Ex. 2, let a; = 2, y = 1.
Then [{x^ + 2y) - 5]2 = (4 + 2 - 5)^ = (1)2 = 1
Also
x^ +4xy +^y^ - lOx -20?/ +25 =4+8+4 -20-20 +25 = 1.
EXERCISE 26
Write by inspection the value of each of the following and
check each result:
1. (n + yy 5. (5a: +1)2
2. {c-xY 6. (0:2+1)2
3. (2x - yf 7. {x — 2/2)2
4. (3a: - 2yf 8. (1 - Iff
I
ABBREVIATED MULTIPLICATION 111
9. (3x4 ^ 5^)2 13 (1,5^ _ .02)2
10. (GarV - lli/V)2 19. [(a + 6) + 4]^
11. (5a:» - 3i/"z-)2 20. [(a + 6) - 3]2
12. (4a:32^s2" + V")2 21. [(a + h) + c]^
13. (|a:^ + f2/)' 22. [(2a - .t) + 37/]2
14. (fa6 - |x2)2 23. [3 + (a + &)]'
-^05. (.2x + 2yf 24. [5a - (a: + 2/)]'
16. (.3a + .0462)2 25. [2a2 - (6 - 2c) ]2
17. (.02a: - myf 26. [{x + 2/) - (« + &)P
27. Find the value of 998^ by multiplying 998 by itself.
This product might also have been obtained in the following
U^ay:
9982 = (1000 -2)2 = [10002 - 2 x 2 X 1000 + 22]
= 1,000,000 - 4000 + 4 .
= 996,004
After practice the part of the work in the brackets may be omitted.
Compare the amount of work in the two processes of finding th?
value of 9982.
By the short method obtain the value of:
28. 9992 31. 512 34 9962
29. 9972 32. 10032 35. 99972
30. 99982 33. 972 36. (99.2)2
37. Make up and work an example similar to Ex. 19. To
Ex.29. Ex.36.
38. How many of the examples in this Exercise can you
answer orally?
112 SCHOOL ALGEBRA
77. III. Product of the Sum and Difference of Two
Quantities.
By actual multiplication, a -\- b
a — b
a^ + ab
-ab-W
q2 _ 52 Product
Or, in brief, (a + b) {a - b) = a^ - b\
which, stated in general language, is the rule:
The product of the sum and difference of two quantities equals
the square of the first minu^s the square of the second.
Ex. 1. (2x H- 32/) {2x - 3y) = 4a^ - 9y^ Product
Ex. 2. Multiply x -^ {a -^ b) hy x - {a -{- b).
We have
lx + {a+ b)] [x -(a+ b)] =x^ -{a+ by, by IIL
= a;2 _ (a2 + 2ab + ¥), by I.
= a;2 - a2 - 2ab - b^ Product
Let the pupil check the work.
It is frequently necessary to re-group the terms of trino-
mials in order that the multiplication may be performed by
the above method.
Ex. 3. Multiply x + y — zhyx — y + z.
{x +y - z) (x -y +z) = [x -{■ (y - z)][x - (y - z)]
= x^ -{y - z)\ by III.
= a^2 _ (^2 _ 2yz + z^), by II.
= a:2 - 2/2 + 2yz - z^ Product
Let the pupil check the work.
EXERCISE 27
Write by inspection the value of each of the following
products, and check the work for each result:
1. (x + z) (x - z) 3. (3a; - y) (Sx + y)
2. (2/-3)(y + 3) 4. (7a: + 41/) (7a: - 42/)
I
ABBREVIATED MULTIPLICATION 113
5. (x^ -2)(x'-h 2) 9. (ia + ib) iia - J6)
6. (ax' - b'y) (ax' + b'y) lo. (2iar - iy) (2ix + iy)
7. (1 - lla:^) (1 _^ iia:^) ^^^ (.2^ ^ 3^) ^2x - .S^/)
8. (2a;» + 5!/-) (2a:" - 52/-) 12. (.05a2-.363)(.05a2+.3fe3)
13. (fa: + .7y) (ix - ,7y)
14. (a»+^ + ^6-1) (a"+i - i6"-i)
15. [(a + 6) + 3] [(a + 6) - 3]
16. [(a: + 2/) + a] [(x + 2/) - a]
17. [(2a: - 1) + 2/] [(2a; - 1) - 2/]
18. [4 + (a: + 1)] [4 - (a: + 1)]
19. [2a: + (32/ - 5)] [2a: - (32/ - 5)]
20. (a + 6 + 3) (a + 6 - 3)
21. (a: + 2/ + a) (a: + 2/ - a)
22. (4 + a: + 1) (4 - a: - 1)
23. (2a: + 32/ - 5) (2a: - 32/ + 5)
24. (4 + a: + 2/) (4 - a: - 2/)
25. (a:2 + 3a: + 2) (a:^ + 3a: - 2)
26. (a 4- 6 + 3a:) (a + 6 - 3a:)
27. (a + 6 - 3a:) (a - 6 + 3a:)
28. (a:2 - a:2/ + y^) (x^ + xy -{- y^)
29. (a'' + a-{- 1) (a2 - a + 1)
30. (2a:2 - 3a: - 5) (2a:2 + 3a: - 5)
31. (2a:2 4- 5xy - if) (2^? - bxy - y^)
32. (x' + xy - 2/2) (x' - xy - y^)
33. l(a + b)-(c- 1)] [(a + 6) + (c - 1)]
34. [(a:2 + f) + (:r22/2 + 1)] [(x' + 2/^) - (0:2^2 _^ i)j
35. (a; + 1/ + z + 1) (a: + 2/ - z - 1)
U4 SCHOOL ALGEBRA
36. Work Ex. 16 in full (see Art. 54, p. 65). How much
of this labor is saved by the short method of multiplication?
37. Make up and work an example similar to Ex. 36.
38. Multiply 93 by 87. This product may also be obtained
thus:
93 X 87 = (90 + 3) (90 - 3)
= 8100 - 9 = 8091
Compare the amount of work in the two processes.
39. Make up and work an example similar to Ex. 38.
Find the value of each of the following in the short way:
40. 92 X 88 43. 1005 X 995
41. 103 X 97 44. 1032 - 972
42. 105 X 95 45. (17.31)2 - (2.69)2
Find in the shortest way:
46. The area of a rectangle 102 ft. long and 98 ft. wide.
47. The cost of 32 doz. eggs at 28f^ per dozen.
48. The cost of 67 yd. of cloth at 73^ a yard.
49. Make up and work two examples similar to Exs. 47^8.
50. Work Ex. 40 in full. How much of this labor is saved
by using the short method of multiplication?
Write at sight the product for each of the following miscel-
laneous examples:
51.
(x + 2a)2
58.
l(x + 2y) + 5]2
52.
(x + 2a) {x - 2a)
59.
(x^2y-\-5){x-h2y-5)
53.
(x - 2a)2
60.
(.3a: + .5y) (.3a: - .5y)
54.
(3a: - 1) (3a: + 1)
61.
9982
55.
(3a: - 1)2
62.
998 X 1002
56.
(3a2 - 26^)2
63.
972
57.
(3a2 - 2¥) (3a2 + 2¥
') 64.
97 X 103
I
■ ABBREVIATED MULTIPLICATION 115
^K 65. Make up and work an example in each principal form
^Bf abbreviated multiplication studied thus far.
66. How many of the examples in this Exercise can you
answer orally?
78. IV. Square of any Polynomial.
By actual multiplication,
a -\-b + c
g 4- 6 + c
a^ -^ ab -}- ac
+ ab +62+ 6c
-^ ac + 6c + c^
a^ + 2ab + 2ac + 6^ + 26c + c^ Product
Or, in brief, (a + 6 + c)^ = a^ + 6^ + c^ + 2ab + 2ac + 26c.
In like manner we obtain
(a + 6 + c + 6^)2 = a2 + 62 + c2 + (Z2 4- 2a6 + 2ac + 2ad
+ 26c + 2bd + 2cd
Or, in general.
The square of any polynomial equals the sum of the squares
of the terms plu^ twice the product of each term by each term
which follows it.
It is often useful to indicate the order in which the products of
the terms are taken as shown in the following diagram. (If the
curved Unes joining the terms are drawn as each product is taken,
the numbers on these lines may be omitted.)
Ex. (a - 26 + c -~3x)2 = a^ + W + c^ + 9x2 - 4a6 + 2ac - 6aa;
-46c+ 126a; - Qcx.
Let the pupil check the work.
116 SCHOOL ALGEBRA
EXERCISE 28
Find in the shortest way the value of the following and
cheek each result:
1. (2a; + 2/ + 1)' 8. {2c? + 5a - 3)^
2. (a: - 2?/ + 2zf 9. {x - y ■\- z - Xf
3. (3a; -2y- 5)^ lo. {2x-\-Zy-Az- hf
4. (2a - 6 + 3c)2 11. (3a;3 _ 4^ _^ ^ _ 2)2
5. (a; -21/ -32)2 ^3. (|a;2 - f a: + 5)^
6. (4a; + 3^ - l)^ 13. {\x^ - \x^ + f a? + 6)^
7. (a;2 - a; + 1)^ 14. (.2a + .3& - .hcf
15. Expand (2a — 36 + c — 4(Z)2 by multiplying in full.
Now obtain the same result by the method of Art. 78, p. 115.
About how much of the work of multiplication is saved by
the latter method?
16. Make up and work an example similar to Ex. 15.
Write at sight the product for each of the following mis-
cellaneous examples:
17. (2a -36)2 21. (3a;2 + 2/3)2
18. (2a + 36) (2a - 36) 22. (3a;2 - ^ff
19. (2 + 36)2 23. (a; + 22/- 3) (a: + 22/ + 3)
20. (3a;2 - 2/3) (3^ ^ 2/3) 24. (a; + 22/ - 3)2
25. (4a; + Ja-f)2
26. (4a; + ia-f)(4a; + ia + f)
27. (a;"+^ - 3x"-V)2
28. (a;"+i - 3a;"-V) (a;"+^ + 3a;"-V)
29. (.02x2 - .3a; + .5)2
I
ABBREVIATED MULTIPLICATION 117
30. Make up and work an example in each principal form
of abbreviated multiplication studied thus far.
79. V. Product of Two Binomials of the Form a? -f a,
a? + 6.
By actual multiplication,
X -h 5 X — by x-\- a
g; +3 a; + 3i/ x + h
x^ -\- bx oP- — 5xy x^ + ax
+ 3a; + 15 + ^xy - 15y^ -\- hx + ab
r^ + 8a; + 15 x"" - 2xy - 152/^ x^ + (a + h)x + ah
By comparing each pair of binomials with their product,
we observe the following relation:
The product of two binomiab of the form x -{- a and x -{- b
consists of three terms:
The first term is the square of the first term of the binomials;
The last term is the product of the second terms of the
binomials;
The middle term consists of the first term of the binomials
with a coefficient equal to the algebraic sum of the second terms
of the binomials.
Ex. 1. Multiply a; - 8 by a; + 7.
-8+7 = - 1. -8x7 = - 56.
.*. (x -S) ix + 7) =x^ -X -66 Product
Ex. 2. Multiply (a; - 6a) (x - 5a).
(- 6a) + (- 5a) = - 11a. (- 6a) X (- 5a) = + SOa\
,'.{x -6a) {x - 5a) = x^ - Uax -{- SOa^ Product
Ex. 3. Multiply x + y + Qhyx-^y-2.
(x+y +6){x+y -2) =l(x-{-y) +6] [(x + y) - 2]
^ {X + yY + 4(rc -^ y) - 12 Product
118 SCHOOL ALGEBRA
EXERCISE 29
Write the product for each of the following and check each
result:
1. {x + 2){x + 5) 10. {x + .2) {x + .5)
2. {x " 5) (a: - 3) li. {x + |) {x + \)
3. {x - 7) (a: + 4) 12. {x + .02) {x + 5)
4. (x - 4) (a: + 8) 13. (a + .02) (a + .5)
5. (a: + l)(a:-7) 14. {x - h) {x + \)
6. {x^-2){x'-Z) 15. (a + f)(a-i)
7. (a:2 + 3) (a:2 + 1) le. {ah + x) {ah + 3a;)
8. {a + 3a:) (a - lOx) 17. (a6 + a:) {ah - 3x)
9. (a: - 7y) {x + 2/) 18. {xy - 7z^) {xy + 32^)
19. (a;" + 5) (a:" - 5)
20. [{x + 2/) + 3] [(a: + ?/) + 5]
21. [(a: + 2/) - 3] [(a: + 2/) + 5]
22. (a; + 2/ - 3) (a: + 2/ + 5)
23. (a + 26 + 5) (a + 26 + 3)
24. {2x + 3?/ + 3a) (2a: + 3?/ - 5a)
25. (a: + a + 6) (a: — a — 6)
26. (2a: + a + 36) (2a: - a - 36)
27. (2a: + a - 36) (2a: - a + 36)
28. Find the product of a: + 2/ + ^ and x -{■ y — ?> by
multiplying in full. Then find the same product by the
method of Art. 79. About how much of the work of multi-
plication is saved by use of the latter method?
2,9. Make up and work an example similar to Ex. 27.
I
ABBREVIATED MULTIPLICATION 119
30. A building lot is 167 ft. wide and 213 ft. deep. If the
width and depth of the lot are each increased by 1 foot, find
the increase in area without multiplying 167 by 213.
Write at sight the product for each of the following mis-
cellaneous examples:
31. {x + 5) (x - 5) 35. (x + 5) {x - 3a)
32. (x + 5)^ 36. (x — 3a) {x + 3a)
33. {x — 5y 37. (x — 3a) {x + 5a)
34. {x -{- 5) (x — 3) 38. {x — 5a)2
39. (a: + 2/ + 5a) (x + y - 5a)
40. (.T + 2/ + 5a)2 ^
41. (x + y + 5a) (x + y + 3a)
42. (x + y + 5a) (x -\- y ^ 3a)
43. (^2 + la; + 3)2
44. (a:2 + i a: + 3) {x' + ix - 3)
45. {x'-i-ix + 3)(x^ + ix-5)
46. (a^ -\- ax -\- x^) (a^ — ax + x^) ^
47. Make up and work an example in each principal form
of abbreviated multiplication studied thus far.
48. How many of Exs. 31-45 can you answer orally?
80. VI. Product of Two Binomials whose Corresponding
Terms are Similar.
By actual multiplication,
2a - 36
4a + 56
8a2 - 12a6
+ 10a6 - 156^
8a'' - 2ab - 156^ Product
120 SCHOOL ALGEBRA
We see that the middle term of this product may be ob-
tained directly from the two binomials by taking the alge-
braic smn of the cross products of their terms. Thus,
( + 2a) ( + 56) + (- 36) ( + 4a) = 10a6 - 12a6 = -2a6.
Hence, in general,
The product of any two binomials of the given form consists
of three terms:
The first term is the product of the first terms of the binomials;
The third term is the product of the second terms of the
binomiah;
The middle term is formed by taking the algebraic sum of
the cross products of the terms of the binomials,
Ex. Multiply lOx + 7y by 8x - Uy.
To show the method of obtaining the middle term of the
product, we write the given expression in the form
{10x+7y){Sx -Uy)
I I
Hence,
(lOx) (-lly) + {7y) {Sx) = - UOxy + 5Qxy = - 5^y
.*. (lOx + 7y) {Sx - lly) = 80a;2 - 5^y - 77y^ Product
EXERCISE 30
Write at sight the product of each of the following and
check each result:
1. {2x + 3) (a: + 4) 7. {bx - 1) {x + 7)
2. {2x -2>){x- 4) 8. {x + 3z/) (3a; - 82/)
3. {2x + 3) (a; - 4) 9. (Sa^ + b) (4a2 - 56)
4. (2a; - 3) (a; + 4) 10. (a; + i) (f x + J)
5. (3a + 5) (2a + 3) 11. (a + .26) (2a - .36)
6. (3a - 5) (2a + 3) 12. {\x + fa) (fa; - |a)
ABBREVIATED MULTIPLICATION
121
13. How many examples in Exercise 9 (p. 41) can you
now work at sight?
EXERCISE 31
Review
Write at sight the value of each of the following and check each
result:
1. (2a; + 3)2 16.
2. {2x ~ 3)2 17.
3. (2x + 3) (2x - 3) 18.
4. {x + 3) (x - 5) 19.
5. (2x + 3) (3a; - 5) 20.
6. {x + Sy) {x + 2y) 21.
7. (2a; + ZyY 22.
8. (2a; + Zy) (3a; - 4?/) 23.
9. (2a; - ZyY 24.
10. (2a; + 3?/) (2a; - Zy) 25.
11. (5a - 3a;) (4a + 5a;) 26.
12. (7a; + 32/2)2 27.
13. (5a2.+ 3?/3) (5a2 - ^y^) 28.
14. {a? + 3a;) {a? - 5a;) 29.
15. (2a + 3a; + 5)2 30.
2a + 3a; + 5) (2a + 3a; - 5)
2a + 3a; + 5) (2a + 3a; - 3)
1 - 2a; - 3a;2 + x^Y
a + 6 + a; + 2/) (a + 6 -a; -?/)
a2 + aa; + a;2) (a2 - ax -\- x'^)
Aa? + 2a + 1) (4a2 - 2a + 1)
3aa;" - 2a"-ia;)2
a;" + 2a;2/"-i)2
1 -a)2
a - 1)2
- 2a; + 32/)2
- 2a; - 3?/)2
a - 6) ( - a + 6)
a; + 3) ( - a; - 3)
3i/)2 is the
31. Why is it that the result of expanding ( — 2a;
same as that of expanding (2a; + ^yY ?
32. Give two expressions similar to those in Ex. 31 for which
the product is the same.
33. Why is (a - 6)2 equal to (6 - a) 2 ? Make up two expressions
similar to these.
34. Make up and work an example in each principal form of
abbreviated multipUcation studied thus far.
122 SCHOOL ALGEBRA
' Simplify, using the methods of abbreviated multipHcation as far
as possible :
35. (a + 26)2 + {a - 26)2
36. (a + 26)2 - (a - 26)2
37. (2x - 1)2 + (1 - 2x)2
38. (2x - 1)2 - (2x + 1)2
39. (3a - 1)2 + (2 - 3a) (2 + 3a)
40. {2x - 7y) {2x + 7y) - 4(a; - 2yy + ISy {by - x)
41. (3a;2 + 5)2 + a;2 (10 - 3a:) (10 + 3a;) - (5 + 13a;2)2
42. (a - c + 1) (a + c - 1) - (o - 1)2 + 2 (c - 1)2
43. {x +y - xy) {x -y - xy) + x'^y - {x - y^) {x + y'^)
44. Show that a2 = (a + 6) (a - 6) + 62.
45. By use of the relation proved in Ex. 44, obtain the value of
(7i)2 in a short way.
SuG. We have (7i)2 = (7i + i) (7i - h) + {W
= 8 X 7 + i = 56i Ans.
Using the method of Ex. 45 find the value of:
46. (8i)2 49. (15J)2 52. (7.5)2 55^ (75)2
47. (19J)2 50. (49i)2 53. (19.5)2 56. (195)2
48. (199i)2 51. (99§)2 54. (99.5)2 57. (995)2
' 58. (9.7)2 (Use (9.7)2 = 10 x 9.4 + .32)
59. (9.8)2 60. (9.6)2 g^ (4,8)2 52. 982
63. Find the value of (a + 6)' by multiplication. Examine the
result obtained. Make a rule for obtaining similar products in a
short way. Treat (a — 6)^ in the same way.
64. By use of the rule obtained in Ex. 63, write out by inspection
the value of {x + yY-
65. Also of (a - xY. 66. Of (6 + yY.
Solve the following equations, using methods of abbreviated
multiplication wherever possible:
67. {2x - 1)2 - 4a:2 = 19
68. {2x + 1)2 - {2x - 1)2 = 16
ABBREVIATED DIVISION 123
Compute in the shortest way:
69. The area of a field 103 rd. long and 97 rd. wide.
70. The area of a square field each side of which is 98 rd.
71. The cost of 62 yd. of cloth at 58^ per yard.
72. The cost of 85 A. of land at $95 per A.
73. How many of the examples in this exercise can you work at
sight?
Abbreviated Division
81. Utility of Abbreviated Division. In certain cases
much of the labor of division may be saved by the use of
mechanical rules. We discover these rules by performing
the division operation in a typical case, noting the relation
between the quantities divided and the quotient, and for-
mulating this relation into a rule.
82. I. Division of the Difference of Two Squares.
Either by actual division, or by inverting the relation of
Art. 77, p. 112, we obtain
= a — 0 and r- = a -{- h,
a-\- 0 a — b
Hence, in general language.
The difference of the squares of two quantities is divisible by
the sum of the quantities, arid also by the difference of the quan-
tities, the quotients in the respective cases being the difference of
the quantities and the sum of the quantities.
Ex. 1. — — = 2x + 3y Quotient
2x — Sy
Ex. 2. ^^ ~ ^!^ "^ ?r = X- (a + b) Quotient
x+ (a-\- b)
Let the pupil check the work in these examples.
124 SCHOOL ALGEBRA
EXERCISE 32
Write at sight the quotient for each of the following, and
cheek each result:
5.
6.
a — x
9-4a:2
3-2x
x^ - 811/2
25a:2 - 362/^
5a; - 6?/2
\%x^ - 49^
At? + 7r
25a:io - y^''
7
a'h^ - mM^
aW + ^(^d!"
8.
\^ - V
i^- h
9.
i%a' - i^
ia2 - i:^
in
.25a2 - .1662
,5a- Ah
n
Mx" - .09^2
.2a; 4- .3^
TO
a;2 - .256*
5x^ - y^ •a; - .562
13. Divide a^ + 2a6 + 62 — 4a;2 |3y ^ _|_ 5 _ 2a; by long
division. Write the result of dividing (a + 6)2 — 4a;2 ]^y
a-\-h— 2x by the method of Art. 82. Estimate how much less
the labor of the second process is than that of the first.
14. Make up and solve an example similar to Ex. 13.
Obtain in the shortest way the quotient for each of the
following:
jg (X + 1)^ - g' ^ (a - hf -(c- lY
X
+ l + a
a2-
- (6 - 2c)2
a -
- (6 - 2c)
4a;4
- (f + D'
16. ^- -4- 19.
17. ^ T n A 20
(a - 6) + (c - 1)
1 - (g + 6 - c)2
1 + (a + 6 - c)
(2a + 36)2 _ (5^ _ 42^)2
2a;2 + (2/2 + 1) . {2a + 36) - (5a; - 4y)
ABBREVIATED DIVISION 125
Write a divisor and quotient for each of the following:
^, a2-4r' ^^ da" - (x + yf
21. ^ 24. =
a^ - 4a:« (a + a;)^ - (6 + i/)^
22. = 25.
^ -f 4(a + 6)2 - 9ir2
23. = 26.
Find two factors for each of the following:
27. 2500 - 16 29. 2491 31. 99.19
28. 2484 30. 9919 32. 6319
33. Divide a^ — 6^ by a — 6. Divide (a — 6)^ by (a — 6).
34. Find the difference in value between (x + yf and
aj^ + y^i when a: = 2 and 2/ = 3.
83. II and III. Division of Sum or Difference of Two Cubes.
By actual division we can obtain,
0? -j- 53 ^3 _ ^3
— —-r- = a} — ab -\- 6^ and r = c? ■\- ah ■\- h^,
a + 0 a — 0
Hence, in general language.
The sum of the cubes of two quantities is divisible by the sum
of the quantities y and the quotient is the square of the first quan-
tity ^ minu^ the 'product of the two quxintities, plus the square of
the second quantity; also
The difference of the cubes of two quantities is divisible by
the difference of the quantities ^ and the quotient is the square of
the first quantity y plus the produ^ct of the two quantities, plus the
square of the second.
126
SCHOOL ALGEBRA
Ex.1.
2x -3y 2x - 3y
= (2xy + (2x) (32/) + (SyY
= 4a^-h Qxy + V Quotient
F,Y 91
(«-^)' + 27 ,„. ,,, ,,„, ,, ,
• • (a - 6) + 3 ' ^ ^ ^ '
= a2-2a6 + 62-3a+36 + 9 Qwo^teni
Let the pupil check the work in these examples.
EXERCISE 33
Write at sight the quotient for each of the following and
check each result:
a^ + 8 27a« + y^
^' - ^ 3a2 + y^
m^T? - f
'2x - y
h^ - h
11. Divide Sa^ + 276^ by 2a + 36 by the method of long
division. Now write out the same quotient by the method
of Art. 83. Estimate how much of the labor of division is
saved by using the second method of obtaining the quotient.
12. Make up and work an example similar to Ex. 11.
13. Treat (a + hf — Sa:^ divided by (a + 6) — 2x as in
Ex. 11.
a+2
x^-l
X- 1
27:x? - 64
3a: -4
l + 8a:«
l + 2ar^
125 - x^
2.
3. —z 7- a
ABBREVIATED DIVISION 127
Obtain in the shortest way the quotient for each of the
following:
c3 + (1 - xy (a - 1)3 - a^
14 ' — ^^ — 17 ^^
• c + (1 - a:) (a - 1) - x"
8 - (a: + yf Sx^ + jx'' - If
' 2- (x-hy) 2x^x^ -\
21x' + 125i/^ 8(a; - yf - z'
^^' 3r^ + 51/3 ^9- 2{x -y) -z
Write the binomial divisor and the quotient for
Sa^ - :x? 8a3 -fl
20. = 24. =
21. = 25. -^ =
22. = 26. ^^-^ =
23. = 27. ^ ^ =
(g + 6)3 + (a; + yf _
28.
Find a factor of each of the following:
29. 203 ^ 33 31; 3Q27 33, 125027
30. 8000 + 27 32. 7973 34. 124973
35. Divide c^ — ¥ by a — b. Also divide (a — by by
a — b.
36. Find the difference in value between 3^ -\- y^ and
(x + yY when x = 2 and y = S.
Write a binomial divisor and the corresponding quotient
for each of the following miscellaneous examples:
o, a' -462 ^^ a3-863
37. = 38. =
128 SCHOOL ALGEBRA
40. ?Z^-^ = 47. g' - 4(a; + y)' ^
41. 9^-1 = 48. g' - Six + y)» ^
^2 16a^ - 9 ^ ^g 8(x + y)» + a? ^
^g 8x» - 27jf' ^ ^ a* - 9{x - yf ^
44. ^ + ^ = 51. 27a» - (a; - yY ^
45. ^^ = 52. (x + y)^ -(X- vY ^
53. (a: + 1)^ + (X - 1)^ _
54. How many of the examples in this Exercise can you
work at sight?
55. How many examples in Exercise 1 (p. 8) can you
now work at sight?
84. IV, V, and VI. Division of Sum or Difference of any
Two Like Powers.
By actual division we can obtain,
a + h
a — b
= a? — a^b + ah'^ — ¥ Quotient
= a^ + a?b + ab'^ + W Quotient
a^ + ¥ is not divisible by either a + 6 or a — 6. But
«^ + ^' _ ^4 _ ^35 _^ ^252 _ ^53 ^ 54 Quotimt
a-\-b
a^ + 0^5 + a2fe2 + a¥ + ¥ Quotient
a — 0
ABBREVIATED DIVISION 129
Hence,
The difference of two like even powers of two quantities
is divisible by the sum of the quantities, and also by their
difference;
The sum of two like odd powers of two quantities is divisible
by the sum of the quantities;
The difference of two like odd powers of two quantities is
divisible by the difference of the quantities.
For the quotient in all these cases —
(1) The number of terms in a quotient equals the degree
of the powers whose sum or difference is divided;
(2) The terms of each quotient are homogeneous (since
the exponent of a decreases by 1 in each term, and that
of b increases by 1 in each term).
(3) If the divisor is a difference, the signs of the quotient are
all plus; if the divisor is a sum, the signs of the quotient are
alternately plu^ and rrmms.
In the above statements the parts in italics should be
committed to memory.
The last statement forms a general rule for signs of a
quotient in all the cases of abbreviated division, including
I-VI.
g^ ^ 320:^ + 2/^ ^ {2xY + l^
' ' 2x + y 2x + y
- (2xy - {2xfy + {2x)Y - {2x)f + t/*
= 16a:4-8a;3|/ + AxY - 2xi^ + y^ Quotient
a^o + x^^ {a'f + {x'f
Ex.2.
a^ -\- x^ a^ -\- x^
130 SCHOOL ALGEBRA
EXERCISE 34
Write at sight the quotient for each of the following and
check each result:
1.
a^-ha:^
a -}- X
2.
a^-3^
a — X
3.
b'-\-y'
h + y
a.
h' - y'
c
a5 + 32
a + 2
6.
a? - 128
a- 2
7.
7^ - 1
X- 1
o
0:^ + 1
9.
10.
11.
12.
Z2x>
-f
2x-
-y
a" + ar^i
a +
X
x^^-
ylS
x"--
f
243 -
-d^
b - y X -^ I
13. Divide 32a^ -]- x^ hy 2a -\- x hy the method of long
division. Now write the same quotient by the method of
Art. 84, p. 128. Estimate how much of the labor of division is
saved by using the second method.
14. Make up and work an example similar to Ex. 13.
Write a binomial divisor and the corresponding quotient
for each of the following:
_ a'-^-y" _ ^^ a^-32_
16.
17.
-io
o'-
y"
fc' +
x'
6'-
x-
20.
21.
a? + 128
1 - 32a:5
2/^ + 1
22.
Obtain a factor of each of the following:
23. 100,001 25. 100,032
6fk. 100,243 26. 99,757
ABBREVIATED DIVISION 131
27. Divide a^ + U" by a + &. Also divide (a + hY by
a + 6.
28. Find the difference in value between ^ — -if" and
{x — y)^, when x = 3 and y — 2,
EXERCISE 36
Review
Write at sight the quotient for each of the following:
1
62 _a:2
6 -X
o
¥ -x^
b-x
•5
¥ -x^
b -X
4
¥ +x^
b+x
(%
¥ +x^
6 H-2x
11.
7 ^' - ^^^ 12
b -2x
¥ - 32a;5
b -2x
13.
x' -
8(a
+ by
X -
2(a
+ b)
Sx' ■
-a3
2x
- a
27a«
-8{
[x + yY
3a2
- 2{x + y)
X'' + y'
x'-\-y^
a;i2-
.yS
9. ^^+^^^ 14.
b +2x
10 ^' - 4(« + by ^g
"'6+0; ■ a: - 2(a + &) ' x^ + y^
16. In Ex. 11 remove the parenthesis in the dividend and divisor,
and divide by long division. The work required is about how
many times that required in the abbreviated process?
Write a binomial divisor and the corresponding quotient for each
of the following:
j^j b' - 82/3 ^ ^^ a? - 4(x + yY
18. ^1^^ = 25.
19 ¥+%y^ _
26.
20. ^^-^^ = 27.
21. §?1±27 ^ 28.
22. ^^^ = 29.
23.^^^=^ 30.
a3 + 8(a:
+ yY
a? - 8(x
+ yy
8x«+2/«
4x* -'f
32a« - t/
5
8X6 _y^
132 SCHOOL ALGEBRA
Divide each of the following by a; - o in a short way:
31. a:3 - a3 + x2 - a^ 34. 3(a;3 - a^) + 4(a; - a)
32. x^ - a? + 5(a;2 - a^) 35. (a; - af + 5(^2 - a^)
33. x' - a' + 5(x - a) 36. 7(a; - a)^ + b{x - a)
Find the value of each of the following in the shortest way:
37. (o + 6) (a + 6) (a - 6) (a - 6)
38. (a + 26) (a + 26) (a - 26) (a - 26)
39. (3x - 22/) (3a; - 2y) (3a; + 2y) (3a; + 22/)
Simphfy:
40. bx - Z{x - 2)2 - 3(3 - 2a;) (1 + x)
41. 7 - 5(a; - 2Y - 3(3 - 2a;) {- x)
Solve:
42. (a; - 8) (a; + 12) _ (x + 1) (a; - 6) = 0
43. (2a; - 1) (a; + 3) - (a; - 3) (2a; - 3) = 72
46. Four times a certain number diminished by 12.07, equals
twice the number increased by 1.13. Find the number.
47. Separate 1000 into three parts such that the second part is
three times as large as the first part, and the third part exceeds the
first part by 100.
f^ 48. The Suez Canal is 100 miles long. This is 2 miles more than
8 times the length of the Simplon Tunnel. Find the length of the
tunnel.
49. The temperature of the electric arc is 5400° F. This is 464°
more than 8 times the temperature at which lead fuses. Find the
temperature at which lead fuses.
50. The velocity of sound in the air is 1090 ft. per second. This
rate is 10 ft. more than 9 times the rate at which sensation travels
along a nerve. Find the rate at which sensation travels. How
does this compare with the velocity of an express train going 60
miles per hour?
51. Who first used the sign + to denote addition, and when?
(Seep. 268.)
ABBREVIATED DIVISION 133
52. Give some other symbols used to represent addition before
fche sign + was invented. Discuss as far as you can the relative
advantages of these signs.
53. Answer the questions in Ex. 52 for the subtraction sign.
54. Answer the questions in Ex. 52 for the sign X.
55. Answer the questions in Ex. 52 for the sign t-.
56. Answer the questions in Ex. 52 for the sign =.
CHAPTER VIII
FACTORING
85. The Factors of an expression (see Art. 11) are the
quantities which, multipHed together, produce the given
expression.
Factoring is the process of separating an algebraic expres-
sion into its factors.
86. Utility of Factoring If it is known that
V - 8a: + 15 = ( a: - 3) ( a: - 5)
and 2x2 _ 13^ + 21 = (2a: - 7) ( a: - 3)
a:^ - 8a: + 15 ( a: - 3) ( a: - 5) x-
^"^ 2a:2 _ 13^. _^ 21 {x- 3) (2a: - 7) 2a: -
-5
-7
The above reduction of a fraction to a simpler form illus-
trates the usefulness of a knowledge of factoring in enabling
us to simplify work and save labor.
Why do we now proceed to make definitions and rules and
to divide the topic, Factoring, into cases?
87. A Prime Quantity in algebra is one which cannot be
^divided l)y any quantity except itself and unity; as a, b,
a" + b^ 17.
In all work in factoring, prime factors are sought, unless
the contrary is stated.
88. Perfect Square and Perfect Cube. When an expres-
sion is separable into two equal factors, the expression is
134
TTACTOHnSIG 135
called a 'perfect square, and each of the factors is the square
root of the expression.
Thus, 9aV = 3ax2 • Sax*.
.*. 3aa;2 jg j;he square root of Oa^x*.
Also, x2 - 4x + 4 = (x - 2) (x - 2), and is therefore a perfect
square, with a; — 2 for its square root.
When an expression is separable into three equal factors,
the expression is called a perfect cube, and each of the factors
is its cube root.
Thus, 27a^xY = ^(^^Y ' ^(^^Y ' ^a^y.
.'. Saxhj^ is the cube root of 27 a^x^y^.
89. The Factors of a Monomial are recognized by direct
inspection.
Thus, the factors of 7aV are 7, a, a, x, x, x.
90. Factors of Polynomials. Multiply ix^ + 2xy + y^ fcy
4:x'^ — 2xy + 2/^. What terms are canceled in adding the
partial products? Because these terms have been thus can-
celed and have disappeared, it is difficult to take the final
product 16a;'* + A.x^y'^ + y^ and from it discover the original
factors which were multiplied together to produce it.
Hence, in factoring polynomials various methods must be
devised to meet different cases, and the cases must be care-
fully discriminated.
Case I
91. A Polynomial having a Common Factor in all its Terms.
Ex. Factor 3x2 ^ e^.^
3x2 4- 6a; = Zx{x + 2) Factors
At first, in working examples of this kind, it is well to put the
work in the following form:
3a;)3a;2 + 6a; = 3x(a; + 2) Factors
X +2
136 SCHOOL ALGEBRA
Check by Substitution If we let a; = 2
Sx{x + 2) = 6(2 + 2) =24
Also 3^2 + 6a; = 12 + 12 = 24
Check by Multiplication
X +2
Sx
3^2 + 6a;
Hence, in general,
Divide all the terms of the polynomial by their largest common
factor;
The factors will be the divisor and quotient,
EXERCISE 36
Factor the following and check the work for each example
cither by substitution or by multiplication, or by both as
the teacher may direct:
1.
2:x^ + 50^2
6.
Sa^x^ - 15a'^s^
11. ia'b - f a62
2.
;i^-2x
7.
ISa:^ - 27x'y
12. U^ - 2x^
3.
x^-^x
8.
x'-7?-x'
13. .2:x? + Aax^
4.
Sa^-a
9.
aH - 2aV
14. max^ - AaV
5.
7a + Ua^
10.
h:^ + \^
15. 1.2mn — .6^2
16.
3a2 - Qax + 9^2
19. a^bh-
- a^tV + 2a26^c2
17.
2x + 4^2 -
63^
20. 2a;V
- Sx^Y + 60:3 Y
18.
lOa^b^ - S5aW
21. a^6V
" + lla-6V»+i
22. 7(a + b)x + 5(a + b)y
23. 7{a-\-b)x^y-\-b{a-{-byxy'^
24. 21 (a; - yf - U{x - yf
25. 9(2a; - a) 3- 12(2a; - af
FACTORING 137
26. 847 X915 - 847 X 913
t27. 312.75 X 87 - 312.75 X 84
28. 8 X 11 X 232 + 7 X 11 X 232 - 5 X 11 X 232
29. 7rE2 -j- 7rr2 when ir = -\2-, R = 8, and r = 6.
30. -n-R^ - 7rr2 when tt = ^j\ R = 410, and r = 60.
Find the value of x in the following equations:
31. ax -\- bx = 10. (What does the value of x become
when a = 5 and b = 15?)
32. ax = 10 — 6a:
33. ax -\- bx -\- ex = 12
34. 2ax — bx + 3cx = 15
Factor the numerator and denominator of each of the fol-
lowing fractions and then simplify the fraction by canceling
factors :
Sa^b — 6ab'^ aV
35. — 37.
3a26 + 6a62 4x^ - Qa^
3g x^ + 2x' ^ gg Spq - 6pV
3x^ - 6x2 12pY _ Qpq
39. From an examination of Exs. 26-38, state the uses or
advantages of being able to factor by the method of Case I.
40. How many of the examples in this Exercise can you
work at sight?
Case II
92. A Trinomial that is a Perfect Square. By Arts. 75
and 76 a trinomial is a perfect square when its jBrst and last
terms are perfect squares and positive, and the middle term is
twice the product of the square roots of the end terms. The
138 SCHOOL ALGEBRA
sign of the middle term determines whether the square root
of the trinomial is a sum or a difference.
Ex. 1. Factor IQx^ - 24xy + V.
16x2 - 24xy + 9?/2 = (4a; - Sy) {4x - Sy) Factors
Ex. 2. Factor (a + by + 4(a + b)x + 4x2.
(a + by + 4(a + fe)x + 4x^ = [{a +b) + 2a:p
= (a + 6 + 2xy Ans.
Hence, in general, to factor a trinomial that is a perfect
square.
Take the square roots of the first and last terms, and connect
these by the sign of the middle term;
Take the result as a factor twice,
EXERCISE 37
Factor and check:
1. 4^^ + 4x2/ + 2/2 9. a5 + 2a^4-a'
2. 16o2 - 2^ay + V 10. ^3? + 44a:y 4- i2lxy^
3. 25a:2 - lOx + 1 11. SWb + \2^a^b'' + 4Qa}^
4. x^ - 20xy + 100y2 12. Sa^y - 40axy + 50x^y
5. 49c+286c2 + 46V 13. 2x^ - Sx^ + Sx^
6. a362 + 4a36 + 4a3 14. 30x2?/ + 3x^ + 75y2
7. x?/2 + 2x1/ + X 15. a^x + ax^ — 2a2x2
8. 2m'^n — 4m?i + 2n 16. x^" + 2x"!/ + y^
17. (a - by - 2c{a - 6) + c^
18. 9(x + yy + 12z(x + 2/) + 4^2
19. 16(2a - 3)2 - 16a6 + 246 + 6^
20. 25(x - I/)' - 120x!/(x - y) + 144x2/
21. a2 + 62 + c2 + 2a6 + 2ac + 26c
FACTORING 139
22. i'r + 4xy-\- V 24. .04a2 - .12a6 + .0962
23. iir2 + ixy + jy 25. 25a2 - 30ax + 90:^
Solve the following equations for x:
26. ax - Sbx = a^ - Qab + 96^
27. aa: + 36a; = a^ + 6a6 + 96^
28. X — 2ax = 1 — 4:ax - 4a2
29. 2ax + 36a; = 4a2 + 12a6 + 96^
Factor the following examples in Cases I and II, and check
each result:
30. a;3 _ 4^.2 ^ 4^. 36 iQa^ - 20a + 10
31. x^ -3x^ + 4x 37. 20a2 - 10a + 10
32. m^ — 2m^n + 'm^n^ 38. IGa^p^ _ 24a^p + 90^
v33. m^ — m% + m^n^ 39. 4a;^ + 4:3^y + a;^?/^
34. a%2 _ 8^3^ _^ 15^3 4Q 83^ _^ 163.22^ ^ ^2/4
35. o''^a;2 — Qa^x + 4a2 41. IGm^Ti^ — 9mn^ + n^
42. How many of the examples in this Exercise can you
work at sight?
Case III
93. The Difference of Two Perfect Squares.
From Art. 77, p. 112, (a + 6) (a - 6) = a^ - 6^
Hence, a^ — 6^ = (a + 6) (a — 6)
But any algebraic quantities may be used instead of a and
6. Hence, in general, to factor the difference of two squares,
Take the square root of each square;
The factors will he the sum of these roots and their difference.
140 SCHOOL ALGEBRA
Ex. 1. Factor x^ - 16?/2.
x^ - 162/2 = (a: + %) {x - Ay) Factors
Ex. 2. Factor x^ - y\
X* -y* = (x^ + 2/2) (x2 - 2/2)
= (a;2 + 2/2) (x + y) (x - y) Factors
Ex. 3. Factor (3a: + Ayf - {2x + ?^yf.
(3x + 42/)2 - (2a; + ZyY = [(3a: + Ay) + (2x + 32/)] [(3x + 42/) -
(2a: + 32/)]
= (3a: + 42/ + 2a: + 32/) (3a: + 42/ -2a; -Zy)
= (5a: + 72/) {x + 2/) Factors
Let the pupil check the above examples.
EXERCISE 38
Factor and check:
1. a:2 - 9 10. x^ - ^arx? 19. a^^ - a^
2. 25 - 16a2 11.' m - 64a27/i 20. a^x - x
3. 4a2 - 4962 12. 242 - 2x'^ 21. 225a:2" - 2/2
4. a:^ - 42/2 13. x^ - X? 22. 2ia:2 - ^2/2
5. 100 - 81m2 14. 2.3? - 75xy^ 23. 2*5^2 - 962
24. .09a:2 _ iQy2
25. .01a2 - .0462
26. .252/2-^^2^2
27. .81a:2 _ .002562
35. (a: + 22/)2 - (3a: + 1)2
29. a:^" - 2/2"26 36. 25(2a - 6)2 - (a - 36)2
30. (a: + 2/)2 - 1 37. a:i22/9 - 2/2^6
31. a:2 - (2/ + 1)2 38. 81a:i2 _ iQyA
32. (a: - 2/)^ - 9 39. x^ - lAAxyh^
33. 4(a: - 2/)2 - 25 40. (a - 6)2 - 4(c + 1)2
34. 1 - 36(a: + 2y)^ 41. 1 - 100(a:2 - a: - 1)2
6.
9a' - x'
15.
a'-x'
7.
1 - 64m2
16.
a^ - 816^
8.
3a:2 _ 122/2
17.
^8 _ y%
9.
X? - Wx
18.
x'-x
28.
x'-f
35.
II FACTORING 141
Solve the following equations for x:
42. ax + 26a; = a^ — 46^ 44. ?>x — ax = ^ — a^
43. ax — 2bx = a^ — 46^ 45. x — bx = 1 — b^
Factor the following miscellaneous examples:
46. a^ — 4:a 53. a^ — 9x
47. a^ — 4 . 54. a^ + 9a^x + 6ax^
48. a^ - 4a + 4 55. a^ + Qa^x + Qax^
49. a^ — 4a 56. a^ — ax'^
50. a' — 4a2 + 4a 57. a^ + ax*
51. a* - 4a3 + 4a2 58. (a + x)^ - 9
52. 0:^-60: 59. (a + 6)2 - (x - 2/ - a)2
60. Make up and work an example in factoring in each of
the cases treated thus far.
61. How many of the examples in this Exercise can you
work at sight?
62. How many of the examples in Exercise 2 (p. 13) can
you now work at sight?
Case IV
94. A Trinominal of the Form oc^ + bx-\-c.
It was found in Art. 79 (p. 117) that on multiplying two
binomials like x + 3 and x — 5, the product, x"^ — 2x — 15,
was formed by taking the algebraic sum of + 3 and — 5 for
the coefficient of x (viz. — 2), and taking their product (—15)
for the last term of the result. Hence, in undoing this work
to find the factors of x^ — 2x — 15, the essential part of the
process is to find two numbers which, added together, will
give — 2 and, multiplied together, will give — 15.
142 SCHOOL ALGEBRA
Ex. 1. Factor a:^ + n^ + 30.
The pairs of numbers whose product is 30 are 30 and 1, 15 and 2,
10 and 3, 6 and 5. Of these, that pair whose sum is 11 is 6 and 5.
Hence, x^ + llx + SO = (x + Q) {x -{- 5) Factors
Ex. 2. Factor x^ - x - 30.
It is necessary to find two numbers whose product is — 30, and
whose sum is — 1.
Since the sign of the last term is minus, the two numbers must
be one positive and the other negative; and since their sum is — 1
the greater number must be negative.
x^ -X -SO = {x -Q) (x +6) Factors
Ex. 3. Factor x^ + Sxy - 10/.
Since 5y and — 2y, added give Sy, and multipUed give — lOy^,
x2 + Sxy - 10?/2 = (a; + 5y) {x - 2y) Factors
Hence, in general, to factor a trinomial of the form
7? -{• hX ■}- Cy
Find two numbers which, multiplied together, produce the
third term of the trinomial and, added together, give the coefficient
of the second term;
X {or whatever takes the place of x) plu^ the one number, and
X plus the other number, are the factors required,
EXERCISE 39
Factor and check:
1. x^-{-5x-\-6 e. x^ + x-SO
2. x^ — X — 6 7. a;^ + 6xy — 16/
3. 7? -^ X — ^ B. x^ — 6xy — 16/
4. x^ + 7x-44: 9. x^ + Sx-i- 16
5. x^ - llx + 30 10. x^ + 5x- 3fi
I
FACTORING 143
x''-5x- 36 19. xY - 23xy + 132
a;^ - 5a;2 - 36 20. x^ - 5ax - 2Aa^
13. x^ + 3x-28 21. a;^ - 9a:2 + 8
14. a:2 - 2a; - 48 22. 2a - Uax - QOax^
15. a:2 - 8a: - 48 23. 20:^ _ 22a:2 - 120a:
16. x^ + 13a: - 48 24. a:^ - 25a:3 + 144a:
17. x^ - 22a: - 48 25. a:^" - a:" - 56
18. a:2 - 4a: - 96 26. aW - llabc^ - 26c^
27. x^ -\- (a + h)x + a6
28. a:2 + (2a - 36)a: - 6a6
29. a:2 + (a + 26 + c)x + (a + 6) (6 + c)
30. a:2 + (a + h)x + (a - c) (6 + c)
31. (a: - yY - 3(a: - 2/) - 18
Factor and check each of the following miscellaneous
examples :
32. a:2 - 4a: + 4 38. a"^ - 4y^
33. a:^ — 4 39. a^ — A^a^y + a^
34. a:^ _ 4a; -I- 3 4q a^ - 1
35. a:^ — a:^ — 6a: 41. x^ + bax + 6a2
36. a:^ — 4a: 42. a: — a:^
37. a:^ + 6a:2 + g^ 43 ^4 _ 7^2 ^ 12
44. Make up and work an example in factoring to illus-
trate each case treated thus far.
45. How many of the examples in this Exercise can you
work at sight?
144 SCHOOL ALGEBRA
Case V
95. A Trinomial of the Form ax^ -\-hx-\- c.
From Art. 80 (p. 119) it is evident that the essential part
of the process of factoring a trinomial of the form ax^ -\-hx-\- c
lies in determining two factors of the first term and two factors
of the last term, such that the algebraic sum of the cross
products of these factors equals the middle term of the
trinomial.
Ex. Factor lOa:^ + 13a; - 3.
The possible factors of the first term are 10a; and x, bx and 2x.
The possible factors of the third term are — 3 and 1, 3 and — 1. In
order to determine which of these pairs will give + 13a: as the sum
of their cross products, it is convenient to arrange the pairs thus:
lOx, - 3 bx, -I
X, 1 2x, 3
Variations md,y be made mentally by transferring the minus sign
from 3 to 1; and also by interchanging the 3 and the 1.
It is found that the sum of the cross products of
5x, - 1 .
IS + 13a;
2a;, 3
Hence, lOx^ + 13a; - 3 = (5a; - 1) {2x + 3) Fact(yrs
Let the pupil check the work.
Hence, in general, to factor a trinomial of the form
ax^ -\-hz -\- Cy
Separate the first term into two siich factors, and the third
term into two such factors, that the sum of their cross products
equals the middle term of the trinomial;
As arranged for cross multiplication, the upper pair taken
together and the lower pair taken together form the two factors.
I
FACTORING 146
EXERCISE 40
Factor and check:
1. 2x2 4- 3a; -f 1 15. 6a;2y - 2xy - 4y
2. 3a:2 - 14x + 8 16. IGx^ - 60:2/ - 21y^
3. 2a:2 + 5a; + 2 17. 12a:2 + a;^ _ 63?/2
4. 3a:2 + lOx + 3 18. 32a2 + 4a6 - 456^
5. 6a:2 - 7a; - 5 19, 4^.4 _ 13^,2 _^ 9
6. 2a;2 + 5a; - 3 20. 9a;^ - 148a;2 + 64
7. 6a;3 + 20a;2 - 16a; 21. 12a;2 - 7xz - 12z^
8. 3a;^ -ij^-4x^ 22. 24a;3 _|_ i04a;V - 18a;?/^
9. 8a2 + 2a - 15 23. 25a4 + 9a^b^ - 16¥
10. 2a;2 + a; - 10 24. 16a;^ - lOa:^?/^ - 9y^
11. 12a;2 - 5a; - 2 25. 3a;2" - 8a;"i/ - 3y^
12. 4x2 + 11a; - 3 26. 25a4 - 41^252 + 166*
13. 5a;2 + 24x - 5 27. 20 - 9a; - 20a;2
14. 93^ - 15a;2 - 6a; ' 28. 5 + ^2xy - 2\xY
29. (a + 6)2 + 5(a + 6) -24
30. 3(a; - yy- + l{x - y)z - ^z^
31. 3(a;2 + 2a;)2 - 5(^2 + 2a;) - 12
32. 4a;(a;2 + 3a;)2 - 8a;(x2 + 3a;) - 32a;
33. 2(a; + 1)2 - 5(a;2 - 1) - 3(a; - 1)2
Factor and check each of the following miscellaneous
examples:
34. 4a;2 - 1 38. a;'* - 1
35. 4ar^ + 4a; + 1 39. a;^ — a;2 — 6a;
36. 3a;2 + 4a; + 1 40. 5a2 + 9a - 2
37. a;2 + 4a; + 3 41. or^ - 9a; + 18
46.
x'-a^
47.
x' -X
48.
x^ - ax - 2a^
49.
27? - bx^ - Zx
146 SCHOOL ALGEBRA
42. a:^ _ 6^2 + 9a;
43. a2 - 4(a: + yf
44. 3a:2 + 7a; _ 6
45. (a + &)2 + 2(a + 6)a: + a:2
50. Make up and work an example in factoring in each
case treated thus far.
51. How many of the examples in this Exercise can you
work at sight?
52. How many examples in Exercise 25 (p. 101) can you
work at sight?
Case VI
96. Sum or Difference of Two Cubes.
From Art. 83 (p. 125), ^5!±_^ ^ a" - ah -\- h\
a -jr 0
Hence, a^ -^ ¥ = (a + b) {a^ - ab + ¥) , . . (1)
In like manner, a^ - ¥ = {a - b) (a^ -^ ab + b^) . . . (2)
But any algebraic expressions may be used instead of a and
b in formulas (1) and (2).
Ex. 1. Factor 27a:3 - Sf,
27x3 -8z/3 = (Sxy - {2yY
Use 3x for a and 2y for h in (2) above.
27a;3 - 8i/3 = (3a; - 2y) {^x^ + ^xy + 4^/^) Factors
In working examples of this type, it is often convenient to call
3a; — 2y the "divisor factor" and Oa;^ + ^xy + 4?/^ the "quotient
factor." Why are these names appropriate in this case?
Ex. 2. Factor a« + 86^
a« + 869 = (^2)3 _|_ (263)3
= (a2 + 263) (a* - 20^6^ + 46^) Factors
I
FACTORING 147
Ex. 3. Factor (a + by - s^.
(a+by -3^ = [{a + b) - x] [{a + by + (a + b)x + x"]
Let the pupil check the above examples.
Hence, in general, to factor the sum or difference of two
cubes.
Obtain the values of a and b in the given example, and substi--
tute these values in formula (1) or (2).
97. Sum or Difference of Two Like Odd Powers.
Since the difference of two like odd powers is alwp.ys divis-
ible by the difference of their roots (see Art. 84, p. 128), the
factors of a" — 6", when n is odd, are the divisor, a — b, and
the quotient.
Ex. 1. Factor a^ - ¥.
a5 - 6^ = (a - b) (a* + a% + a%^ + a¥ + ¥)
Since the sum of two like odd powers is divisible by the
sum of the roots (see Art. 84, p. 128), the factors of a" + &",
when n is odd, are the divisor, a -{- b, and the quotient.
Ex. 2. Factor a^ + 32yK
x^ + 32?/5 =x' + {2yy
= (x + 2y) [x* - xK2y) + x\2yy - x{2yy + {2yy]
= {x + 2y) {t^ - 2x^y + ^"^y"^ - 8xy^ + IQy*) Factors
98. Sum or Difference of Two Like Even Powers.
The difference of two like even powers is factored to best
advantage by Case III (p. 139).
Ex. 1. x^ — y^,
= {x' + y') (x' + y') (x -\-y){x- y) Factors
The sum of two like even powers cannot in general be fac-
tored by elementary methods unless the expression may be
148 SCHOOL ALGEBRA
regarded as the sum or difference of two cubes (Art. 96), or
other Uke odd powers.
Ex. 2. a« + 66 = {a^f + Qy'f
= (a2 + 62) (a^ - a^h^ + 6^) Factors
But a^ + 6^, a* + 6^ and a^ -\- ¥ cannot be factored by any
elementary method, and are therefore prime expressions.
Let the pupil check the examples of Art. 97 and 98.
I*
I
EXERCISE 41
Factor and check:
1. m' — n^
14.
a« - 64n^
27.
a" + x^i
2. c^ + 8d^
15.
250a: - 2a:7
28.
a« + 69
Z.TI -7?
16.
8a:« + 2/3
29.
32a:5_ i
4. a^ + 86^03
17.
(a + 6)3 + 1
30.
a^i - 611
5. a:^ - 125
18.
125 + (26 - af
31.
243 -a:^
6. 64!/3 _ 27
19.
8 - (c + d[)3
32.
64 - (a - 6)3
7. aW + 1
20.
{x - yf - 21^
33.
8(0: - 2yf + 1
8. 1 - lOOOa:^
21.
16a:y - 54a:z3
34.
^10 _ 510
9. 27a;^ + a^x
22.
x^ + 2/^
35.
a}^ + 610
10. 512x3 - y^
23.
0:^-2/'
36.
32a:5 _ ^10
11. a + 343a4
24.
a« + m«
37.
a« + ^9
12. a^ — x^
25.
a:i2 + 2/i2
38.
8.t:12 + 2/9
13. a;i2 _ y^
26.
a^ - 1286^
39.
512a:3 _ (a + 6)
40. Make up a binomial expression whose terms contain
unlike exponents and which can be factored as the sum of two
cubes. Also one that can be factored as the sum of two 5th
powers.
41. Make up a binomial the exponents in whose terms
are even numbers, and which can be factored as the sum of
I
FACTORING 149
two cubes. Also one that can be factored as the sum of two
5th powers.
42. State which of the following expressions can be fac-
tored :
^ ^y^ a:^ + 2/^ ^ -y^^ x^-\- y^°
3^ + y^ a:^ _ yi2 ^e _p yS ^e _^ yU
3^ + y^ x^-y^ x^-\- y^ x^ - y^^
Factor and check each of the following miscellaneous
examples :
43.
a:2 -. 4a2
51.
x^-^a^
44.
o^'-8a'
52.
x' -a^
45.
x^ — 4:ax + 4a2
53.
^12 _^ y9
46.
x'^ - ^ax + 3a2
54.
^12 _ 2^12
47.
x'-a'
55.
6a2 - 13a + 6
48.
x' + a'
56.
16x' - Sxy + 1/2
49.
x" - 4(a + 6)2
57.
x^ + 27a^x
50.
a:^ - 8(a + hf
58.
x^'^ + f
59. Make up and work an example in factoring to illus-
trate each case treated thus far.
60. How many of the examples in this Exercise can you
work at sight?
Case VII
99. A Polynomial whose Terms may be Grouped so as to
be Divisible by a Binomial Divisor.
Ex. 1. ax — ay — bx +by = (ax — ay) — (bx — by)
= a{x - y) - b{x - y)
= {a — b) (x — y) Factors
The last step in the preceding process is sometimes clearer to the
pupil when written in the following form:
(x-v) )a(i-y)-b(x-y) ^ (^ _ ^) („ _ j) p^^,
a — 0
15a SCHOOL ALGEBRA
Ex. 2. 1 + 15a* - 5a - 3a3 = 1 - 3a' - 5a + 15a*
= (1 - 3a3) - 5a(l - 3a»)
= (1 - 3a3) (1 - 5a) Factors
Ex. 3. a» + 3a2 - 4 = a' + 2a^ + a^ - 4
= a2(a + 2) + (a + 2) (a - 2)
= (a + 2) (a2 + a - 2)
= (a + 2) (a + 2) (a - 1) Factor*
Let the pupil check the above examples.
EXERCISE 42
Factor and check:
1. ax-}- ay + bx -\- hy 14. a:^ - a;^ - 4x + 4
2. x'^ - ax-\- ex - ac is. a^o^-h^x^-a^y^-^-h'^f
3. 5a:!/ - lOy - 3a; + 6 16. a;(a: + 4)^ + 4(a; + 4)
4. 3am-4mn-6a2/+8ny 17. a2(a + 3) - 3(a + 3)
.. 5. a^x + 3aa: + oca; + 3ca: is. 2(ar^ - y^) _ (3. _ y)
6. 3a2 + 3a6 — 5an — 56n 19. 4a:(a; — Vf -^ x — \
1. x^^^^2^-\-2x 20. a:^ - 1 + 2(a:2 _ 1)
8. 2a:4-2ar'-2aV+2a2a: 21. ar^ - y^ _|. ^ _ ^3
9. 2/^ + 2/^ + y + l 22. a^-y + ar^-y*
10. a^ — 2a^x — x-\-2a 23. a:^ - y^ _ ^^ _|_ ^2
11. x^ -{-Zy -Zx- xy 2^. 3? - f - {x - yY
12. ^ - z^ - z-^l 25. 4a2 - a^a:^ 4- a:2 _ 4
13. ab -by - a-{-y 26. a^^-y^+ar^-y^+a^-y
27. 4aa:3 _j_ g^^ - 8a - 4aa:2
28. a(3a - a:)2 - Qax^ + 2a:3
29. a:^ _ 8 _ 7(^ _ 2)
30. 4(a:3 _!_ 27) _ 313. _ 93
31. (2a: + 1)3 - (2a: + 1) (3a: + 4)
32. (2a: - 3)3 + 2a:2 - 9a: + 9
FACTORING 151
If 33. a:^ - 7a: - 6
i 34. x3 - Sx^ - lOo: + 24
[ 35. :t^ -Sx^ + 17a: - 10
Factor and check each of the following miscellaneous
examples :
36.
a'-Sa^
44.
a:^ - a:2 + a: - 1
37.
a" - 16a:2
45.
a:^ - 9a:2 + jg^
38.
a2 - Qax + 9x^
46.
0:6 + (a + ^)3
39.
ax — hx -\- ay — by
47.
a^ - y^
40.
a^-a-2
48.
(a + by - 2(a + b)p + p2
41.
3^-a^-\- 2(x - a)
49.
(a + 6)2 _ (a + 6) - 2
42.
3a2 - 4a - 4
50.
x^-\-a^-a^-a^
43.
x' + y'
51.
x + a^ — a — a^
52. Make up and work an example to illustrate each case
in factoring treated thus far.
53. How many of the examples in this Exercise can you
work at sight?
Special Cases Under Case III
100. By the Grouping of Terms we may often reduce an
'Qxpression to the difference of two perfect squares.
Ex. 1. Factor a:^ - ixy + 4i/2 - 9z\
a;2 _ 4xy + 4?/2 - 9^2 = (x^ - ^xy + 41/2) _ 9^2
= (x - 2yY - 9^2
= [{x - 2y) + 32] [{x - 2y) - Sz]
= {x -2y + Sz) {X -2y - Sz) Factors
Ex. 2. Factor a^ - a:^ - i/^ + 6^ + 2ah + 2xy.
o2 - a:2 - 1/2 + 62 + 2db + 2xy = {a^ + 2ah + 62) -(a;2 - 2xy + 1/^)
= (a + 6)2 -{x- 2/)2
= (a + fe + ic - ?/) (a + 6 - a: + 1/)
Let the pupil check the above examples. r actors
152 SCHOOL ALGEBRA
EXERCISE 43
Factor and check:
1. a^-]-2ab + b^-x^ ll. 2ab -\- x^ - a^ - b^
2. a2 - 2a6 + 62 _ 4^2 ^g. x'^ -\- a^ - y^ - 2ax
3. a^ — x^ — 2xy — 2/^ 13. a^ -\- y'^ — x^ -\- 2ay
4. 9a2 - a:^ - 4a:y - 4?/2 i4. a^ - a:^ - 2a:2!/ - 1/2
5. 16a2 - ar^ + 2a;!/ - 2/2 15. a:^ _ ^/Z _ i _ 22/
6. m^ — a;2 — 2/2 — 2x1/ 16. 1 + 2xy — ^ — y^
7. a2 + 62 + 2a6 - 4a;2 17. c^ - a2 - 6^ + 2a6
8. a2 + 62 _ 4a;2 4- 2a6 is. a2 + 62 - c2 - 2a6
9. a2 - 4a:2 + 62 + 2a6 19. 2a6 + a262 + 1 - a:2
10. a:2-2a6-a2-62 20. 222-42-22^ + 2
21. 2O2/2 + a:2 - 4?/2 - 2522
22. o2 + 2a6 + 62-c2-2c(i-(^
23. a;2 + 42/2 _ 9252 _ 1 _ /^^y _ 52
24. 9a2 - 25a;2 + 462 - 1 - 10a; - 12a6
25. a2 - 962^2 - 1 + 66a: - 10a6 + 2562
Factor and check each of the following miscellaneous
examples :
26. ax — bx -^ ay — by 3^. a^ — ¥ + x^ — W
27. a2 - a:2 - 2x2/ - !/2 35. a^ + W - y'^ -\- 2ab
28. a2 + ax — a6 — 6x 36. a^ — 27?/^
29. a2 - 2a6 + 62 - x^ 37. a^ - Qay + 9y^
30. 2a + 26 - 3a - 36 38. a^x - 16x2/^
31. 4a2 + 4a + 1 - 62 39. 3a2 - 4a + 1
32. 9x2 - 4a2 - 4a6 - 62 40 ^^_g(^a + bf
33. 9x2 _^y2^^__y 41 a^ + 2/^
I
FACTORING 153
r42. Make up and work an example in each case in factor-
ing treated thus far.
43. How many of the examples in this Exercise can you
work at sight?
101. The Addition and Subtraction of a Square will some-
times transform a given expression into a difference of two
perfect squares.
Ex. 1. Factor a' + a%^ + b\
Add and subtract a%'^.
a* + a262 4_ 54 = (j4 4. 2^262 + 6* _ ^^252
= (a2 + 62)2 _ ^^252
= (a2 + 62 + ab) {0? + 62 - ah) Factors
Ex. 2. Factor ic^ - IxY + 2/^-
Add and subtract ^x^y"^.
x^ - 7a;2?/2 + 2/4 = 3,4 _^ 2xhf + y^ - 9xY
= (x^ + y^y - 9xY
= {x"^ +y^ + Sxy) (a;2 + y^ - Sxy) Factors
Let the pupil check the above examples.
EXERCISE 44
Factor and check:
1. c' + c'^x' + x' 6. 49c^ - llcW + 25(i^
2. 3:^ + 0:2+1 7. 16a:4-9a;2 + l
3. 4x^ - 13x2 + 1 8. lOOx^ - 61a;2 + 9
4. 4a4 - 21a262 + 954 9 225a^64 - 4a262 + 4
5 93.4 _|_ 33,y _}_ 4^/4 iQ 32a^ + 26^ - 56a262
11. a^ + 46* 12. l + 64a:^ 13. a:y + 324
154 SCHOOL ALGEBRA
Factor and check each of the following miscellaneous
examples :
14. a^ + 2a^x^ -\- a^ 20. 4a'^ - ^a^x^ + a^
15. a' + a^x'^ + x' 21. a^ - ax - Qx"
16. a^ + 4aV + 3a:* 22. ¥ -^ a^b^ -{- a"^
17. a^-4x^ 23. a^ + 2a^b^ -\- b^
18. a^ + 4x* 24. a^-x^
19. 4a^ - Ua^x^ + a:^ 25. a« + 64a:«
26. Make up and work an example in each case in factoring
treated thus far.
102. Other Methods cf Pactoring algebraic expressions will
I e treated later. Thus it will be found that
a2 + 62 = (a + 6 -f V2ab) (a^ + b^ - V2ab)
Also, a2 + 62 = (a + V^b) (a - \/^6)
Factoring by use of the Factor Theorem is treated in
more advanced text-books.
103. General Principles in Factoring. It is important in
factoring to reduce each expression to its prime factors.
Therefore it is important to use the different methods of
factoring in such a way as to obtain prime factors most
readily.
Hence, in factoring any given expression, it is useful to
1. Observe, first of all, whether all the terms of the expression
have a common factor {Case I); if so, remove it.
2. Determine lohich other case in factoring can be used next
to the best advantage.
3. If the expression comes under no case directly, try to dis-
cover its factors by rearranging its terms; or by adding and sub-
tracting the same quantity; or by separating one term into two
terms.
155
I FACTORING
. Continue the process of factoring until each factor can be
resolved no further,
EXERCISE 46
Review
Factor:
1. 3a;3 - Sx 26.
2. 2a;3 - 8x^ + Sxy^ 27.
3. x^ - lla:2 + SOx 28.
4. 4x* + 5xh/ — Qxy^ 29.
5. 12a2 - 2ab - SOb^ 30.
6. a;^ - 1 - "/ + 2y 31.
7. 40a3 - 5 32.
8. IQx* - ^Ox^y + 25xY 33.
9. x^ -^Sax - Sa - X 34.
10. Sx"^ - Sx 35.
11. 4a^ - 5a2 + 1 36.
12. 2x8 _ 32 37,
13. a:2 + 4x - 45 38.
14. 4x2 + 2a - a2 - 1 39.
15. 5ax^ - 5a 40.
16. 18x3 _ 3^.2 _ 363- 41
17. x* + 3xV + 42^ 42.
18. a^x^ - 9x2 - a2 -I- 9 43
19. 110 - X - x2 44.
20. 3x2 ^ i^jcy _ 302^2 45
21. 7a - 7a%* 46.
22. 6x2 4. 14a; -[- 8 47.
23. x^ - (x - 2)2 48.
24. 3a + Sa* 49.
25. a^ - a2 + 2a - 2 50.
6x5 _ 2x - 4x2
1 - 23^2 + z*
128 - 2y^
1 - a2 - 62 _ 2a6
21a2 - 17a - 30
3.12 _j_ yl2
8x3 + 729^9
405xV - 45x4
a5 _ 4^3 ^ 5^2 _ 20
(c-hdy -1
(x - yy + 2(x - 2/)
24x2 + 5xy - 36i/2
x3 - 2x^2/ - 4x2/2 + Sy*
(a2 - 6)2 - a2
24+2:2 + 1
(a2 - 62 - c2)2 - 462c2
21x2 - 40x1/ - 21t/2
32 +n5
5x^ + 5xy^
m} + n^
2ax3 + iai/'
1 + X — x^ — x^
x2 - 9 - 7(x - 3)«
4a4 - 37a2 + 9
x« -64.
156 SCHOOL ALGEBRA
51. a;3 - 27 ^ 7{x - 3) 56. 4(^2 - 52) - 3(a + 6)*
52. 32a;^i/ - 2/2^0 57. (a - 6)3 + (a: - yY
53. (a:2 + j/^)^ - 16xV 58. (a^fe - aV^Y
54. x^ + a;2?/2 - y'^z'^ - ^ ^ 59. x^^ + a»
55. aa:^ - ax — x^y + y 60. a;^ + 2/^ + (a; + t/)'
61. (fl - by {x +y) -\-{a - b) {x + yY
62. (a - by + 4(a - 6) (x + ?/) + 4(a; + 2/)2
63. ai2 - 1 65. 4a2 -962-1-66
64. 4a2 - 962 + 4^^ _ 55 gg (3,2 _ ^^2)2
67. (a:2 - 1)2 + (2a; + 3) (x - 1)2
68. a2 - 6^ - a^x^ + b^^
69. 3a;2 _ 27 + aa;2 - 9a
70. a? + 3a26 + 3a62 + 6^
71. a? - 3a2j + 3aa;2 - 2?
72. a26ca; — amnpx + mhipy — abcmy
73. 4a; + ian + x^ - Aa^ - n^ + 4t
74. 2{x^ - 8) + 7a;2 - 17a; + 6
75. a* - 46^ + a2 + 262
76. (Sx^y - Sxy^y
77. 18a;2 + 52xy - Qy^
78. (x + 1)' - x^
79. (1 - 2a:)2 - x*
80. aa;2 — ex + ax — c
85. a;4 - 492/2 + 9
86. xY - 4x2 + 4 - 2/2 - 4^22^2 _^ 4py
87. o27ix — bcni^yz + ocmxz — abmny
88. How many of the examples in this Exercise can you
work at sight?
89. Work again the odd-numbered examples on p. 88.
81.
7^ - 79x2 _|. 1
82.
a2 - 9 + 962 - 606
83.
x« - 4x* - 16x2 + 64
84.
(x2 + 3)3 - 64x«
6x2
FACTORING 157
104. Factorial Method of Solving an Equation.
Ex. 1. Solve a:2 + 5a: - 24 = 0.
Factoring the left-hand member, we obtain
(x +8){x -3) =0
If any factor of a product equals zero, the entire product equals
zero. Hence to obtain the roots for the above equation, we may
let each factor in the left-hand member equal zero and obtain the
value of X from the two resulting simple equations.
Hence we have for the above equation
x -}-8 = 0
a; = — 8 Root
Check for a; = — 8
x^ -^5x - 24
= 64-40-24
= 0
Also a; - 3 = 0
X = S Root
Check for x = 3
x^ + 5x - 24
= 9-1-15-24
= 24-24
= 0
Ex. 2. Solve x(x - 2) (3a; -h 4) (a; + 1) = 0.
Using the above method, we obtain
a; = 0, 2, - t, - 1 Roots
Check for x = 0. x{x — 2) (3a; -1- 4) (a; -j- 1) Let the pupil
= 0(0 - 2) (0 -t- 4) (0 -h 1) apply the checks
= 0(-2)4xl=0 for the other
values of x.
Ex. 3. Solve 7? -x^ = ^x-4:.
Transposing all terms to the left-hand member, we have
a;3 - a;2 - 4a; + 4 = 0
Hence, x\x - 1) - 4(a; - 1) = 0
{x ■- 1) (a;2 - 4) = 0
(x -l)(x +2)(x -2) =0
a; = 1, 2, - 2 Roots
Let the pupil check the work.
158 SCHOOL ALGEBRA
EXERCISE 46
Solve and check each of the following:
1. x^ - 5x + 6 = 0 14. a:2 + 2a: = 0
2. 0:2 _ a. _ 2 = 0 15. a:2 + ao: = 0
3. x^ -7x = - 12 16. a^ - a^x = 0
4. a:^ ~ a: = 6 17. a:^ + a:^ = 4a: + 4
5. a:2 = a: + 12 18. a:^ + a:^ - 9x - 9 = 0
6. a:2 - 16 = 0 19. a:^ - 5a:2 + 4 = 0
7. a:2 = 9 20. a:^ - a:2 - 4a: + 4 = 0
a xix^ - 4) = 0 21. 3(a:2 _ i) _ 2(3; + 1) = 0
9. a:^ _ 25a: = 0 22. 1 + a:^ = 2ar^
10. a:^ = 9a; 23. y* — 9y^ = 0
11. 2a:2 _ 33. ^ 1 = 0 24. p2 _ 3p -f 2 = 0
12. 3ar^ - 4a: = 4 25. 3m^ -4m + l = 0
13. a:^ — a:^ — 6a: = 0 26. z^ — 4z + 4 = 0
27. y* - 13!/2 + 36 = 0
Form the equation whose roots are
28. 3 and 4 30.-3,-7 32. 0, 2
29. - 5, 2 31. 1, 2, - 2 33. 2, 3, 0
34. The square of a certain number diminished by 4 times
the number equals 45. Find the number.
35. The square of a certain number increased by 6 times
the number equals 40. Find the number.
36. What number plus its square equals 12?
37. The square of a certain number diminished by 9 times
the number equals zero. Find the number.
FACTORING 159
38. The square of what number equals 25 times the
number?
39. The cube of what number equals 25 times the number?
40. Find two consecutive numbers whose product is 72.
41. If to 3 times the square of a certain number we add 4
times the number, the result equals 4. Find the number.
42. The depth of a certain lot equals three times the front,
and the area of the lot is 7500 sq. ft. Find the dimensions
of the lot.
43. The temperature at which iron fuses is 2800° F.,
which is 332° more than 4 times the temperature at which
lead fuses. Find the temperature at which lead fuses.
44. The area of Texas is 265,780 sq. mi. This is 29,240
sq. mi. less than 6 times the area of New York. Find the
area of New York.
45. How many of the examples in this Exercise can you
work at sight?
46. How many examples in Exercise 30 (p. 120) can you
work at sight?
CHAPTER IX
HIGHEST COMMON FACTOR AND LOWEST
COMMON MULTIPLE
105. Utility in the Highest Common Factor and Lowest
Common Multiple. The advantages in the knowledge and
use of the largest factor common to two or more expressions
and of their lowest common multiple are similar to those
found in arithmetic for the same principles. They aid in
reducing fractions to a simple form, in adding and subtract-
ing fractions, and in multiplying and dividing fractions.
Other advantages will appear later.
Why do we now proceed to make definitions and rules?
Highest Common Factor
106. A common factor of two or more algebraic express
sions is an expression which divides each of the given expreS'
sions without a remainder.
The highest common factor of two or more algebrai<
expressions is the product of all their prime common factors
Thus, the highest common factor, or H. C. F., of 4x2, 12x', anc
IQx^y is 4x2.
107. The Method of Finding the H. C. F. is to
Factor the given expressions, if necessary;
Take the H. C. F. of the numerical coefficients;
Annex the literal factors common to all of the expressions^
giving to each factor the lowest exponent which it has in any,
expression.
160
HIGHEST COMMON FACTOR 161
Ex. 1. Find the H. C. F. of ^x'y - I2xy'^ + ^f and 3a:y
f ^xjf - 12y\
6xh/ - 12xy^ + 61/ = 6y(x - yY
SxY + 9xy^ - 12y' = Sy^{x^ + Sxy - ^y^) = ^yKx + 4i/) {x - y)
:. H. C. F. = Sy(x - y)
Ex. 2. Find the H. C. F. of ?^a{a^h - ah^f and a\h - a)\
Zaia^h - ab^Y = Sa[ab{a - b)Y = Sa%^{a - by
a\b - aY = a\ - {a - b)Y = a\a - bY
:, H. C. F. = a^{a - bY
P EXERCISE 47
Find the H. C. F. of
1. 4:a%Qab^ 6. x^ - 3x, x'^ - 9 \
2. 5a^y, 15a:V 7. Ax^ + 6x, Ga:^ + 9a;
3. 24aV, 56a3a:2 8. a^ - 3^, a^ - x^
4. 24:xy, 48fla;2, 36a; 9. xy -y^ot^ - x
5. 34a5a:3^ Slrta;^ 10. ^a^ + 2a^ Aa^ - a
11. a;2 + a;, a:2 - 1, a:2 - a: - 2
12. ^a^x - ^ax?, Sa^^ - Sax\ ia^x\a - xf
13. 2a:3 _ 2x, 2x^ - 3a:, 4a;(a: - l)^
14. ^x^ H- ^xy — 42/2, ^x? + 4iXy — 3y^
15. Ss^ - 5a:2 _ 2x, 4a? - 5a;2 - 6a;, a:^ _ 4^ *
16. b - a% 36 - a'b - 2a'b, ¥ - a'b^
17. 1 - a\l - a^ 3a + 3a2 + 3a^ 1 + a^ + a^
18. Find the H. C. F. of the numerator and denominator
of the fraction iii Ex. 5 (p. 170).
19. Beginning with Ex. 17 (p. 170), treat each example
through Ex. 22 in the same way.
162 SCHOOL ALGEBRA
Find the H. C. F. of
20. {a'^b - aW)\ - aW(a - by
21. 9(x^ - xyf, 12a;2(x2 - y'^f
22. (a + b){x- y), (a - b) (y - x)
23. (a + b)(x- yy, {a - b) (y - x)^
24. 4.-x\x^-x-2,(2- xy
25. 3a2 - 10a + 3, 9a - a\ (3 - ay
26. x^{x — a)^ a*(a2 — x^)
27. In Exs. 1 and 6, name some common factor of the two
given expressions which is not their H. C. F.
28. Write two expressions whose H. C. F. is aV.
29. Write also two expressions whose H. C. F. is Sx{x — 1).
30. Write three expressions whose H. C. F. is a{x — b).
31. How many of the examples in this Exercise can you
work at sight?
Lowest Common Multiple
108. A common multiple of two or more algebraic expres-
sions is an expression which will contain each of them with-
out a remainder.
The lowest common multiple of two or more algebraic
expressions is the expression of lowest degree which will
contain them all without a remainder.
Thus, the lowest common multiple, or L. C. M., of Sa^, Wx, and
4ax2 is 12a^x'^.
109. The Method of Finding the L CM. is to
Factor the given expressions , if necessary;
Take the L. C. M. of the numerical coefficients;
Annex each literal factor that occurs in any of the given eX"
LOWEST COMMON MULTIPLE 163
pressions, giving the factor the highest exponent which it has in
any one expression,
Ex. 1. Find the L. C. M. of Sx^ - 9^^, a^ - 9x, and
x^-6x + 9.
Sx* - 9x' = 3x\x - 3)
x^ -9x = x(x +S) (x - 3)
x^ -Qx -\-9 = (x - 3)2
. • . L. C. M. = SxKx +S)(x - 3)2
Ex. 2. Find the L. C. M. of {a'b - ab^f, 2ab{b - af, and
a\a^ - 62)2.
{a'h - a¥y = [ab{a - b)Y = a?b\a - by
2ab(b - ay = 2ab[ - {a - 6)]^ = 2a6(a - 6)^
^2(^2 _ ^2J2 ^ Q2(a + 6)2 (a _ 6)2
. • . L. C. M. = 2a%'{a + by (a - by Ans.
EXERCISE 48
Find the L. C. M. of
1. 3a26, 2a62 6. 12a26, 16a62, 24a262
2. 12a2a:2, 9ay 7. 2x{x + 1), a:2 - 1
3. 2aCySbc,4ab 8. 3a2 + 3a6, 2a6 + 26^
4. 3a26, 4ac2, 662c 9. 7x^ 2x^ - Qx
5. 42a:y, 2Syh^ 10. a:^ - 1, a;2 - 1
11. x^ — 1/2, 0^2 — Sxy + 2!/2
12. 3x3 _ 3^^ 5^2 _ 12a: + 6
13. 5aa:2(ar — yY, 3bxy(x^ — y^)
14. :t^ -Sx^- 40a:, a:2 - 9x + 8
15. a2 — 62, a^ — 6^ a^ + 6^
16. 6a:2 + 6a:, 2a:3 _ 2x2^ 3a;2 _ 3
17. 4a26 + 4a62, 6a26 - 6a62, 3a2 - 362
18. 2a:2 + a; - 1, 4a:2 - 1, 2a:2 + 3a: + 1
19. 30:^ - 3, 6a:2 _ i2a; + 6, 20:^ ^ 2a:2 + 2x
164 SCHOOL ALGEBRA
20. 12a:3 _ 2x^ - 140a;, \M + 6a; - 180, ^x^ - 39x2 ^ 53^
21. 1 — a: 4- a:^ — a:^^ ;[ _l_ ^ _l_ ^2 _|_ ^^ 2a; — 2x^
22. {x - 1)3, 7a:!/3(ar^ - 1)2, \^x^y{x + 1)^
23. ISa:^ - 12a;2 + 2a:, 27a:^ - ?>x', \%^ - 24^2 _^ g^.
24. (a: - 1) (a: + 3)2, (x + 1)2 {x - 3), (a:2 - 1)2, ar^ _ 9
25. Find the L. C. M. of the denominators of the fractions
in Ex. 18 (p. 181).
26. Find the L. C. M. also of the denominators of the
fractions in each example from Ex. 21 to Ex. 28 (inclusive),
p. 181.
Find the L. C. M. of
27. (a^h - a¥)\ aW{a + 6)2
28. {abc - body, (3a^c - 3acd)^ Qa^c^ - 6aW
29. (a26 - a62)2, (^2 _ ^5)2^ (^3 ^ ^2^)2
30. 9(ar^ -xy)\ 12(a:2 _ ^^2^2^ jg^^ ^ ^2^)2
31. a — 6, 6 — a
32. 9(a - 6)2, 12(6 - a)2
33. (a + 6) (a: - 2/), (a - h) {y - x)
34. (a + 6) (a: - y)\ {a - h) {y - xf
35. 4 - a:2, a:2 - a: - 2, (2 - a:)2
36. a:2(a; — of, x(a^ — x^),
37. Find two consecutive numbers the difference of whose
squares is 5.
38. Make up and work an example similar to Ex. 37.
39. The reclaimable swamp land in the United States and
the land that is capable of irrigation equal 178,000,000 acres
all together. If the irrigable land exceeds the swamp land
by 22,000,000 acres, how many acres of each of these kinds
of land are there?
LOWEST COMMON MULTIPLE 165
40. The distance from New York to Havana is 1410 mi.
If a steamer leaving New York travels at the average rate
of 260 mi. per day, and one leaving Havana at the same time
travels at the average rate of 280 mi. per day, how many
days and hours will elapse before the two steamers meet?
41. The distance of the sun from the earth is 92,800,000
mi. This distance exceeds 107 times the diameter of the sun
by 95,200 mi. Find the diameter of the sun.
42. A man bequeathed $20,000 to his wife, daughter, and
son. To his daughter he left $2000 more than to his son, and
to his wife three times as much as to his son. How much
did he leave to each?
43. The distance of the moon from the earth is 238,850
mi. This exceeds 110 times the moon's diameter by 1030 mi.
Find the diameter of the moon.
44. If 10 m. exceeds 10 yd. by 33.7 in., how many inches
are there in a meter?
45. Write a common multiple of the expressions in Ex. 1,
which is not their L. C. M.
46. Write a common multiple of the expressions in Ex. 10
which is not their L. C. M.
47. Write two expressions whose L. C. M. is 24a^6V.
48. Write two expressions whose L. C. M. is 12a:^(a; — 2)^
(X - 1).
49. Make up and work an example similar to Ex. 27. To
Ex.31. To Ex. 47.
50. How many of the examples in this Exercise can you
work at sight?
51. How many examples in Exercise 35 (p. 131) can you
work at sight?
CHAPTER X
FRACTIONS
110. TTtility of Fractions. In algebra, as in arithmetic,
fractions are useful in indicating new units, and in indicating
quotients and thus opening the way to save labor by cancel-
lation.
In algebra fractions also have other uses besides those
which appear in arithmetic. Thus, in algebra, a fraction is
often useful in expressing a general formula.
Ex. If an automobile goes a miles in h hours, how far
would it go in c hours at the same rate?
Y = no. of miles the automobile travels in 1 hour
0
— = no. of miles the automobile travels in c hours
b
Why do we now proceed to make definitions and rules?
111. A Fraction is the indicated quotient of two alge-
braic expressions. This quotient is usually indicated in the
form -.
b
The fraction - is read "a divided by 6," or, for brevity,
b
"a over b."
Note that the dividing line of a fraction takes the place of
a parenthesis and is in effect a vinculum.
166
FRACTIONS 167
Another method of writing the preceding fraction is a/6. This is
called the solidus notation. It is convenient in printing mathematical
expressions, and is much used in European mathematical hterature.
X 4- 1
— — — written in the solidus notation would be {x + l)/{Sx — 5)
3x — 5
The numerator of a fraction is the dividend and the de-
nominator is the divisor of the indicated quotient.
Terms of a fraction is a general name for both numerator
and denominator.
EXERCISE 49
1. If three boys weigh a, b, c pounds respectively, what is
their average weight?
2. If four boys can run the quarter mile in p, q, r, s sec-
onds respectively, what is their average time?
3. How many acres are there in a field a feet long and h
feet wide?
4. How many acres are there in a field c rd. X ^ rd.? In
one/ yd. X e ft? p ft. X g rd.?
5. If sugar is worth a cents a pound, how many pounds
can be obtained in exchange for b pounds of butter worth
c cents a pound?
6. If coal is worth c dollars a ton, how many tons of coal
can be obtained in exchange for p tons of hay worth b dollars
a ton?
7. Make up and work a similar example concerning c
calves, worth a dollars each, exchanged for chairs worth d
dollars each.
8. If coal is worth c dollars a ton, how many tons can be
obtained in exchange for / bushels of wheat worth h cents a
bushel and for w bushels of corn worth y cents a bushel?
168 SCHOOL ALGEBRA
9. Who first used the letters a, 6, c to represent known
numbers? (See p. 268.) Tell all you can about this man.
10. Before the use of a, b, c, what other symbols were used
to represent known numbers? Discuss the relative advan-
tages in these different sets of symbols.
11. As a notation, in what respects is a/b superior to
a H- 6? To --? In what respects is it inferior to each of these?
b
12. How many examples in Exercise 45 (p. 155) can you
now work at sight?
112. An integral expression is one which does not contain
a fraction; as 3x^ — 2y.
An expression like 5x^ + fa; + } in which fractions occur only
in the numerical coefficients is sometimes regarded as an integral
expression.
A mixed expression is one which is part integral, part
fractional.
Thus, 3^2 + X - 5 + A^
' 3x2-2
113. Sign of a Fraction. A fraction has its own sign, which
is distinct from the sign of both numerator and denomina-
tor. It is written to the left of the dividing line of the fraction.
The sign of — t is — , and the sign of -^^ is + understood.
0 0
General Principles
114. A. // the numerator and the denominator of a fraction
are both multiplied or divided by the same quantity, the value of
the fraction is not changed.
For if a dividend is denoted by D, its divisor by d, and the quo-
tient by Q T)
~ = Q, Siiid D = d X Q
a
I
TRANSFORMATIONS OF FRACTIONS 169
If m denotes any multiplier, Dxm=dXmX Q, or
^=Q (Art. 15, p. 18)
Also if m denotes any divisor except zero,
D^m =d ^mXQ, or f "^ ^ = Q (Art. 15)
a -j- m
115. B. law of Signs. By the laws of signs for multipli-
cation and division (see Arts. 50, 62, pp. 59, 77),
a— aa— aaa a a
b -}) h b -V be -bXc -bX -c
x + y ^ x + y ^ _ X + y
y - X - {x - y) X - y
Or, in general,
The signs of any even number of factors of the numerator
and denominator of a fraction may be changed without changing
the sign of the fraction.
But if the signs of an odd number of factors are changed, the
sign of the fraction mu^st be changed.
Transformations of Fractions
I. To Reduce a Fraction to its Lowest Terms
116. A Fraction in its Lowest Terms is a fraction whose
numerator and denominator have no common factor.
To reduce a fraction to its lowest terms, as in arithmetic,
Resolve the numerator and the denominator into their prime
factors, and cancel the factors common to both.
Ex. 1. Reduce ,^ , , „ to its lowest terms.
Divide both numerator and denominator by 12a^x^ (see Art. 114).
. SQa^x^ Sa
^Sa^x^y^ ^y^
Ans,
170 SCHOOL ALGEBRA
E 2 Q^^ - 12^^ 36(3a - 46) ^ 36 .
"^^ • 12a2 - 16a6 4a(3a - 46) ~ 4a
Notice particularly that in reducing a fraction to its lowest terms
it is allowable to cancel a factor which is common to both denomina-
tor and numerator, but it is not allowable to cancel a term which
is common unless this term is a factor.
Thus, — reduces to - ;
ac c
but in , a of the numerator will not cancel a of the denominator.
a +y
This is a principle very frequently violated by beginners.
EXERCISE 60
Reduce each of the following to its simplest form:
^ 27 g 3a' - 6a'b ^^ 45(a: - yY
36 4aW - 8a¥ 18(a; - yY
2 108 g 2a ^g a^b + ab^
144 - 4a2 - 2a 2a^b - 2ab''
150 Gax - \2ay ^x'y - \2xy^
My^ ^^ 4x-h4y ^^ 6aW-\-12a¥
11. 18.
12aV iax + 4ay 9a'b + 18a262
g 12a:V a:^-y^ .^ 2a;^ - Sxy
15s^y^^ ' (x + 2/)2 • 4a:3 _ g^y2
^ Sa^x 13 12aV - Sa^xy ^ 49x^ - 64^^
^ 127?f^ ^^ 8{7? - 1) 2^ ^-21
mxy^'^ \2x - 12 a:2 - 6x + 9
22 (3; - y)^ (a: + y)^ 24 6a;^ - a^y - 2^^^
' (x2 - 2/2)3 • 6a:2 _ 7^^ _j_ 22^2
23 20:^ - 8y2 ^5. ^^ + ^)' " ^'
4a:2 _ 2xy - 12y^ a^ - (6 + c)^
TRANSFORMATIONS OF FRACTIONS 171
26. l-C"-'^)' 29 ^-^^
a^-(i
X-l)2
12a;2-
2oa; -
24a2
4a:2_
2ax -
6a2
a^-
-8
30.
31.
x^ - :(? - 6a:2
a:^ — 2/^
x' + xy + 2/^
ax — bx — ay -j- by
x^ — z^ — 4: — 2xy — 4:Z -\- y^
32.
x^y^ + 2a:!/2 + 4?/^
x^ — 4 — 22/2 — 4a: + !/^
1+4
What is the correct value of the fraction ^ . ^ ? If
6 + 4
the 4's are struck out, what does the value of the above
fraction become? Is it allowable, therefore, to strike out the
4's in the above fraction?
34. Make up and work an example similar to Ex. 33.
35. Which of the following fractions can be simplified by
striking out the 4's?
1+4 a; + 4 4a; 3X4 4a 4a
11 + 4 2/ + 4 4(1 + 2/) 4 + 11 a; + 4 4a;
36. Make up and work an example similar to Ex. 35,
involving 3's.
37. Which of the following can be simplified by striking
out the 62's?
b^-\-x b^ b^-4 a^b^ o^
b^ + y b'^y 36^ + 4 a^ + 6^ 6V
38. Which of the following can be simplified by striking
out a + 6 in both numerator and denominator?
a + 6 5(a + b) 3x(a + b) 3(a + b)
a + b + c 3(a + 6) + c 4y{a + b) 5(a + 6) + c
39. Make up and work an example similar to Ex. 38 con-
cerning the striking out of x. Of (p + q)^.
172 SCHOOL ALGEBRA
40. Why is it allowable to subtract 4 from each member
of the equation a; -h 4 = a + 4 and not from each term of
the fraction ?
a + 4
41. How many of the examples in this Exercise can you
work at sight?
EXERCISE 61
(x — 2Y
1. Reduce ^^- ^ to its lowest terms.
4 — or
(x - 2y ^ {x -2)(x- 2) ^ (2 -x){2- x) ^ 2 -x
4-x^ {2-i-x){2 -x) {2+x){2 -x) 2+x
Check. Let a: = 1, then, ^f ~ ^]' = 1-=^ = 1
' '4-^2 4-1 3
., 2-^2-11
^^' 2T^=2-iri = 3
Reduce to simplest form and check the work:
(a — by ^ a -{- b — c
3.
4.
5.
6.
b'
-a"
(2a-
-yY
f
-^
9- to''
m^
- 7to + 12
9-
-r'
(.X-
-3)^
2-
■ y
-4
a —
h
h — a
c2 - (a + bf
9
6?/ -3a:
12ai/ - 6aa;
10.
6a26 - 3a3
4a262 - 8a63
n
a:3-27
9 - 6x + ar^
19
4 - (a + 6)2
(a - 2)2 - 62
T»
ax — bx — ay -]- by
TRANSFORMATIONS OF FRACTIONS 173
Without changing the value of the fraction
14. Change each of the following so that the denominator
of the fraction shall be a — b.
-3 3 X X- y - Sx 2a - 36
b — a b — a b — a b — a b — a b — a
15. Change each of the following so that the denominator
shall be {x — y) {x — y).
3 -4 a-b
(y - x){y - ^) (y - x) (x - y) {y - x)
16. Show that — r equals
{y -x){z- y) {x - y){y - z)
17. By changing signs of factors, write each of the follow-
ing in three different ways:
5 a — b _ a — b a — b (b — a) {c — d)
a-b X - y c - d (x - y) (y - z) {y - x) {y - z)
Solve the following equations, after reducing the fraction
in each equation to its simplest form:
18. ^^^ + 2a: = 5 20. ^^^^ = 7 + ar^
x — I a: + 3
19. - a:^ = 7 21. — = b — X
a: — 1 x^
^^ ax — Za -\- bx — Zb ^ ^
22. ; = 17
23. — ^±JIli^ = 15 - 3a;
a + 6
ax -
- 5a - 6a: + 56
a-b
a^ -
-8 a^ + S
24. - ~ — -^ = 20
a: -2 a: + 2
25. How many of the examples in this Exercise can you
work at sight?
174 SCHOOL ALGEBRA
II. To Reduce an Improper Fraction to an Integral or
Mixed Quantity.
117. An Improper Fraction is one in which the degree of
the numerator equals or exceeds the degree of the denomi-
nator.
Since a fraction is an indicated division, to reduce an im-
proper fraction to an integral or mixed expression,
Divide the numerator by the denominator;
If there is a remainder, write it over the denominator, and
annex the residt to the quotient with the proper sign.
3^ + 4x^-5
Ex. Reduce
x^-j-x + 2
x^ +4x2 - 5
x^ + x2 + 2x
x^ +x +2
X +S
Sx^ -2x - 5
Sx^ + 3a; + 6
- 5x - 11
= X +S Ans,
x^ +x +2 a:2+x+2
When the remainder is made the numerator of a fraction with
the minus sign before it, as in this example, the signs of terms of
the remainder must be changed, since the vinculum is in effect a
parenthesis (see Art. 41, p. 50).
EXERCISE 52
Reduce each of the following to a mixed quantity:
« 121 ^ 181
2. 3.
9 17
^ 10a^3^ + 5ax -7 - a
1.
32
5
4,
0:2-
-2x + S
X
ti
4a:3 + 6a: - 5
7.
5ax
2x x-h 1
I
TRANSFORMATIONS OF FRACTIONS 175
g x' + ?.xy- 2y^ - 1 ^^ 9a'
3a:* - 13x -
28
+ 2- a
X —
ar*+l
1
x^ + x-1
a;2 - a: - 1
2x^ + 7
^^-{-x+l
x^-\-x^ - X
- 1
16.
10. -^ 17.
11. -i: ■ 18.
3a2 - 26
a:^ + a;^ - 4a; + 7
a: + 3
2a2
a-{-b
0^ - x!^ -h 3^ - 2x
a^ + 1
12. — — ^^ 19. (To three terms.)
13. — ■ 20.
14. t: 21
1 -\-x
1
l-\-x-
8
a:2 + 2 2 + a:-a:2
22. Make up an improper fraction with a monomial de-
nominator and reduce it to a mixed number.
23. Make up an improper fraction with a binomial de-
nominator and reduce it to a mixed number.
24. How many examples in Exercise 1 (p. 8) can you
now work at sight?
III. To Reduce a Mixed Expression to a Fraction
118. To Reduce a Mixed Expression to a fraction, it is nec-
essary simply to reverse the process of Art. 117. Hence,
Multiply the integral expression by the denominator of the
fraction, and add the numerator to the result, changing the signs
of the terms of the numerator if the fraction is preceded by the
minv^ sign;
Write the denominator under the result.
176 SCHOOL ALGEBRA
g^ + y'
Ex. x + y- ^^
^ (a; + y) (a^ - y) - {^'^ + j/^)
a: - 1/
a;2 _ 2/2 _ a;2 _ ^2 _ 2^2
a; - 2/ X - 2/
Ans.
I
Reduce to a fraction
y - X
EXERCISE 63
1. 3^ . 2. 12| 3. 13iV
4. a - 1 + - 10. a - a: + 1 - ^-^
a a -\- X
-±- 11. ?^^
X — 1 a — 2
5. a: + l+^L_ 11. ^?LZ^^a-l
6. a:2 + a;-l ^ 12. a: - a - ^^^^ ^' + y
a; — 1 X -\- a
,. 4.-2-f^ 13. 1-^--^ + "'
2a: + 1 26c
7.-262 ,. A J. (2 - 3a)2
a + 26 4
16. M_li)! + 2a6
4
IB TRANSFORMATIONS OF FRACTIONS 177
IB 20. The distance from New York to Chicago is 912 mi.,
which is 100 mi. more than one fourth of the distance from
New York to San Francisco. Find the latter distance.
21. A running horse with a rider has gone 1 mi. in 1 min.
35^ sec, which is 13 J sec. more than three times the time in
which an automobile has gone one mile. Find the latter
time.
22. Make up two mixed numbers of your own and reduce
them to improper fractions.
23. How many of the examples in this Exercise can you
work at sight?
IV. To Reduce Fractions to Equivalent Fractions of
THE Lowest Common Denominator
119. To Reduce Fractions to their lowest common denomi-
nator, as in arithmetic, we
Find the lowest common multiple of the denominators of the
given fractions;
Divide this common multiple hy the denominator of each
fraction;
Multiply each quotient hy the corresponding numerator; the
results will form the new numerators;
Write the lowest common denominator under each new
numerator,
2 3 5
Ex. Reduce - — , ^-t-, and - — - to equivalent fractions
6ax 4:a^x oax^
having the lowest common denominator.
The L. C. D. is 12aV.
Dividing this by each of the denominators, we get the quotients
^ax, 3x, and 2a.
178 SCHOOL ALGEBRA
Multiplying each of these quotients by the corresponding numer-
ator and setting the results over the common denominator, we obtain
Sax 9x 10a ,
Ans.
! 12a^x^' 12aH^' 12a%2
i
' EXERCISE 64
Reduce the following to equivalent fractions having the
lowest common denominator:
1.
5 7
8' 12
3 4 9
2.
5^ 15' 20
2x 5x
3.
9' 6
12a 7 a
4.
56 ' 10' b
1 2 1
b.
2a¥' a^V ah
6.
2 3 1
3a2' 4aa:' ^""^ x
ac ab be ad
' M'^'^'Yc
s. / ,2, %
a^ — a a — 1
3 ^ 1 1 1
1 + x' ' a;' x + or^
10 ^ 1
^° ^_1'^_1
11 ^ 11
"' 4x2 _ 9' 2a; + 3' X
1
'x
1+x 1-x
2-2x'^'3 + 3;r
13.
3
4 4
l'x2 + a.+ i'^
14.
3 4
a% + a¥' a^h - ab^
15.
1
5 3
3a:-
- 6' 2x + 4' x2 - 4
ifi
2
X X
a; - a:^' 3 _|_ 3^' 2 -2ar
i ADDITION AND SUBTRACTION OF FRACTIONS 179
Processes with Fractions
I. Addition and Subtraction of Fractions
120. The Method of Adding or Subtracting Fractions^ as in
arithmetic, is to
■Reduce the fractions to their lowest common denominator;
Add their numerators^ changing the signs of the numerator of
any fraction preceded by the minus sign;
Set the sum over the common denominator;
Reduce the result to its lowest terms.
Ex. 1. - a + -. h -
a — 1 a^ — a a
a a 1 1
"1 Zo ~ '
a — 1 1 a^ — a a
a^ - a^ +0^ -\-l +a -1
a{a — 1)
- a' + 2a2 + a - a^ + 2a jM ^^
a{a — 1) a
Ex. 2. Simplify -^ +, "^
x'' -1 x-hl I - X
The factors of x^ — 1 are x -{- 1 and x — 1. Hence, if the sign of
the denominator, 1 — a:, is changed, it will become x — 1, and be a
factor of a:2 - 1. But by Art. 115 (p. 169), if the sign of 1 - x is
changed, the sign of the fraction in which it occurs must also be
changed. Hence, we have
x"^ X X _ x"^ + x^ — X + x^ -\- x _ Sx^ .
x^ -l'^ x + 1'^ ^"^ ^~^ ^^^
Where the differences of three letters occur as factors in the va-
rious denominators, it is useful to have some standard order for
the letters in the factors. It is customary to reduce the factors so
that the alphabetical order of the letters is preserved in each factor, .
except that the last letter is followed by the first. This is called
the cyclic order. ^ I j
Thus, a — 6, 6 — c, c — a are written in the cyclic order.
180 SCHOOL ALGEBRA
Ex. 3. Simplify
(a — b) (c — a) (a — b) (c — b) (c — b) (a — c)
Changing c — b io b — c, and a — c to c — a where they occur,
we obtain
1 1.1
(a - 6) (c — a) {a - b) {b - c) {b - c) {c - a)
- b— c— c-{-a+a— b
{a — b) {b — c) (c — a)
^ 2a -2c ^ -2 ,
\ (a -b)(b - c) (c -a) (a - 6) (6 - c)
EXERCISE 66
Find the value of
3.
,3,21 ^ a h
1. 1 — D. — r-
2x X '^x a — b a -\- 0
„2 3,1 , 3a; + 1,1- 3a;
3a ^ax X 8 6
J 2__l Q g + l g-l
2ac 3a6 6c '22
a + 26 _ 6a- 1 ^ a; - 1 _ a; + 1
■ 2a6 6a2 " a; + 1 x - 1
^ 2a2a; + 3 , , 3a + a; ,^ x + 1 ^ . 7-3a;
3a — 46 2a — 6 — c , 15a — 4c
2x^y — 33 _ arz^ — i/^g y — 3x3^ _ 2
^^' Sx^y 2xy^ Qxh 3
a:-2 a; + 2 (m - 1)^ m^ - 1
ADDITION AND SUBTRACTION OF FRACTIONS 181
(a + 6)2 26 ,^ 3a: 2a; , lOx
4(a-6)2 a-6 a: + 2 a:-2'ar^-4
a:^ + a: a;^ — a;
1 1 . a:
3x-3 2a; + 2 6a:2_6
2a: - 1 4a; + 2 4x^-1
a: ,^ a: — 1 a: — 2
20. h 2
a;2 - 1 ^ a: + 1 a; - 1
3 4,2
21. —— — +
22.
23.
a: + l a; + 2 a; + 3
a: + 2 a;-3. 2a: + 5
2a;2 + a: - 1 4a;2 - 1 ' 2a;2 + 3x + 1
6 _ a6 _ ab^
^T6 (a + 6)2 ~ (a + 6)^
24. 2a:2/ J 3y ^ 3a: 3a;2 - Sy^
a;2 — i/2 2a: 2?/ 2a;i/
25 ^^~y _L l^^y _ 3a; + 2/
a; + 22/ a;2 — 4i/2 x — 2y
2 , 3a: - 1 2a: - 5
26. 1 + X —
27.
28.
a; — 1 a; + 1
2 a:-3
a; + 4 a;2 - 4a: + 16 a;^ + 64
5x 7 26
' 2(a:-3)2 3a: + 9 4a;2 - 36
Reduce each of the following fractions to its lowest terms
and collect:
3? -9 x-i x^-l^ (x + iy
182 SCHOOL ALGEBRA
31 46^ (a-hy g 3a + 36
• a2 - 62 a2 - 62 "^ 3a - 36
3a: , 4 , 1
32. -.^ 7 + ;^ +
0:2 - 1 ' 1 - a; 1 + a:
33. 2a -1 ^ 2
a^ — b^ a + b b — a
34. 3a:t/ y - a: ^ y + a:
Q? — 4y^ 2y -\-x 2y — X
a: - 1 1 + a: ' 1 - r^
36. ^^!±^'-_^+ 2/
a:2-2/2 a: + i/ t/-a:
^n '3 , 5 7a
37. h
8 -8a 4a + 4 Sa^ - 8
38.3+^-+ 5- 1 3
39.
X X — 1 1 — x^ x-\-l a: + a:^
1 1
(a: - 2) (3 - a:) 10 - 7a: + a:^ (5 _ 3.) (^ __ 3)
.n 2 3^4
+
(a - 3) (6 - 2) (a - 2) (2 - 6) (a - 2) (3 - a)
5
(a - 3) (2 - 6)
26 + a 26 - a 46a: - 2a2
41
x-^ a a — X x^
42 ^ + 1 2a:-l 2 7^
^ ' 6a:-6 12a: + 12 3 - 3^^ 12a:
a:^ - a: - 6 a:^ + 4a: + 3 _ 15a:
a:2 4- 5a: + 6 a:^ - 4a: + 3 9 - x"
a^ + 2a6 + 6^ _ 4a2 - 6^ a^ - 2a6 + 36^
a2 - 62 2a2 - 3a6 - 262 "^ a^ - 3a6 + 262
43.
MULTIPLICATION OF FRACTIONS 183
45. z +
9-0:2 ■ 3(^_j_3)2 5(x-3y 5x2-45
^^ 3a: + 2 , x 4 - x
a:2 - 5a: + 6 8a: - a:2 - 15 7a: - a:^ - 10
(a — 6) (a — c) (6 — c) (6 — a) (c — a) (c — 6)
{a — b) (a — c) (b — c) (b — a) (c — a) {c — b)
49. y + ^ I ^ + ^ I ^ + y
(a: -y) (x- z) (y - z) {y - x) {z - x) {z - y)
50. y^ + ?^- - + ""y
{x- y) {x -z) {y - z) {y - x) {z - x) (z - y)
SJ" 71 TTl T + Z T^ t: +
52.
(I — m) {I — n) (m — n) {m — I) (n — /) (n — m)
> y i ^ r l-a: 1 1 1? ^
a: ^a:+l La:2_3,_|_i a:4-lJ ) 3:^ + 1
53. Make up and add three fractions with monomial
denominators.
54. Also three with binomial denominators.
55. How many examples in Exercise 2 (p. 13) can you
now work at sight?
II. Multiplication of Fractions
121. The Method of Finding the Product of two or more
fractions, as in arithmetic, is to
Multiply the numerators together for a new numerator, and
multiply the denominators together for a new denominator, can-
celing factors that are common to the two products.
This method reduces the multiplication of fractions to
the multiplication of integral expressions, and enables us to
use again our knowledge of the latter process.
184 SCHOOL ALGEBRA
Ex. ^+J^xf~^lx ^
-*:+ 2/ ^ (x + 1/) (x - y) ^^
X a:(a;2 + y^) ^ (aL±4/) (a; ^y)
4:(x - y)
a;2 +?/2
Ans.
II. Division of Fractions
122. The Method of Dividing one fraction by another is
the same as in arithmetic. For
a c aXd b X c
b ' d bXd ' bXd
aXd
(see Art. 114, p. 168)
©(^^
bXc
Hence, to divide one fraction by another,
Invert the divisor and proceed as in multiplication,
x{x-{-l) x^-\-x+l ' {x + 1)2
_ (x -l){x^ +X + 1) {x^ - 1) {x^ - 1) (x + ly-
X{X + 1) ^ X^ +X + 1 ^ (X - 1)3
Ans.
{x + ly
X
The reciprocal of a number is the result obtained by divid-
ing unity by the given number.
Thus, the redprocal of 2 is 1 -J- 2 or ^; oi x la -.
z x
Hence, the reciprocal of a a fraction is the fraction inverted.
2 2.33
Thus, the reciprocal of ^ is 1 -^ q) that is, 1 X qj or ^.
„. ., , , . . .a . h ^a — h . X + y
Similarly, the reciprocal of -r is -; of ^ is , .
L
H 1 ^x^y
DIVISION OF FRACTIONS 185
EXERCISE 56
bx^y 2SaW ^ a^b^ + Sab ^ ab-\-3
Ua'c 15xy^ ' 4a^ - I * 2a + 1
21xf^28^ 7 a;^-9 . a; - 3
^ U^ ' 39s4 ' x^ + x ' x^-l
9a% ^ 28ax'^ . 21a^x (a - If ^ x + 1
ScPx 15b^c lOftc' o(a; + 1)^ (a - 1)^
492/" 40a:3 9a:2 - 1 i2a: - 18
15a; ^ 2a:(x +1) ,^ 2x' - x - I ^^x^ - \
2a:(2a: - 1) ^x" 2a;2 + a: - 1 a:^ - 1
a^i/ — ax^y . a^i/ — 2aa:^ + x'^y
a^x^ -\- a^x^y a? + ay
12. ^ — t-tt: -^ ^ — t-tt; 14. a; H X
(a +1)2 (a+1)' \ a;-l/ x= + l
3x^+x-2^ 4^-J /a; \ . /x^
2/^'
16. ^X^^X^^X^^
5a:i/ x-\-y {x - yf 2xy
^^ 3(a - 6)2 7(a2 - 62) ^ i4a&
4(a + 6)2'^ 9(a-6)3 8(a + 6)
a;2 + 2a: - 3 a;2 + 2a; - 15 . a:^ + 5ar^
a;2 4- a; - 12 a;2 + 2a; - 3 * a;^ + 4a;2
^^ 6a;2y - 4a;i/2 ^ 30a; + 20y ^ xy
45a;2 — 202/2 4a;22/2 x-\-y
2^ 6a;2 - 5a; - 4 6a;2 + a; - 2 2a;2 + 5a; - 12
* 2a;2 + 7a; - 4 4a;2 - 4x - 3 9^2 - 6a; - 8
a;«-l
3X.(. + 1 + 1)(.-1 + 1).
a;2(a;2 - 1)
186 SCHOOL ALGEBRA
«a-)(-Dx(-^)
23. ^ + 2/" X fl + ^ ^ . 'a:' -xy-\-y^
' 3^ + x^y -i-xy^ \ X - yj ' x^ - f
' a'-{x-\-iy l-{a-xy ' a-x-1
25 2-6-aa^-62-46-4 5^-a^-46+4
* 6-2-aa2 + 62 + 2a6-462_^2^4^_4
\a6 6c acj\b c a) ' a^H^c^
3s, (\ + l, + l-l)4^±^(lz^x(l+? + a)]
\a^ a:^ax/[_ ox \ x J _]
Vm -\- 2n m — 27i~| _^ Vm -{-2n _m — 2?i~|
L^Ti — 2n m + 2/1 J ' L^ — 2/1 m + 2n J
30. Write the reciprocal of each of the following: 3, a, 2x,
4 1 a_ J^ a + 2a; 1
5' 5' 2a;' 2x a - 2V a" - 2b'
31. Make up and work two examples involving both mul
tiplication and division of fractions.
32. How many of the examples in Exercise 15 (p. 60) can
you now work at sight?
IV. Reduction of Complex Fractions
123. A Complex Fraction is one having a fraction in its
numerator, or in its denominator, or in both.
In simplifying any complex fraction, it is important to
write the entire fraction at each step of the process.
29
I
REDUCTION OF FRACTIONS 187
X.1 -^ ^=;,X-^ = ^^^«*.
;
y y
When the numerator and denominator of a complex frac-
tion each contain fractions, the expression is often simplified
most readily if we
Multiply both numerator and denominator by the lowest
common denominator of the fractions contained in them,
i + i + i
Ex.2. Simplify ^ — ^ — -■
- + y + i
y z X
Multiplying both numerator and denominator by xyz, obtain
m xH + xy^ + yz^
124. A Continued Fraction is a fraction whose denomina-
tor is a mixed expression, having another mixed expression
in its denominator, and so on until the fraction ends.
1
Ex. Simplify
x-\-
1 -
2
1
1 , 1 ) , X -2
1 X -h
x"^ - 4x -2 x^ -4x
Ans.
X — 5
Hence, in general, to simplify a continued fraction,
Reduce the last mixed expression in the fraction to an im-
proper fraction (see the brackets in the examples) ;
188 SCHOOL ALGEBRA
Then invert the last fraction and multiply it into the numer-
ator under which it is placed (see the brace) ;
Thu^ alternately reduce a mixed number and invert a divisoi
fraction until the simplification is completed.
Simplify:
1
2x
4^2
1
1 -
'¥x
1
3
2
X
0^3
x'
(-i)
EXERCISE 67
4 1,1
-- X x- - 1 -—
? 3. ^ 5. ^+^
2 X a — 1
« 1 ab ^^ 2
2 - - 2d X- ~
i 4. ^ - 6. §.
. 1 2cd 6
a —
0:2 5
11.
J/ X
a; 1 — X
12.
1 + a: a;
9 _ ^ X 1 — X
1 -\- X X
9. ^^ y^. 13. ° + i
a — 1
a , a:
10. ^ °' 14.
\a 0 c J
J. _ _1_ J. c2(a + 6)2 - a^l
n I o
a26V
I
REDUCTION OF FRACTIONS 189
5 + «_2-l 21. 2a-l- "-^
.^ a X ax o ^
a; a 2 , 1 a
+ — a
a X a ax , 1 + a
,. 2(i^ - i) 1 , 1
16- ^^irr~^ 22. 3
1 .- 2x
I
l_(a6-c(^)2 ^^^ 3^
17 (a6 - 1)^ - c'cP x + 1
{cd + 1)^ - a^6^ _,
(ab + cdy-l 23. l-^4~
a:-l
18. 3--^^ a + b
ox
3
a — b
24. 1
a?
1 „ 8
'^-fT-a ^+2+4
i+- ^
26.
g 6 + c ^ A , 6^ + c^ - g^
1 1 ^ V "^ 26^ y
g 6 + c
4
a:
27. ; — X
a: + 2 a; — 2 oif
28.
1 1 + -^
' 1 + ^-^ 1-
1 -X 1+X
190
29.
30.
xy^ + x^y
SCHOOL ALGEBRA
^^ ~ U ^ c y w h^^U ~c) \l^~c^h)'
i_ _ /I 1 Y 1^ _ /I _ 1 Y / 1 _ 1 Y _ i"
62 V« ^y c2 \a h) \a cj b'
1 -2
31.
1 - 2x
1+2
1 + 2x |(i + a; + i;c2) - I
1 -2a;
Find the value
r.* 22 , 2
32. Of :; — when v = -•
l-\-2v 7
33. Of
-22 , 1
-^-^,when.= --.
34. Of i^ when F = — , m = 10.2, t) = 5, and r = ^•
35. Make up and simpHfy a continued fraction.
36. How many examples in Exercise 19 (p. 78) can you
now work at sight?
EXERCISE 68
Oral Review
1. Give the value of each of the following:
(1) 1-1.
(2) i - 1.
(3)l+&+£.
(4)
a -^b a —b
a 0
a 0
a.E^a,.,e.l)- (.)g-2)' (3,(1-D'
PROCESSES WITH FRACTIONS 191
3. On the foot rule show the meaning of i in. -^ 2. Of J in. 4- 2.
4. Divide each of the following fractions by 2: |, f, i, f, f,
2a a^ a a +b 6 7 3^ 5^ 462 362
V 26' 6' 2 ' a' a' 26' 46' a ' a *
5. Divide^ by 2. By 2a. 36. 46.
6
6. Divide 1 by each of the fractions -, -, -, ^, ^r-
7. Give the reciprocal of 3, -, 4, tj t, r; - •
8. Give the value of - when ^ = ^- When ^^ = q' !» o' o' o' 5»
__2 _1 6
3' a' c*
9. State the value of - when x= -. When x = -, -, — , — , —,
"" 3' 2' 5
15 1
10. Give the value of ;r when x = — -.
2 + X 5
-11 wu^' l-4„^-334a-la4al 2a 1«
11. What IS -of-? Of-, ^, ^, -2-, ^, y, 2"' -3, 4^?
12. If 4 is subtracted from both numerator and denominator of
■<fij, is the value of the fraction changed? By how much?
13. If a = - - and 6 = - -, state the value of -r-. Of - tt-
2 2 6 46
14. What is the value of 1 ^ 2/3? Of 1 4- a/6? Of 2 -^ x/2y?
15. Simplify those of the following fractions which can be reduced
to lower terms :
4x 4a; 4a 4 a — 6 a -f 6 a^ + 6^ d^b
4a +6' 4a +46' 46' a + 4' 6 - a' a^ + ¥' a^ + x^' a^x
Give the value of
_ -3-1 -2 8 _8^ x" -\
8 ' - 4 - 6' - 2/3' a/6' \ - x'
17. Of ia' 4- \(^ Ja' ^ Ja2 ^x^ 4- Jx^ Jx' ■¥ \x 1-5- Jx«
1 -5- §x2 2a2 ^ fa 2 -^ 5/6a;
192 SCHOOL ALGEBRA
18. Make up an illustration to show the value of 5 X 0 (for
instance, in connection with a pupil's mark for 6 examples which
he failed to work correctly).
19. Give in the briefest form the product of (a — b) (b —a). Of
(x - 2y) {2y - x). Of (a - b) {a + b) (a^ + b^) (a* + ¥) (a* + 6«).
20. Does 7—^3 = - , ^ J
{y - xy {x - yY
EXERCISE 69
Written Review
1. Indicate by a parenthesis that 2a — 36 + c is to be subtracted
from 5a + 26 — 3c. Then remove the parenthesis and simplify.
2. Subtract the sum oi^x ■\-2y — z and a — x — Zy from — 5.
Also from 0.
3. Write by inspection the value of [(3a — 6) — c + 2d\^,
4. Factor {a +b - cY - {x + y - xY ^ \- -\ . I -f
x^ y* 5
1 ' 1 ' 1
5. Change — h -;; so that it shall be a perfect square.
x^ xy 2/2
6. What is the difference between an exponent and a power?
Give an illustration.
7. Subtract (5a + 1) (2a - 3) from (a + 2) (a + 1) + (a + 2)^.
8. Find the value of 3(a: - l)^ - 3(x + 1) (x + 2) - x{x - 2)
{y — 2x) when x = — 2 and 2/ = — 5.
9. Find the value of ^B — when A = 5a and B = 2a.
10. By factoring find the roots of a;" - 5j + 6 = 0. Prove your
answer.
oil ,1 , f^ ~ b . . ^ b — a
11. bhow that -\ IS equal to -^ *
12. If a = 12i, 6 = 37^, c = 33 J, and d = 10, find in the shortest
4i(P 4:(P 4c^
way the numerical value of each of the following: — , — , -r--
13. From Y.OSa^ take - 4Ja2.
(^2 _^ 52) 2
14. Reduce - 1 + — j-^ — to an miproper fraction.
PROCESSES WITH FRACTIONS 193
15. When we change a;-3=5tox=5+3, what is the change
called? What right have we to make this change? Why do we
transpose — 3 instead of adding 3 to each member of the given
equation?
16. What is the use or advantage in being able to find the H. C. F.
of two given expressions? In being able to find their L. C. M.?
Illustrate.
3.7. Show that the sum of two numbers (as of a and h), divided by
the sum of their reciprocals, equals the product of the given numbers.
18. If s = :; , find the value of s when a = 2 and r = — 7:.
1 — r' 2
2 3
Also when a = — ^ and t = — -7.
If s = r, find the value of s when r = jj, Z = -77, and
W
1
a =
2'
20. If a = 3, which is greater, .^ _ or
3a •
21. Divide 2a^ + 10 - 16a - 39a2 + 15a* by 2 - 5a^ - 4a.
22. Give an illustration to show why 3x0 gives zero. Also
why ^ gives zero.
23. Show that a common factor of any two algebraic expressions
is also a common factor of their sum and difference. Of the sum
and difference of any multiples of the given expressions.
24. Prove that if half of the sum of any two numbers (as of a
and b) is added to half their difference, the result will equal the greater
of the two numbers. Illustrate by two numerical examples.
25. Prove that if half the difference of any two numbers is sub-
tracted from half their sum, the result will be the smaller of the two
numbers.
26. Write an example of a continued fraction and reduce it.
27. Why is it allowable to change both minus signs to plus in
— X = — 3, and not in — a; — 3?
28. Collect in a, short way ^^-^ + j—^ + ^-33 + ^^^f^-
Sua. Collect the first two fractions first.
194
29. Collect in a short way
30. Also
SCHOOL ALGEBRA
3 2
X +2 X -S X -2 ' X +3'
2xy 4^y^
X — y X + y x^ + y^ x* + y*
Simplify:
mx' + ^x - 2)
31
33.
34.
35.
36.
t(ix2 + ^x-i)
x^ + xy — 2?/2
32.
X +2 6x
2 -X
1 - 2x 2a; + 1 4x2 - 1
4x -3
1 "^ x - 2 X -S^ {x'' -x){x -2)'
' 4--^
1 ' 1
a — - a + - a
a a a
+
1 ' . ^
-3 «^+;;
+
+
1
3x + 2 ' {x -1){S -x) ' (2 -x){x - 3)
37. (^+^-2)(^+^+2).(^-^y.
\x a / \x a / \x a/
^ ll+a:^ 1 +a:'J "^ U +x3 1 + x*l
39.
40.
41.
-^
2
X
■+11
i'4
"-a
(M)-
rc -4 -
a; —
-2
X
a; -4
a; -4
a; —
X - 5
9^ ^
a:2 +9
a;2 9 J
X
e-i)(i+>) •
42.
PROCESSES WITH FRACTIONS 195
1 + 8x» 1 - 27x'
1- -^^^ 1+ '^
••■11 o^ •••
1 - 2a; 1 + 3x
a:^ - 5x' + 4 a; a; - 2 ^ x
*^' a;2 + i ^ 1 ":; r'
a; a;2
44. Given a + b + c = 2s,
show that a +b — c =2{s — c) and that a — 6 + c = 2(s — 6).
45. Also show that
a^ +0^ - 62 2(s - a) (s - c)
I
2ac ac
6. Also show that 1 + ttt = — ^:
2ab ab
7. Show that
(2a - Sxy [Sx'ja + 2xy + 5x* (a + 2a;)^] +-27x^ {a + 2xy (2a - 3a;)»
(2a - Sxy^
reduces to
2ax^ (24a; + 5q) (a + 2a;)^
(2a - 3a;)io
48. The distance from New York to San Francisco by way of
Cape Horn is 13,800 mi. This is 1920 mi. less than three times tho
distance from New York to San Francisco by way of Panama.
Find the latter distance.
49. Make up and work an example similar to Ex. 48, using the
fact that the distance from London to Bombay by way of the Cape
of Good Hope is 11,220 mi., but by way of the Suez Canal is 6332
mi.
CHAPTER XI
FRACTIONAL AND LITERAL EQUATIONS
125. A fractional equation is an equation that contains
an unknown number in a denominator.
Ex. - + 5 = 3x.
X
Equations containing binomial numerators and numerical de-
nominators are frequently termed fractional equations, since they
are solved in the same manner as fractional equations proper. See
Ex. 1. of Art. 126.
An integral equation is an equation which does not con-
tain an unknown number in a denominator.
126. The Method of Solving a Fractional Equation. If an
equation contains fractions, it is necessary first to multiply
the members of the equation by such a number as will remove
the fractions.
^ , CI 1 a; + 1 2a; - 5 llx + 5 a; - 13
Ex.1. Solve ^ 5- = -To S^
The L. C. D. of the denominators is 30.
Multiplying both members of the equation by 30 (see Art. 70,
3), we have
15(x + 1) - 6(2x - 5) = 3(lla; + 5) - 10(a: - 13)
Hence, 15x + 15 - 12x + 30 = 33a; + 15 - 10a: + 130
\hx - 12a; - 33a; + lOo; = - 15 - 30 + 15 + 130
- 20a; = 100
a; = — 5 Root
^ a; + 1 2a; - 5 -5 + 1 -10-5 ^ , o i
Check. — 5" = ""2 ^-=-2+3 = 1
llo; + 5 • a; - 13 -55+5 -5-13 _ f- , ^ _ .
^-10 3~ =~"10 3 -5+6-1
196
FRACTIONAL AND LITERAL EQUATIONS 197
.X. 2. Solve -— h ^ :; -^ = 0.
1 -\- X 1 — X 1— ar
Multiplying by the L. C. D., 1 - x^,
4(1 -x) +{x+iy -x^-^S =0
4 - 4x + aj2 + 2a; + 1 - x2 + 3 = 0
-2x = - 8
a; = 4 i2oo<
Let the pupil check the work.
Hence, in general,
Reduce each fraction in the equation to its lowest terms;
Clear the equation of fractions by multiplying each member
by the L. C. D. of all the fractions;
Complete the solution by the methods of Chapter VI.
I
EXERCISE 60
Solve and check each result:
1.
X 3a; 7a; 34
3 5 5 15*
2.
2a;-3 a; + l 5a; + 2
4^6 12
3.
1 3 5 _ 3 _ 19^
2a; a; 3a; 4a; 24
4.
2a; 2a; + 1 1
3 5 3
5.
3a; + 5 a; + 4
4 6
6.
7 = f (a; - 2).
7.
2a; - 8 - j(24 - 2a;) = 0.
. 8.
l(x - 1) = iix - 2).
9- 3(fx-i)(ia; + f)=a^.
198 SCHOOL ALGEBRA
3- 2a; a;-3 , a; + 4,l
^°—8 6--' = ^- + 2i-
3a; - 1 x + 1 4x + 1 _ 3(a; - 1)
11. -^ g 2i 4 ^•
2a; + 5 x+lj _,_ _ 5a; - 10^ 1
"■ 5 10 "^''~ 20 5'
13. ^-!(5 + :r)+^-i(2a; + 5)=^^.
2(a; + i) + a;(l-l)
14. 2(a; + *) + a;(l-^)=^^ + ^.
x + 5_a; + 7x+l_2a:-5 ^ a; + 22
^^■7.52 10 ~ 70 ■
16. f (5a; + 2) - f (7a; - 2) + i(3a; - 2) = x - ?•
17. .5a; — Ax = .3.
18. 1.5a; — 5 = a;.
19. .6a; - 1.5 = .2 - .15a;.
1.5a;-1.6_3.5a;-2.4 24. g^TTs = 4^qr4-
1.2 .8 *
,.^«. » . . X x^ — 5x 2
3.2a;-3.4 .6a; + 4 25. --
23.
5
8
2a;-
6a;-
1
5
3a; + 1
_8a;-7
21.
4.5
22.
a;-l 3
a; + l 5
27.
2.5
3 3a; - 7 3
5 , 6
26. , h
1 — a; 1 + a; 1— a;
-2
3 4 8a; + 3
28.
3-a; 3 + a; 9 - x^
2a; + 1 10 _^ 2a; - 1
2a; - 1 4a;2 _ 1 =" 2a; + l'
29. — 7-^ +
a; + l a;-l a; + 2
FRACTIONAL AND LITERAL EQUATIONS 199
30. ^:zl = _?_+ 1-3^
a^-S x-2 x^ + 2x + 4:
31. a;^ - a: + 1 _ o^ __ x^-\-x-\-l
32.
X — 1 x-h 1
x-S jx^ + l _ a; + 3
2a: + 6 x^ - 9 Sx - 9
33. _3_ + ^ L_ = ^l_ +
34.
a:-la: + l 2a: -2 3a: + 31
a: + la:^ + 7_^ 2 a:-l
2a: -3 4a:2 - 9 2a: + 3 6 - 4x"
35. ??lzJ___L__2 ^
36.
3a: - 6 6a: + 12 2a:2 - 8
6a; + 6 2a: + 1 2a:
2ar^ + 5a: + 3 2a:2 - a: - 1 ~ a:^ + 2a:'
Reduce each' fraction in the following to its lowest terms
and then solve:
^'•-2 = ^- 39. ^^^ = 4-6(.-3).
Find the value of the letter in each of the following: -
41. _i_ 4- I = ?. 43. J^ ? 14 = 0
« + 2 3(u + 2) 3 3-25 6-45
42 __1_ + 1=^_44 4 31 ^1
3(p-7) 6 2p-14 2^ + 2 3^ + 3 6*
.. r + 6 2r - 18 , 2r + 3 .i , 3r + 4
*'• ^i 3-+"4- = ^^ + '-12-
46. If A = Iw, A = 600 and w = 20, find the value of I
Do you know the meaning of this process in arithmetic in conr
nection with the rectangle?
200 SCHOOL ALGEBRA
47. In like manner, find / when ^ = 80 and w = 11^.
48. If F = Iwh, V = 720, / = 10, and w = Q, find h.
Do you know the meaning of this process in arithmetic in con-
nection with the study of volumes?
49. In like manner if F = .3Q,w = .8, and h = .9, find I.
50. lip = br,p = 9 and b = 45, find r.
Do you know the meaning of this process in arithmetic in con-
nection with the subject of percentage?
51. If p = hr, p = 760 and r = .05, find 6.
52. If i = prt, i = $66, p = $440 and t = 3, find r.
Do you know the meaning of this process in arithmetic in con-
nection with the subject of interest?
53. If i = prt, i = $66, p = $360, and t = 3|, find r.
54. If i = prt, i = $15.75, p = $75, r = .06, find t
55. If C = |(F - 32) find F when C = 50. When
C = 100.
Do you know the meaning of this process?
56. If LW = lwsindL = S,W = 100, and w = 40, find L
Can you find out the meaning of this formula and process?
57. lfR = ^^, find s when R = 10 smd g = 32.
58. If F = (l + -^\, find V when F = 20 and ^ = 13.
59. If Z = ih(b + 6'), K = 280, h = 12, and b = 10,
find b\
60. If F = TrUm, V = 1540, TT = V-, and E = 7, find H.
61. If r = 7ri^(i? + i), T =^ 1144, TT = \S R = 14,
findL.
I FRACTIONAL AND LITERAL EQUATIONS . 201
62. Make up and work an equation containing fractions
with the denominators 4, 6, and 12. Can you form the equa-
tion so that the root shall be 1? 2? 4?
63. Make up and work an equation containing fractions
with the denominators a: + 2, a; — 2, and x^ — 4.
64. Work again Exercise 24 (p. 99).
127. Special Methods. The work of solving an equation
may often be lessened by using some special method or device
adapted to the peculiarities of the given equation.
First Special Method. If in a given equation the denomi-
nators of some fractions are monomials, and of others are poly-
nomials, it is best to make two steps of the process of clearing
the equation of fractions: (1) remove the monomial denom-
inators and simpUfy as far as possible; (2) remove the re-
maining polynomial denominators.
^ , ^ . 2a: + 8J 13a: - 2 , a; 7x a: + 16
Ex.1. Solve -^ _____ + _____.
Multiplying by 36, the L. C. D. of the monomial denominators,
8a; + 34 - ^^^jf^.'gl^ + 12a; = 21x - x - IQ
Transposing all terms except- the fraction to right-hand side,
36(13a; - 2)
17a; - 32
-50
Dividmgby-2, 17-^ ,32 = ^5
234a; - 36 = 425a; - 800
191a; = 764
a; = 4 Root
Let the pupil check the work.
Second Special Method. Before clearing an equation of
fractions, it i^ often best to combine some of the fractions
into a single fraction.
202 SCHOOL ALGEBRA
X — 1 X — 2 X — 3 X — 4
Ex. 2. Solve
X — 3 X — 4: X — 5
In this equation it is best to combine the fractions in the left-
hand member into a single fraction, and those in the right-hand
member also into a single fraction, before clearing of fractions. We
obtain
-1 -1
(x -2){x-S)~ {x- 4) {x - 5)
7
Clearing and solving, ^ = o ^^^^
Let the pupil check the work.
EXERCISE 61
Solve the following and check each solution:
1 ^^ ~ ^ _i_ 4^ X -j- 5
2.
6 3a: + 2 2
3-2x . X l-6x 2-Sx
3. 2J-
6 15 - 7a: 9
2a: — 1 X
2a: + 4 4 2
^ 5a: + 13 _ 2a: + 5 23 5 - jx
12 6 4a: -36 3 *
5 5 _ 2ia: - 3 _ a: + 11 _■ 11a: + 5 _ ^
' 7-x 4 8 16
g 3a:- 1 4a:- 7 ^ x _ 2a: - 3 7a:- 15
30 15 4 12a: -11 60
7 6a: - 7 _ a: + 1 2a: - 1 _ 199
' 11a: 4- 5 15 30 10*
- ^, a: + 4 4- 3a:, 4a: + 9 4-a;,5
^7a: + ll 8 ^12 24 4
3 3a: - 2^ _ 7 x-j _ 2a: - 1| 2a: + 3|
9 12"^|a:+ll 3 6 '
I FRACTIONAL AND LITERAL EQUATIONS 203
Q 2r2a:_lf3a:-l 2a:-5 n 19^^_
^^ 1.2a; - 1.5 Ax + 1 ^ .4a; + 1
1.5 .2a; -.2 .5
1111
12.
13.
14.
15.
o; — 2 x — 3 X — 4: x — 5
X — 1 _x — 3 _x — 5 _x — 7 ^
X — 2 X — 4: X — 6 X — S
X — 7_x — 8_x — 4_x — 5
x — 8 x — 9 x — 5 a; — 6
3 2 2 3
3a; - 2 2a; - 3 2a; + 3 3a; + 2
16.* 2^ + 1 . 2^ + 9 _ 2a; + 3 2a; + 7
a; + l a; + 5 a; + 2 a; + 4 *
^7 4a; - 17 10a; - 13 8a; - 30 5a; - 4 _
*a;-42a;-3 2a; -7 a;-l
18. Work Ex. 2 by clearing all denominators at once. Then
work the same example by the method of Art. 126. About
what fraction of the work is saved by the second process?
19. Treat Ex. 13 in the same way as you treated Ex. 2.
20. On the average, the distance one must go below the
surface of the earth to get an increase of 1° in temperature,
is 62 ft. This is 1 ft. more than one third the distance one
must go above the earth's surface to get a decrease of 1° in
the temperature. Find the latter distance.
21. Who, so far as we know, first invented transposition
in solving equations, and when? Who first brought the use
of transposition into prominence?
* Transpose the second and third fractions.
204 SCHOOL ALGEBRA
22. From what language does the word algebra come?
What does the word algebra mean?
23. Work the odd-numbered examples on p. 101. How
many examples on that page can you work at sight?
128. Two Equivalent Equations are equations which have
identical roots; that is, each equation has all the roots of the
other equation and no other roots.
Thus, X? - 4:X = 0, and x(x -\- 2) (x - 2) =0 are equivalent,
since each is satisfied by the values x = 0, 2, — 2, and by no other
values of x.
If we multiply the two members of an equation by the
same expression, the resulting members are equal, but the
resulting equation may not be equivalent to the original
equation.
Thus, if we take the equation x = S and multiply each mem-
ber by a; — 2, we obtain x{x — 2) = 3 (x — 2) or
{x -S){x -2) = 0,
which is not equivalent to the original equation, since it has the
root X = 2, which the original equation does not have (Art. 103).
In general, if the two members of an integral equation are
multiplied by x — a, the root a is introduced and the resulting
equation is not equivalent to the original equation.
129. An Extraneous Boot is a root introduced into an
equation (usually unintentionally) in the process of solving
the equation.
The simplest way in which an extraneous root may be
introduced is by multiplying both members of an integral
equation by an expression containing the unknown number.
See the example of Art. 128. <
A more common way in which extraneous roots are intro-
duced during a solution — and one more difficult to detect-^
FRACTIONAL AND LITERAL EQUATIONS 205
I is by a failure to reduce to its lowest terms a fraction con-
tained in the original equation.
2a; — 4
Thus, in solving z tyt ^ =1, the first step should be to
2x— A
reduce the fraction 7 rr-? o^ *o i^^ lowest terms. If this is done,
{X — L) (X — z)
2
we obtain the equation r = 1, whence x= S.
X i
If, however, we should fail to reduce the fraction to its lowest
terms and should multiply both members by (x — 1) (a; — 2), we
obtain 2x — 4: = x^ — Sx + 2, whence x^ — dx + Q = 0,
or {x - 3) (x- 2) = 0, anda: = 2, 3.
On testing both of these results, we find that the extraneous root
2 has been introduced.
Often the fraction which can be reduced to simpler terms
occurs in a disguised and scattered form. In this case it is
best to solve the equation without attempting to collect the
parts of the fraction. An extraneous root may then be
detected by checking the results obtained.
Thus, the fraction in the above equation might be changed
in the following way so as to make it difficult to detect its
presence in the equation:
We have —^ — • = 1
{x -l)(x - 2) '
2a; — 2 2
''^''"^ (»-l)(x-2) - (x-l)(x-2) = 1'
whence ' _^_^__^_^=1.
There is nothing in the appearance of this last equation to indi-
cate that it impUcitly contains a fraction which should be simplified
before proceeding with the solution proper.
Hence it is important constantly to remember that a root of an
equation is not such because it is the result of a series of operations,
as clearing an equation of fractions, transposition, etc., but because
it satisfies the original equation.
206 SCHOOL ALGEBRA
130. Losing Roots in the Process of Solving an Equation.
If both members of the equation (x — 2) {x — S) =0 are divided
by a: — 2, we obtain a; — 3 = 0.
The resulting equation is not equivalent to the original equation
since it does not contain the root x = 2, which the original equation
contains.
Hence, in general,
// both members of an equation are divided by an expression
containing the unknown quantity, write the divisor expression
equal to zero, and obtain the roots of the equation thu^ formed
as part of the answer for the original equation.
EXERCISE 62
1. Multiply each member of the equation a; — 2 = 1 by
X — 2. Is the resulting equation equivalent to the original
equation? Why?
2. Make up and work an example similar to Ex. 1.
3. Multiply each member of the equation a; = 2 by x — 5.
Is the resulting equation equivalent to the original equation?
Why?
4. Divide each member of the equation x^ — 9 = x — 3
by a: — 3. Is the resulting equation equivalent to the orig-
inal equation? Why?
5. Make up and work an example similar to Ex. 4.
X — S
6. Solve the equation — — - = 1 after first reducing the
fraction to its lowest terms. Now solve the equation without
reducing the fraction to its lowest terms. Do the two meth-
ods of solution give the same result? Which result is correct?
Why?
7. Make up and work an example similar to Ex. 6.
Solve each of the following, check each result, and point •
I
FRACTIONAL AND LITERAL EQUATIONS 207
out each extraneous root, giving the probable reason for the
occurence of such a root:
8. ?+ 1 ^
2 a;+l x+1
9
^' x+1 ^ (a;+l)(a;-2)
10. ; — 1 — ^+1= 5
(x + 2){x + 3) • x + 2
11. = X
0^2-1 6
12. ^^^ H ^ ^ = 3.
a:2 - 1 ^ a; - 1 a: + 1
13. Form an equation in which 3 is the extraneous root.
14. How many examples in Exercise 31 (p. 121) can you
now work at sight?
131. A numerical equation is an equation in which the
known quantities are expressed by figures. Thus, all the
equations on p. 199 are numerical equations.
A literal equation is an equation in which some or all
of the known quantities are denoted by letters; as by a, by
c . . ., or m, n, p . . .
The methods used in solving literal equations are the same
as those used in solving numerical equations.
Ex. Solve a{x — a) = h{x — b).
ax — a^ =bx — b^
ax —hx = a? —h"^
{a - b)x =a^ -¥
X = a -]-b Root
Check. a{x — a) = a{a +b — a) = ah
b{x -b) =b{a +b -b) =ab
208 SCHOOL ALGEBRA
EXERCISE 63
Solve for x and check:
1. 3a; + 2a = a; + "^a.
11.
a^a; = (a - 6)2 + 62a;.
2. 9ax - 36 = 2ax + 46.
12.
(a— 6)a; = a2— (a + 6)a;,
3. 5ax — c = ax — 5c,
4. ax + 6 = 6a; + 26.
13.
ax hx _a h
b a b a
5. 3ca; = a-(26- a + ca;).
14.
a-\- X a — X
6. 5a; - 2aa; = 3 - 6.
a-2x a-\-2x
7. 2ax - 3b = ex -\- 2d,
8. (x + a)(x-h) = x^.
15.
4a; — a ^ x-\- a
2x — a X — a
9. a6(a; + 1) = a^ + 62a;.
10. (a;-l)(a;-2) = (a;-a)2.
16.
X .X .X ,
- + 7 + - = a.
a 6 c
f(^»)=l(^«)•
18
^g ax — b bx — OCX — a_^
ab be ae
20 ot^ , 6a; _ 3a2a; + 62
3M^ 3a - 6 ~ 9a2 - 62 *
21 ^ — ^ = 1 _ X
a a — b a-\-b b
a^ — X 62 — a; c2 — a; a2
22.
a
„, 5a2-7a; , a62 + 10a; 10c2 + 3a; , 5(a - c) , 62
23. . -1- ^ —I -f- — .
3a6 5ae 66c 36 5c
FRACTIONAL AND LITERAL EQUATIONS 209
a-\- X ^ X X — 1 X
„, a a ^^ a a — 1
24. = 25. =
X — a 1 a+1 a
— X
a X a-\- X
26. Make up and solve two literal equations.
27. How many examples in Exercise 35 (p. 131) can you
now work at sight?
EXERCISE 64
Oral
Solve the following orally, without transposing any term contain-
ing x:
1. 4a; = — 12 ^. ax =b 7. hx + c = d
2. Sx = a 5. 2x - 4: = Q 8. ax +bx = c
3. ax = 5 6. ax — 5 = 7 9. ax = c + 5x
10. 6
= 3a;
11.
10 =
= -5a;
12.
1 = 3.
21.
i=2
X
30.
X 1
3"4
13.
a = bx
22.
i-3
31.
*-i
14.
c = dx
23.
i-'
32.
--f
15.
3a; =2
.24.
--i-'
33.
-1
16.
2x= -
4
5
25.
'4'=--
34.
bx
a= —
c
17.
^-\
26.
a
35.
X 3
2 "5
18.
1
27.
|x=2
36.
1 4
?=7
19.
h-
28.
h^
37.
X 2
4~ 3
20.
l-^
-f>
29.
-lx=6
38.
1 1
x~ 2
210 SCHOOL ALGEBRA
39.
2 3
x~5
41.
2 3
S~ X
40.
7 1
S~x
42.
2 1
3^=5
45.
a+h
P+q- ^
47.
46.
'-J=p+q
48.
43.
ax
c
b
d
4
^?,
44.
X
15
m
X — a
a+ b _ c+ d
X 5
49. How many examples in Exercise 45 (p. 155) can you now
work at sight?
EXERCISE 65
Review
Solve for x and check:
_2 5 2_ X x'^-l
x-2 x+2~x^-4:' 2 a;-l~^-
6x4-1 2x- 1 2a;- 4
15 5 7a;- 16
a;- 6 x+b ^a^-b"^^
x-2a x+2a x^-4a^
g 3pa;-3ga; p-2q ^ P-Q_Q
x^— (f- x-\- q q— X
x—2 X— 3_ X— 5 X— 6
^* a;-3~a;-4~a;-6 a;-7*
a;— 5_ X
~5"~2'
7. 6+i(^-9-^) + 3 +
Find the value of x in the shortest way, when
8. ^a; = ^X 19 + ^X41.
9. 3.1416a; = 3.1416(723) - 3.1416(476).
,^ ^ ^ 9 11 2a; a: -
10. 3-^=2. 11. 3-2-5-
12. If 5 - - = 4, find the value of x when ?/ = ^' Also when
X y o
= - -• When 2/ = c*
4 0
FRACTIONAL AND LITERAL EQUATIONS 211
13. Solve for I: jt^ = _ ,
r. 1 r -640 f-l2S\
14. Solve for a: -^ = a[-^Y)
15. In adding —r — h -^ we retain the L. C. D. 24. In solving
or ~~ 1 ^x
the equation —r— = -^ and clearing of fractions, the L. C. D.
24 disappears. What is the reason for this difference?
16. Make up an example similar to Ex. 15.
17. Make up and solve an equation which contains fractions
with the denominators 8, 2{x — 1), and 4.
18. Make up and solve an equation which contains fractions
with the denominators a + 6, a — 6, and b^ — a^.
EXERCISE 66
1. Find the number the sum of whose third, fourth, and
fifth parts is 94.
2. Make up and work a problem concerning one fourth and
one sixth of some number.
3. State ^ — - = 28 as a problem concerning a number
and find the number.
4. A certain number exceeds the sum of its third, fourth,
and tenth parts by 38. Find the number.
5. A piece of bronze weighs 415 pounds. It contains twice
as much zinc as tin, and 8 times as much copper as tin. How
many pounds of each material are in the bronze?
6. Find two consecutive numbers such that one seventh
of the greater exceeds one ninth of the less by 1.
7. Express in symbols 15% of x. 5% of x. 115% of b.
212 SCHOOL ALGEBRA
8. Two men kept a store for a year and made $4800.
The man who owned the store building received 40% more
of the profits than the other. How much did each receive?
9. In building a macadam road the county pays twice
as much as the state, and the township pays 50% more than
the state. How much does each pay if the road costs $18,000?
10. Separate $770 into two parts so that one shall exceed
the other by 20%. By 33j%.
11. The difference of two numbers is 9. 3 increased by
Yi of the less of the two numbers equals f of the greater.
Find the numbers.
12. The iron ore in the United States is J of the iron ore
in the rest of the world. If there are 75,000,000,000 tons of
iron ore in the entire world, how many tons of iron ore are
there in the United States?
13. The population of India is f that of China, and the
population of the rest of the world is 3| times that of India.
What is the population of India and China, if that of the
entire world is 1,500,000,000?
14. A man bequeathed $60,000 to his wife and three chil-
dren. In a first will he bequeathed his wife three times as
large a share as one child received. Later he changed his
will and bequeathed his wife $10,000 more than the share of
a child. By which of the two wills would she have received
the larger amount?
15. In one kind of concrete, the parts of cement, sand,
and gravel are 1, 2, and 4. In another kind of concrete these
parts are 1, 3, 5. How many more cubic feet of cement are
needed to make 5600 cu. ft. of concrete of the first kind than
of the second?
FRACTIONAL AND LITERAL EQUATIONS 213
16. A girl's grades are, arithmetic 87, reading 92, and
j>c()graphy 85. What grade must she have in spelUng to
make her general average 90?
17. The average wheat crop of the United States for
four years was 660 millions of bushels. What would the
crop for the fifth year need to be in order to make the
average for the five years 700 million bushels?
18. A pupil has worked 15 problems. If he should work
9 more problems and get 8 of them right, his average would be
.75. How many problems has he worked correctly thus far?
19. A baseball nine has played 36 games of which it has
won 25. How many games must it win in succession to
bring its average of gam-es won up to .75?
20. Make up and work an example similar to Ex. 19.
21. A baseball nine has won 19 games out of 36 games
played. If after this it should win f of the games played,
how many games would it need to play to bring its average
of games won up to .66f ?
22. A baseball nine has won 25 games out of 36 played.
It still has 12 games to play. How many of these will it need
to win in order to bring its average of games won up to .75?
23. How much water must be added to 50 gallons of
milk containing 8% of butter fat to make a mixture contain-
ing 5% of butter fat?
SuG. The 50 gal. of milk contain 50 X .08 or 4 gal. butter fat.
4 5
If X denotes the number of gallons of water, — = — — , etc.
50 + a; 100
24. A certain kind of cream is f butter fat, and a certain
kind of milk is 3% butter fat. How many gallons of the
cream must be added to 40 gallons of milk to make a mixture
which is 5% butter fat?
214 SCHOOL ALGEBRA
25. Of what type is each of the above problems an example
or variation?
26. A mass of copper and silver alloy weighs 120 lb. and
contains 8 lb. of copper. How much copper must be added
to the mass in order that 100 lb. of the resulting alloy shall
contain 10 lb. of the copper?
27. A mass of copper and silver alloy weighs 120 lb. and con-
tains 8 lb. of silver. How much silver must be added to the
mass in order that 1 lb. of the resulting alloy shall contain
2j oz. of silver?
28. If 100 lb. of sea water contains 2| lb. of salt, how much
fresh water must be added to it in order that 100 lb. of the
mixture shall contain 1 lb. of salt?
29. How much fresh water must be added to 100 lb. of sea
water in order that 20 lb. of the mixture shall contain 4 oz.
of salt?
30. How much water must be evaporated from 100 lb.
of salt water in order that 8 lb. of the water left shall con-
tain 1 lb. of salt?
31. How much water must be added to a gallon of alcohol
which is 90% pure, in order to make a mixture which is 80%
pure?
32. If it takes a man 9 days to do a piece of work, what
part of it will he do in one day? If it takes him x days to
do the work, what part of it will he do in one day?
33. If a boy can do a piece of work in 15 days which a
man can do in 9 days, how long would it take both working
together to do the piece of work?
SuG. What fractional part of the work will the boy do in 1 day?
The man? If together the boy and man can do the piece of work
in X days, what part of the work can they do together in 1 day?
FRACTIONAL AND LITERAL EQUATIONS 215
B 34. A can spade a garden in 3 days, B in 4 days, and C in
' o days. How many days will they require working together?
35. A and B together can mow a field in 4 days, but A
alone could do it in 12 days. In how many days can B mow it?
|B 36. A and B in 5y days accomplish a piece of work which
I A and C can do in 6 days or B and C, in 7j days. If they all
work together, how many days will they require to do the
same work?
37. One pipe can fill a given tank in 48 min. and another
can fill it in 1 h. and 12 min. How long will it take the pipes
together to fill the tank?
38. Two inflowing pipes can fill a cistern in 27 and 54
min. respectively, and an outflowing pipe can empty it in
36 min. All pipes are open and the cistern is empty; in how
many minutes will it be full?
SuG. Since emptying is the opposite of filling, we may consider
that a pipe which empties jj of a cistern in a minute will fill — yV
of it each minute.
39. A tank has four pipes attached, two filling and two
emptying. The first two can fill it in 40 and 64 min. respect-
ively, and the other two can empty it in 48 and 72 min.
respectively. If the tank is empty and the pipes all open, in
how many minutes wifl it be full?
40. At what time between 3 and 4 o'clock are the hands
of a watch pointing in opposite directions?
Solution. At 3 o'clock the minute-hand is 15 minute-spaces
behind the hour-hand, and finally is 30 spaces in advance: therefore
the minute-hand moves over 45 spaces more than the hour-hand.
Let X = the number of spaces the minute-hand movea
Then X - 45 = " " " " " hour-hand
But the minute-hand moves 12 times as fast as the hour-hand;
hence, x = 12(x — 45). Solving, x = 49xt.
Thus the required time is 49 jj min. past 3.
216 ' SCHOOL ALGEBRA
41. When are the hands of a clock pointing in opposite
directions between 4 and 5? Between 1 and 2?
42. What is the time when the hands of a clock are to-
gether between 6 and 7? Between 10 and 11?
43. At what instants are the hands of a watch at right
angles between 4 and 5 o'clock? Between 7 and 8?
44. The planet Mars is in the most favorable position to
be observed from the earth when it is in
line with the earth and on the opposite
side of the earth from the sun (Mars
is then said to be in opposition). If
the year is taken as 365 days, and it
takes Mars 687 days to make one revo-
lution about the sun, how long is the
interval between two successive opposi-
tions of Mars?
SuG. If it takes the earth x days to overtake Mars and thus put
Mars again in opposition, how many revolutions about the sun does
the earth make in x days? How many revolutions does Mars make
in X days? In the interval from one opposition to the next, how many
more revolutions about the sun does the earth make than Mars?
45. It takes the planet Jupiter 12 yr. to make one revolu-
tion about the sun. How long is it from one opposition of
Jupiter to the next?
46. The interval between two successive oppositions of
Mars is 780 days. Determine the time it takes Mars to make
one revolution about the sun (i. e. the length of the year on
Mars).
47. A courier travels 5 mi. an hour for 6 hours, when an-
other courier starts at the same place and follows him at the
rate of 7 mi. an hour. In how many hours will the second
overtake the first?
FRACTIONAL AND LITERAL EQUATIONS 217
SuG. If a; = the number of hours the second courier travels,
how many hours does the first courier travel? How many miles (in
terms of x) does the first courier travel? The second? Do the two
couriers travel equal distances?
48. A courier who travels 5 J mi. an hour was followed
after 8 hours by another, who went 7 J mi. an hour. In how
many hours will the second overtake the first?
49. A woman can write 15 words per minute with a pen,
and a girl can write 40 words per minute on the typewriter.
The woman has a start of 3 hours in copying a certain manu-
script. How long before the girl using the typewriter will
overtake the woman?
50. A train running 40 mi. an hour left a station 45 min.
before a second train running 45 mi. an hour. In how many
hours will the second train overtake the first?
51. A gentleman has 10 hours at his disposal. He walks
out into the country at the rate of 3 J mi. an hour and rides
back at the rate of 4 J mi. an hour. How far may he go?
52. A and B start out at the same time from P and Q, re-
spectively, 82 mi. apart. A walked 7 mi. in 2 hours, and B
10 mi. in 3 hours. How far and how long did each walk
before coming together, if they walked toward each other?
If A walked toward Q, and B in the same direction from Q?
53. A certain room is 20 ft. long and 12 ft. wide. The
walls and ceiling of the room together have an area of 752
sq. ft. How high is the ceiling?
54. A rifle ball is fired at a target 1100 yd. distant and
4j sec. after firing the shot the marksman heard the impact
of the bullet on the target. If the bullet traveled
at the rate of 2200 ft. per second, what was the rate
at which the sound of the impact traveled back to the
marksman?
218 SCHOOL ALGEBRA j
55. A rifle ball is fired at a target 1000 yd. distant and 4
sec. after firing the shot, the marksman heard the impact of
the bullet on the target. If sound traveled at the rate of
1100 ft. per second, at what rate did the bullet travel?
56. A 21 lb. mass of gold and silver alloy when immersed
in water weighed only 19 lb. If the gold lost yV of its weight
when weighed under water, and the silver yV of its weight,
how many pounds of each metal were in the alloy?
SuG. If X denotes the number of pounds of gold, how many
pounds of silver were there in the mass?
The law involved in the above example is that when any object
is weighed in water, it loses in weight an amount equal to the weight
of the water which it displaces. Hence, if the specific gravity of
gold is approximately 19, the weight of the water displaced by the
gold = yV of the weight of the gold.
Find out if you can who first used this method of determining
the relative amounts of metal in an alloy and what use he first
made of the method.
57. An alloy of aluminum and iron weighs 80 lb., but
when immersed in water it weighs only 60 lb. If the spe-
cific gravity of aluminum is 2| while that of iron is 7|, how
many pounds of each metal are in the alloy?
58. A mass of copper and tin weighing 100 lb. when im-
mersed in water weighed 87.5 lb. If the specific gravity of
copper is 8.8 and that of tin is 7.3, how much of each metal
was in the mass?
59. If a bushel of oats is worth 40f^ and a bushel of
corn is worth 55^, how many bushels of each grain must
a miller use to produce a mixture of 100 bu. worth 4S^ a
bushel?
60. A man has $5050 invested, some at 4%, and some at
5%. How much has he at each rate if the annual income is
$220?
FRACTIONAL AND LITERAL EQUATIONS 219
increased by 2, the second diminished by 2, the third multi-
plied by 2, and the fourth divided by 2, will all produce equal
results.
62. Find three consecutive numbers such that if they be
divided by 2, 3, and 4 respectively, the sum of the quotients
will equal the next higher consecutive number.
63. In the United States the gold dollar is 90% gold and
10% copper. If a mass of gold and copper weighing 24 lb. is
75% gold, how many pounds of gold must be added to it to
make it ready for coinage into gold dollars?
64. My annual income is $990. If J of my property is in-
vested at 5%, I at 4%, and the rest at 6%, find the amount
of my property.
65. If one pipe can fill a swimming tank in 1 hour and an-
other can fill it in 36 minutes, how long will it take the two
pipes together to fill the tank?
66. At what time are the hands of a watch at right angles
between 10 and 11 o'clock?
67. If one baseball nine has won 16 games out of 42
played, and another has won 18 out of 40 played, how many
straight games must the first team win in order at least to
equal the average of games won by the second team?
68. If the interval between two successive oppositions of
the planet Saturn is 378 days, how long is the year on
Saturn?
69. If A, B, and C together can do in 5^ days a certain
amount of work, which B alone could do in 24 days, or C
alone in 16 days, how long would A require?
70. How much water must be added to 1 gal. of a 5%
solution of a certain chemical to reduce it to a 2% solution?
220 SCHOOL ALGEBRA
71. A baseball player who has been at the bat 150 times
has a batting average of .240. How many more times
must he bat in order to bring his average up to .250, pro-
vided that in the future his base hits equal half the number
of times he bats?
72. A girl has worked a certain number of problems and
has f of them right. If she should work 9 more problems and
get 8 of them right, her average would be .75. How many
problems has she worked?
73. If the sum of two consecutive integers is 4p + 5, find
the integers.
74. A man has a hours at his disposal. He wishes to ride
out into the country and walk back. How far may he ride
in a coach which travels b miles an hour, and return home in
time, walking c miles an hour?
75. Generalize Ex. 33; that is, make up and work a similar
example where letters are used instead of figures for the
known numbers.
76. If E denotes the number of days it takes the earth to
revolve once around the sun, P denotes the number of days
it takes a planet (as Mars) to complete a revolution about
the sun, and S the number of days between two successive
oppositions of the planet, show that y? — is = o*
77. The fore wheel of a carriage is a feet in circumference
and the hind wheel is h feet. What distance has been passed
over when the fore wheel has made c revolutions more than
the hind wheel?
78. Make up and work three examples similar to such of
the examples in this Exercise as the teacher may point out.
FRACTIONAL AND LITERAL EQUATIONS 221
EXERCISE 67
1. Given V = Iwh, find h in terms of the other letters.
Also solve for I. For w,
2. Given i = prt, find each letter in terms of the others.
Find each letter in terms of the others in the following
formulas used in geometry:
13. K = ibh 6. S = ttRL
4. Z = ihib + 6') 7. T = irRiR + L)
5. C = 2'TrR 8. r = 2ttR{R + H)
Also find each letter in terms of the others in the following
formulas used in mechanics and physics:
9. S -= vt 11.
10. LW = Iw 12.
15. By use of the formula in Ex. 2 determine in how many
years $325 will produce 184.50 interest at 5 per cent.
16. Also find the rate at which $176 will yield $43.56 in-
terest in 5 yr. 6 mo.
17. Change the following temperatures on the Centigrade
scale to Fahrenheit readings:
(1) 50° (2) 0° (3) 2700°
18. Metals fuse at the following temperatures on the Cen-
tigrade scale. What are the temperatures at which they fuse
on the Fahrenheit scale?
Tin 228° Lead 325° Copper 1091° Iron 1540°
6c_-M
19. Solve the following equation for h: d _ 2d
Also solve for c. For d. be a
C-I(F-
-32)
13. R- ^»
9 + s
-i
14. i = i+L
f p p'
222 SCHOOL ALGEBRA
20. A boy who weighs 80 lb. is on a teeter board at A,
6 ft. from -the fulcrum F. He just balances a boy who
^ jp ^ is at ^ on the same board, 8 ft.
*^ Y^ jft — v^~~^ from F. What does the second
m boy weigh? (Use the formula of
— * — Ex. 10.)
21. Make up and work an example similar to Ex. 19.
22. How many examples in Exercise 48 (p. 163) can you
now work at sight?
CHAPTER XII
SIMULTANEOUS EQUATIONS
132. Need and Utility of Simultaneous Equations.
Ex. A farmer one year made a profit of $2221 on 27
acres of corn and 40 acres of potatoes. The next year with
equally good crops, he made a profit of $2028 on 36 acres of
corn and 30 acres of potatoes. How much did he make per
acre on his corn and on his potatoes?
Let X = no. of dollars made on 1 acre of corn
2/ = " " '' " " 1 " " potatoes
Then 27a; + 40?/ = 2221
36a; + 30^ = 2028
From these equations the value of x may be found by combining
the equations in some way which will get rid of, or eliminate, y.
(See Arts. 136-138.)
Try to solve the above problem by the use of only one unknown,
as X. If you succeed at all, you will find the method awkward and
inconvenient.
Why do we now proceed to make definitions and rules?
133. Simultaneous Equations are a set or system of equa-
tions in which more than one unknown quantity is used, and
the same symbol stands for the same unknown number.
I Thus, in the group of three simultaneous equations,
a; + 2/ + 20 = 13
X -2y -\-z ={)
2x+y -z =S
X stands for the same unknown number in all of the three equations,
y for another unknown number, and z for still another.
223
224 SCHOOL ALGEBRA
134. Independent Equations are those which cannot be
derived one from the other.
Thus, x-{-y = 10,
and 2x = 20 - 2y,
are not independent equations, since by transposing 2y in the sec-
ond equation and dividing it by 2, we may convert the second equa-
tion into the first.
But Sx —2y = 5| are independent equations, since neither one
5x +y = Q] of them can be converted into the other.
135. Elimination is the process of combining two equa-
tions containing two unknown quantities so as to form a
single equation with only one unknown quantity. Or, in
general, elimination is the process of combining several sim-
ultaneous equations so as to form equations one less in
number and containing one less unknown quantity.
There are three principal methods of elimination: I, ad-
dition and subtraction; II, substitution; and III, comparison.
These methods are presented to best advantage in connec-
tion with illustrative examples.
136. I. Elimination by Addition and Subtraction.
Ex. Solve (9,_4 = 33 (3)
In order to make the coefficients of y in the two equations ahke,
we multiply equation (1) by 4, and (2) by 5,
48x + 2% = 300 (3)
45a; - 20?/ = 165 (4)
Add equations (3) and (4), 93a; = 465
Divide by 93, a; = 5 Root
Substitute for x its value 5, in equation (1),
60 + 5?/ = 75
.' .y =3 Root
Check. 12a; + 5^ = 12 x 5 + 5 X 3 =75
9a; -4?/ =9X5-4X3= 33
H SIMULTANEOUS EQUATIONS 225
Hpince y was eliminated by adding equations (3) and (4)
the above process is called elimination by addition.
The same example might have been solved by the method
of subtraction.
Thus, multiply equation (1) by 3, and (2) by 4,
36a; + 15?/ = 225 (5)
36a; - 162/ = 132 (6)
Subtract (6) from (5), 31?/ = 93
2/=3
and a; = 5
»
It is important to select, in every case, the smallest multipliers
that will cause one of the unknown quantities to have the same
coefficient in both equations.
Thus, in the last solution given above, instead of multiplying
equation (1) by 9, and (2) by 12, we divide these multipUers
by their common factor, 3, and get the smaller multipliers, 3
and 4.
Hence, in general,
Multiply the given equations by the smallest numbers that
mil cau^e one of the unknown quantities to have the same co-
efficient in both equations;
If the equal coefficients have the same sign, subtract the corre-
sponding members of the two equations; if the equal coefficients
have unlike signs, add,
EXERCISE 68
Solve by addition and subtraction:
1. Sx -2y = 1 4. 5a; - 32/ = 1
X -{- y = 2 3a; + 5y = 21
2. 2x - 7y = 9 5. X -\- 5y == - 3
5x-{-3y = 2 7x-\-8y = 6
3. 4a; + 3i/ = ^1 6. Sx - 2y = 4:
2a; -62/ = 3 5a; - 42/ = 7
226 SCHOOL ALGEBRA
7. 2y + X = 0
4a: + 61/ = - 3
13.
3 5-^
8. 9a: - 8y = 5
5 + 2~^
15a: + 12y = 2
9. 4a: - 6i/ H- 1 = 0
14.
3 ^4
5a: - 72/ + 1 = 0
10. 8a: + 5?/ = 6
3a: 5i/ _
2^8
dy + 2x = 11
11. 5a: - 3i/ = 36
7a: - 52/ = 56
15.
4^4. 32/ _ 7
5 "^ 2
3a:, 22/ 7
12. ^-1 = 1
2 3
16.
5a: Sy
6 9
^ 2/_ i
4 9
3a: 5?/
4 6
17. Find two numbers whose sum is 12 and whose differ-
ence is 2.
18. The half of one number plus the third of another
number equals 13, while the sum of the numbers is 33.
Find the numbers.
19. State Ex. 1 as a problem concerning two numbers.
20. State Ex. 2 as a problem concerning two numbers.
21. 7 lb. of sugar and 3 lb. of rice together cost 57^; also
5 lb. of sugar and 6 lb. of rice cost QO^. Find the cost of a
pound of each.
22. Make up and work an example similar to Ex. 18. To
Ex. 21.
23. How many examples in Exercise 50 (p. 170) can you
now work at sight?
SIMULTANEOUS EQUATIONS 227
137. II. Elimination by Substitution.
Ex. Solve 5a;+22/ = 36 . . . (1)
2a; + 3!/ = 43 (2)
From (1) 5x = 36 - 2?/
.-. x = ^^ (3)
In equation (2) substitute for x its value given in (3),
^^+3. = 43
72 -4?/ +152/ =215
11?/= 143
2/ = 13 Root
36—26
Substitute for !/ in (3), x = — z = 2 Root
Let the pupil check the work.
Hence, in general,
In one of the given equations obtain the value of one of the
unknown quantities in terms of the other unknown quantity;
Substitute this value in the other equation and' solve.
EXERCISE 69
1. Work the examples of Exercise 68 (p. 225) by the
method of substitution.
Find out which of the following sets of equations are worked
more readily by the method of addition and subtraction, and
which by the method of substitution, and work each example
accordingly:
2. X = Sy - 5 4. ir-3=0
2x-\-5y = 12 2y-\-Sx = 5
3. Sx- Ay ^ 1 5. 2x-{-Sy = 1
4x- 5y = 1 3a; + 4y = 2
228 SCHOOL ALGEBRA
e. 7x + 8y = 19 8. y = 3
5x + 62/ = 13J 2x = Sy - 17
7. a; = 2?/ — 3 9. y = 3x
y = 5x - 21 4a; + 5?/ = 38
10. Make up and solve an example in simultaneous equa-
tions which is solved more readily by the method of addition
and subtraction than by the method of substitution.
11. Make up and solve an example of which the reverse
of Ex. 10 is true.
12. How many examples in Exercise 51 (p. 172) can you
now work at sight?
138. III. Elimination by Comparison.
Ex. Solve 2x-3y = 23 (1)
5x-^2y = 29 (2)
From(l) 2z=23+Sy (3)
From (2) 5x=29-22/ (4)
From (3) x=?^ (5)^
From (4) x = ?^^ (6)
Equate the two values of x in (5) and (6), //^^^"^
23+%.. 29 -2y
2 "" 5
Hence, 115 + 15y = 58 - ^y
19y = - 57
2/ = — 3 Root
23—9
Substitute for 2/ in (5), x = — ^ — = ^ ^^^^
Let the pupil check the solution.
Hence, in general.
Select one unknown quantity, and find its valus in terms of
the other unknown quantity in each of the given equations;
Equate these two values, and solve the resulting equation.
I
SIMULTANEOUS EQUATIONS 229
EXERCISE 70
1. Work the examples of Exercise 68 (p. 225) by the
method of comparison.
Ascertain by which of the three methods of eUmination
each of the following examples can be worked most readily,
and solve accordingly:
2. x = Sy-\-9 7. 9x + 122/ = - 6
a; = 5i/ + 13 6x - 5y = - 17
3. a; = 3i/ + 9 8. a; = 5
Sx - 5y = 10 Sx - 2y = 13
4. 6a; + 5?/ - 8 = 0 9. 5a: + 3i/ = 8
4a; - 32/ - 18 = 0 5x-4y = 7
5. y = 2x 10. 2/ = f (ic - 3)
3a; + 22/ = 21 2/ = f a: + 1
6. 2/ = 6a; - 3 11. y = 2x + 1
8-5x = y 3x + § = 8
12. Make up and solve an example in simultaneous
equations which is solved more readily by the method of
comparison than by either of the other two methods of elim-
ination.
13. Make up and solve an example in simultaneous equa-
tions which is solved more readily by the method of substi-
tution than by either of the other two methods.
14. Make up and solve an example solved more readily
by the method of addition and subtraction than by the other
two methods.
230 SCHOOL ALGEBRA
EXERCISE 71
Solve and check each result:
3 4 40 ^
1-3^ = 0 ~-' + ^y-h
4y _ ^+10 = 3 3y-x = 2
3
2x - y , Sx + 2y _„
2a; 4a: + y - 1 _
3 "^ 4
6. — — ^^— = 0
y + 3 a; + 4
j/(a; - 2) - a;(y - 5) + 13 = 0
7. f (x + 32/) - i{x + 2y) = -jV
32/-f(a; + 4y + f) = 0
8. .4a; - .32/ = .7 10. .5x + 4.5t/ = 2.6
.7a; + .2y = .5 1.3a; + 3.1y = 1.6
9. 2a; + 1.52/= 10 1^- -Sa; - .7y = .005
.3a; - my = .4 2a; = 82/
12. ^-3^^ =;r + 13
, 2-3a;
IO2/+I ^ _ 2z + 3
5 =' . + 3-i±^
SIMULTANEOUS EQUATIONS ' 231
x-\-y 5
y _ 3^ % _ 2^
3 2^_12 3^^2
' 114 If
14. (x-5)(2/ + 3) = (a:-l)(2/ + 2)
xy + 2x=: x(y + 10) + 72y
x-2 x-\-10 lO-y_.^
2i/ + 6 4a: + i/ + 6 ^_Q
3 8
61/ + 5 3a: + 54 _ % - 4
8 5a: -2^ 12
2y + 3 x + y _4y + 7
4 3x - 22/ 8
3a: -2 ^ 6a: - 5 _ a: + y + 64
^^* 5 10 Qx-\-y
3i/-2 ^ 2?/ -5 3 + 7a:
12 ~ 8 lOy-Sx
18. Practice oral work with small fractions as in Exercise
58 (p. 190).
139. Literal Equations.
Ex. Solve ax -^by = c (1)
ax -\-l>y = c (2)
Multiply (1) by a!, and (2) by a,
aa!x + a%y = dc (3)
adx + ah'y = ac' (4)
Subtract (4) from (3), {a!h - ah')y = a!c - ad
dc — ad ^ .
232 SCHOOL ALGEBRA
Again, multiply (1) by 6', (2) by b,
ah'x +bh'y =h'c (5)
a'hx + hh'y = b(f (6)
Subtract (6) from (5), (a6' - a'b)x = b'c - be'
b'c - bc^ _ ,
ao — ao
Let the pupil check the work.
In solving simultaneous literal equations, observe that if the
value obtained for the first unknown is a fraction containing a
binomial term (or the value is complex in other ways), it is better
not to find the value of the other unknown as in numerical equations,
i. e. by substituting the value found in one of the original equations
and reducing. A better method is to tak6 both of the original equa-
tions and eliminate anew. See the solution of the preceding example.
EXERCISE 72
Solve and check each result:
1. 3x + 4?/ = 2a 7. ax -\-hy = c
5x-{-6y = 4:a mx -\- ny = d
2. 2ax + Zby = ^ah 8. 6a: + ay = a + 6
5ax 4- Ahy = Sab ab{x - y) = a^ - b^
3. ax -\-by = 1
9. c^x — <Py = c — d
ax + 6V - 1 ^^^2dx - cy) = 2d''-c'
4. x — y = 2n
mx-ny = m'-\- n^ lo. ^ + ^ = i
5. 26a; + a!/ = 46 + a 2/-^ ^
abx-2aby = 46 + a x + y = 2n
6. ax-by = a^-\-b'' H- (a + l)x-by = a-\-2
bx-\-ay = 2(a2 + 6^) {a - l)x + 3by = 9a
12. (a-6)a:- (a + 6)2/ = a2 + 62
bx-{- ay = 0
13. -4- + -i^ = 2 15. la7%+(a + % ^^
a + 6a-6 a^ + b^
x-y = 2b ax-2by = a^ - 26^
14. ax — hx = ay — dy 16. (a + b)x -\- cy = 1
x-y = l CX+ {a-j-b)y = I
SIMULTANEOUS EQUATIONS 23L
17. (a + b)x - (a- h)y = Sab
(a — b)x — (a + b)y = ab
a + o a — 0 0 — 1 0 — a
x-{-2a y-2b_a^ + b- x-\-l y-1 1
a — b a-\-b a^—b^ b 1 — a b
20. (x -l)(a-\-b)=a(y + a+ 1)
(y +l){a-b) = b(x-b- 1)
21. Make up and work an example similar to Ex. 7. To
Ex. 11. ^
140. Three or More Simultaneous Equations.
3a: + 4!/ - 52 = 32 (1)
Ex. Solve 4a; - 52/ + 3s = 18 (2)
5a: - 3^ - 42 = 2 (3)
If we choose to eliminate z first, multiply (1) by 3, and (2) by 5,
9x + 12y -15z =m (4)
20a: -^5y + 15z =90 (5)
Add (4) and (5), 29a: - 13?/ = 186 (6)
Also multiply (2) by 4, (3) by 3,
16a: - 20?/ + 122 = 72 (7)
15a; -9y -12z =6 (8)
Add (7) and (8), 31a: - 29?/ = 78 . (9)
We now have the pair of simultaneous equations,
1 29a; - 13?/ = 186
l31a: - 29?/ = 78
Solving these, obtain a: = 101 „
y =8 \
Substitute for x and y in equation (1),
30 + 32 - 52 = 32,
2=6 Root
Check. 3a: +4?/- 52 =3x 10 +4x8-5x6= 32
'4a: -5?/+ 32 =4x 10 -5X8+3X6 = 18
5a: -3?/- 42 =5X 10 -3X8-4X6=2
11.
234 SCHOOL ALGEBRA
In like manner, if we have n simultaneous equations con-
taining n unknown quantities, by taking different pairs of
the. 71 equations, we may eliminate one of the unknown quan-
tities, leaving n — \ equations, with n — 1 unknown quanti-
ties; and so on.
EXERCISE 73
Solve and check:
1. a: + 2/ + z = 6 9.
3a; + 22/ + z = 10
3a; + 2/ + 33 = 14
2. 3a; - 2/ - 2z = 11 10.
4a; - 2!/ + 2 = - 2
6a; - 2/ + 33 = - 3
3. 5a; - 62/ + 22 = 5
8a; + 42/ - 5z = 5
9a; + 52/ — 6z = 5
4. 3a;-i2/ + 2 = 7|
2x-\{y-?>z) = b\
2a;-i2/ + 42; = ll
5. 2a; + 32/ = 7 12. a; + 2/ + 22 = 2(a + h)
32/ + 42 = 9 a; 4- 2 + 2i/ = 2(a + c)
5a; + 62 = 15 2/ + 2 + 2a; = 2(6 + c)
6. 2a; + 42/ + 32 = 6 13. a; + 2/-2 = 3-a-6
62/ - 3a; + 22 = 7 a; + 2-2/ = 3a-6-l
3a; - 82/ - 72 = 6 2/ + 2-a; = 36-a-l
7. a; + 32/ + 32 = 1 14. 3a; + 22/ = V-a
3a; - 52 = 1 62 - 2a; = f 6
92/ + IO2 + 3a; =1 52/ - 132 + a; = 0
Q. u-\-x — w = 4t 15- — a; + 2/ + 2 + 2) = a
2z + 2) — a; = 1 a; — 2/ + 2 + 2J = 6
2J + 2(; + a; = 8 a; + 2/ — 2 + 2J = c
2/ — 2(J + a; = 5 a; + 2/ + 2 — 2J = (Z
ix
+ h + h
= 2
ia;
^\y + \z
= 9
1^
+ hy + ¥
= 3
2a; + 22/ - 2 =
■2a
3a;
-y-Z =
46
5a; + 32/-32 =
= 2(a + 6)
2a;
3
32/ 42
^4 5
= 18
5a;
6
52/32
8 "^ 4
= -5
3a;
2
72/32.
5 10
= -41
I
SIMULTANEOUS EQUATIONS
235
Ex. 1. Solve
16. Practice the oral solution of simple equations as in
Exercise 64 (p. 209).
141. The Use of ^ and ;^ as Unknown Quantities enables us
to solve certain equations which would otherwise be diffi-
cult of solution.
^^ + - = 49 (1)
X y
- + - = 23 (2)
KX y
Multiply (1) by 7, and (2) by 5,
(55+91 = 343 (3)
5-^^=115 (4)
x^ y ^ ^
Subtract (4) from (3), — = 228, /. -= 3, or ?/= i Root
Substitute the value of y in (2), hence, x = \ Root
Let the pupil check the work.
Ex. 2. Solve
I- + #-=11
2x Zy
(1)
(2)
X 4y 4
When X and y in the denominators have coefficients, as in this
example, it is usually best first to remove these coefficients by mul-
tiplying each equation by the L. C. M. of the coefficients of x and y
in the denominators of that equation. Hence,
Multiply (1) by 6, and (2) by 4,
x y
5-1=29
X y
(3)
(4)
Solving (3) and (4) by the method used in Ex. 1,
' X =^]
i\ Roots
Let the pupil check the work.
236 SCHOOL ALGEBRA
EXERCISE
74
Solve and check;
1. ^ + Z = 3
7.
a b _a-h
X y
X y a
3 + 5 = 2
h a _h — a
X y
X y a
.i-?=7
8.
X y
X y
3+-^ = i
1 1 ^ .2
- + - = m + ^
X y
X y
'■l^h='
9.
ox a?/
M-3
X y ab
*• ^+1=1
2x^iy
10.
« + 6 + «-6^2a
ar y
^+3--5
a; y
5. A-A = iij
11.
by -?,x = Ixy
Ax Zy ''==
15a: + my = IQxy
M='«i
12.
W~+'~ = 2
X y z
6.1 + 1.1
?-i+i=7
X y n
X y z
1-1-
3 2 5
- + - + - = 14
X y '^
^
1
a: 2/ 2
13. - -
— t
^i
X
y
5 ,
3
- +
= -
-7
y
2
2
1
'
= c
1
a:
2
SIMULTANEOUS EQUATIONS 237
2 1,4^1 ,1111
14. = 3q77 15. = -
? + !-l = 12 l_l_i = i
X y 2z y z X b
5 3 1.. 1111
2x 4i/ 3s z X y c
X y z
a , c b
-H = m
X z y
b , c a
-H = n
y z X
17. 5yz + 6icz — 3x1/ = 8a:i/z
4:yz — ^xz -\- xy = l^xyz
yz — 12a;2 — 2xy = ^xyz
18. Make up and work an example similar to Ex. 1,
To Ex. 4. Ex. 13. Ex. 15.
19. Work again such examples on pp. 212 and 213 as the
teacher may point out.
142. In the Solution of Problems Involving Two or More
Unknown Quantities, it is necessary to obtain as many inde-
pendent equations as there are unknown quantities involved
in the equations and to eliminate. (See Art. 134, p. 224.)
Ex. Find a fraction such that if 2 is added to both nu-
merator and denominator, the fraction becomes | ; but if 7 is
added to both numerator and denominator, the fraction be-
comes f .
Two unknown numbers occur in this problem, viz, : the numera-
ator and denominator of the required fraction. Hence two
equations must be formed in order to obtain a solution of the
problem.
238 SCHOOL ALGEBRA
X
Let - represent the fraction.
Then, ^^=2 ^^^ ^+T = 3
Clearing these equations, and collecting like terms,
2x -y = - 2
Sx -2y = -7
The solution gives x = S and ?/ = 8.
Therefore f is the required fraction.
Let the pupil check the work.
EXERCISE 76
1. Find two numbers whose sum is 23 and whose difference*
is 5.
2. Twice the difference of two numbers is 6, and | of their
sum is 3|. What are the numbers?
3. Find two numbers such that twice the greater number
exceeds 5 times the less by 6; but the sum of the greater num-
ber and twice the less is 12.
4. 2 lb. of flour and 5 lb. of sugar cost 31 cents, and 5
lb. of flour and 3 lb. of sugar cost 30 cents. Find the value
of a pound of each.
5. A man hired 4 men and 3 boys for a day for $18; and
for another day, at the same rate, 3 men and 4 boys for $17.
How much did he pay each man and each boy per day?
6. In an orchard of 100 trees, the apple trees are 5 more
than f of the number of pear trees. How many trees are
there of each kind?
7. One woman buys 4 yd. of silk and 7 yd. of satin, and
another woman at the same rate buys 5 yd. of silk and 5|
yd. of satin. Each woman pays $17.70. What is the price
of a yard of each material?
SIMULTANEOUS EQUATIONS
239
8. Solve Ex. 7 without using x and y to represent unknown
numbers (see Art. 1). About how much of the labor of writ-
ing out the solution is saved by the use of x and i/?
9. 1 cu. ft. of iron and 1 cu. ft. of lead together weigh
1180 lb.; also the weight of 3 cu. ft. of iron exceeds the weight
of 2 cu. ft. of lead by 40 lb. What is the weight of 1 cu. ft.
of each of these materials?
10. In an athletic meet, the winning team had a score of
26 points and the second team had a score of 21 points. If
the winning team took first place in 7 events and second
place in 5 events, while the second team took 6 firsts and 3
seconds, how many points does a first place count? A second
place?
11. In an athletic meet, the three winning teams made
scores as follows:
Team
Ists
2ds
3ds
Total Score
A
B
C
5
3
1
2
3
4
2
1
6
33
25
23
What did each of the first three places in an event count in
this meet?
12. Make up and work an example similar to Ex. 10.
13. Two partners agree to divide their profits each year in
such a way that one partner receives $1000 more than f of
what the other receives. If the profits for a given year are
$10,000, what does each partner receive?
14. Separate 240 into two parts such that twice the larger
part exceeds ^yg, times the smaller by 10.
240 SCHOOL ALGEBRA
15. If the cost of a telegram of 14 words between two
cities is 62^, and one of 17 words is 71^, what is the charge
for the first 10 words in a message and for each word after
that?
16. Make up and work an example similar to Ex. 15
concerning telegraph rates between two cities near yom:
home.
17. A farmer one year made a profit of $1640 on 20 acres
planted with wheat and 30 acres planted with potatoes.
The next year, with equally good crops, he made a profit
of $1210 on 30 acres planted with wheat and 20 acres
planted with potatoes. How much per acre on the average
did he make on each crop?
18. In three successive years, the farmer raised crops with
profits as follows:
(1) 20 A. wheat, 30 A. corn, 40 A. potatoes; profits $1720
(2) 30 A. wheat, 40 A. corn, 20 A. potatoes; "profits $1520
(3) 40 A. wheat, 20 A. corn, 30 A. potatoes; profits $1440
What were his average profits per acre for each kind of
crop?
19. The freight charges between two cities on 400 lb. of
first-class freight and 600 lb. of second-class freight were
$14.24, while the charges on 500 lb. of first-class freight and
800 lb. of second-class were $18.48. What was the rate per
100 lb. on each class?
20. The freight charges on shipments between two places
were as follows: 800 lb. of 4th class + 500 lb. of 5th class +
700 lb. of 6th class, $17.11; 1000 lb. of 4th class + 600 lb. of
5th class + 800 lb. of 6th class, $20.66; 600 lb. of 4th class +
1000 lb. of 5th class + 900 lb. of 6th class, $20.52. Find the
rate per 100 lb. for each of the classes named.
SIMULTANEOUS EQUATIONS 241
21. The corn and wheat crops of the United States in the
year 1909 were together 3,509,000,000 bu.; the corn and oat
crops 3,779,000,000 bu.; and the wheat and oat crops,
1,744,000,000 bu. How many bushels were in each
crop? •
22. One cubic foot of iron and one cubic foot of aluminum
weigh 636 lb.; a cubic foot of iron and one of copper weigh
1030 lb.; a cubic foot of copper and one of aluminum weigh
706 lb. How much does one cubic foot of each of these ma-
terials weigh?
23. In boring holes in a metal plate, three circles touching
each other are to be drawn, the distances
between their centers being .865 in.,
,650 in., and .790 in., respectively.
Find the radius of each of the three J_
circles.
24. The Eiffel Tower is taller than
the Metropolitan Life Building of New York, and the latter
building is taller than the Washington Monument. If the
difference between the heights of the first two is 284 ft.;
between the first and last is 429 ft.; and the sum of the
first and last is 1539 ft., find the height of each.
25. A ton of fertilizer which contains 60 lb. of nitrogen,
100 lb. of potash, and 150 lb. of phosphate is worth $21.50;
a ton containing 70, 80, and 90 lb. of these constituents in
order is worth $19; and one containing 80, 120, 150 lb. of
each in order is worth $25.50; what is the value of one pound
of each of the constituents named?
26. If a bushel of oats is worth 40^ and a bushel of corn is
worth 55f^, how many bushels of each must a miller use to
produce a mixture of 100 bu. worth 48ff a bushel?
242 SCHOOL ALGEBRA
27. How many pounds of 20^ coffee and how many pounds
of 32^ coffee must be mixed together to make 60 lb. worth
28j;i a pound?
28. Make up and work an example similar to Ex. 27.
•
29. If two grades of tea worth 50}Z^ and 75f!f a pound are to
be mixed together to make 100 lb. which can be sold for 72^
at a profit of 20%, how many pounds of each must be used?
30. A farmer wishes to combine milk containing 5% of
butter fat with cream containing 40% of butter fat in order
to produce 20 gal. of cream which shall contain 25% of
butter fat. How many gallons of milk and how many of
cream must he use?
31. A man has $5050 invested, part at 4%, and the rest
at 5%. How much has he invested at each rate if his annual
income is $220?
Can you work this example by use of one unknown quan-
tity?
32. A man wishes to invest part of $12,000 at 5% and the
rest at 4% so that he may obtain an income of $500. How
much must he invest at each of the rates named?
33. Make up and work an example similar to Ex. 32.
34. If a rectangle were 3 in. longer and 1 in. narrower it
would contain 5 sq. in. more than it does now; but if it were
2 in. shorter and 2 in. wider its area would remain unchanged.
What are its dimensions?
SuG. Draw a diagram for each rectangle considered in the prob-
lem. See Ex. 30, p. 104.
35. If a rectangle were made 3 ft. shorter and 1| ft. wider,
or if it were 7 ft. shorter and 5^ ft. wider, its area would
remain unchanged. What are its dimensions?
SIMULTANEOUS EQUATIONS 243
36. A party of boys purchased a boat and upon payment
for the same discovered that if they had numbered 3 more,
they would have paid a dollar apiece less; but if they had
numbered 2 less, they would have paid a dollar apiece more.
How many boys were there, and what did the boat cost?
SuG. Let X = the number of boys, and y = the number of dollars
each paid. Then xy represents the number of dollars the boat cost.
37. After going a certain distance in an automobile, a
driver found that if he had gone 3 mi. an hour faster, he would
have traveled the distance in 1 hr. less time; and that if he
had gone 5 mi. faster, he would have gone the distance in
1| hr. less. What was the distance?
38. Make up and work an example similar to Ex. 37.
39. If a baseball nine should play two games more and
win both, it will have won | of the games played. If, however,
it should play 7 more and win 4 of them, it will then have
won I of the games played. How many games has it so far
played and how many has it won?
40. If a physician should have 12 more cases of diphtheria
and treat 10 of them successfully, he will have treated f of his
cases successfully. But if he should have 32 more cases and
succeed with 30 of them, he will have succeeded with | of his
cases. How many cases has he had so far and how many has
he treated successfully ?
41. If 1 be added to the numerator of a certain fraction,
the value of the fraction becomes J; but if 1 be subtracted
from its denominator, the value of the fraction becomes J.
Find the fraction.
42. There is a fraction such that if 4 be added to its numer-
ator the fraction will equal f ; but if 3 be subtracted from its
denominator the fraction will equal f . What is the fraction?
244 SCHOOL ALGEBRA
43. Make up and work an example similar to Ex. 42.
44. A certain fraction becomes equal to ^ if if is added to
both numerator and denominator. It becomes | if 2J is
subtracted from both numerator and denominator. What is
the fraction?
45. Find two fractions, with numerators 11 and 7, respec-
tively, such that their sum is 3y^, but when their denomina-
tors are interchanged, their sum becomes 3^2 •
46. If f is added to the numerator of a certain fraction, its
value is increased by 21 ; hut if 2^ is taken from its denomi-
nator, the fraction becomes f. Find the fraction.
47. The sum of two numbers is 97, and if the greater is
divided by the less, the quotient is 5 and the remainder 1.
Find the numbers.
SuG. The divisor multiplied by the quotient is equal to the divi-
dend diminished by the remainder.
48. Divide the number 100 into two such parts that the
greater part will contain the less 3 times with a remainder
of 16.
49. The difference between two numbers is 40, and the
less is contained in the greater 3 times with a remainder of
12. Find the numbers.
50. Separate 50 into two such parts that J of the larger
shall exceed | of the smaller by 2.
51. A tank can be filled by two pipes one of which runs 4
hr. and the other 5; or by the same two pipes if the first runs
3 hr. and the other 8. How long will it take each pipe running
separately to fill the tank?
52. Two persons, A and B, can perform a piece of work in
16 days. They work together for 4 days, when B is left
SIMULTANEOUS EQUATIONS 246
alone, and completes the task in 36 days. In what time could
each do the work separately?
53. A and B can do a piece of work in 8 da.; A and C can
do the same in 10 da.; and B and C can do it in 12 da. How
long will it take each to do it alone?
54. 37 means 10 X 3 + 7. Does xy mean lOo; + y"i Why
this difference?
55. How would you write a number whose digits in order
from left to right are /, m, and n? Why may not such a
number be expressed as Imnt
56. Express in symbols a number whose digits in order are
a, 6, c, and d. Whose digits are Xy y, and z. x and y.
57. A number consists of two digits whose sum is 13, and
if 4 is subtracted from double the number, the order of the
digits is reversed. Find the number. !
58. The sum of the digits of a certain number of two
figures is 5, and if 3 times the units' digit is added to the
number, the order of the digits will be reversed. What is
the number?
59. Twice the units' digit of a certain number is 2 greater
than the tens' digit; and the number is 4 more than 6 times
the sum of its digits. Find the number.
60. In a number of 3 figures, the first and last of which are
alike, the tens' digit is one more than twice the sum of the
other two, and if the number is divided b'y the sum of its digits,
the quotient is 21 and the remainder 4. Find the number.
61. An oarsman can row 12 mi. down stream in 2 hr., but
it takes him 6 hr. to return against the current. What is his
rate in still water and what is the rate of the stream?
Make up and work a similar example.
246 SCHOOL ALGEBRA
62. A boatman rows 20 mi. down a river and back in 8
hr.; he can row 5 mi. down the river while he rows 3 mi. up
the river. Find the rate of the man and of the stream.
63. A man rows down a stream 20 mi. in 2| hr., and rows
back only f as fast. Find the rate of the man and of the
stream.
64. 3 cu. ft. of cast iron and 5 cu. ft. of wrought iron to-
gether weigh 3750 lb.; also 7 cu. ft. of the former and 4 cu.
ft. of the latter weigh 5070 lb. What is the weight of 1 cu. ft.
of each?
65. Regarding the orbits of the earth and of the planet
Mars as circles whose center is the sun, the greatest distance
between the earth and Mars at any time is 234,000,000 mi.,
and the least distance between them is 48,000,000 mi. How
far is each of them from the sun?
66. 2 lb. of tea and 5 lb. of coffee cost $2.50. If the price
of tea should increase 10% and that of coffee should diminish
10%, the cost of the above amounts of each would be $2.45.
Find the cost of a pound of each.
67. Two bins contain a mixture of corn and oats, the one
twice as much corn as oats, and the other three times as
much oats as corn. How much must be taken from each
bin to fill a third bin holding 40 bu., to be half oats and half
corn?
68. If A gives B $10, A will have half as much as B; but if
B gives A $30, B will have | as much as A. How much has
each?
69. Two grades of spices worth 25fi and 50^ a pound are
to be mixed together to make 200 lb. which can be sold at
52^ per lb. at a profit of 30%. How many pounds of each
grade must be used?
SIMULTANEOUS EQUATIONS 247
70. A train maintained a uniform rate for a certain dis-
mce. If this rate had been 8 mi. more each hour, the time
)ccupied would have been 2 hr. less; but if the rate had been
[0 mi. an hour less, the time would have been 4 hr. more,
^ind the distance.
71. If the greater of two numbers is divided by the less,
the quotient is 3 and the remainder 3, but if 3 times the
greater be divided by 4 times the less, the quotient is 2 and
the remainder 20. Find the numbers.
72. Why are we able to solve problems like Exs. 70 and
71 by algebra and not by arithmetic?
73. Find two numbers whose sum is a and whose difference
is b.
74. If a pounds of sugar and b pounds of coffee together
cost c cents, while d pounds of sugar and e pounds of coffee
together cost / cents, what is the price of one pound of each?
75. If a bushel of oats is worth p cents, and a bushel of
corn is worth q cents, how many bushels of each must be
mixed to make r bushels worth s cents per bushel?
76. Find a fraction such that if a be added to both nu-
merator and denominator the value of the fraction is p/q;
but if b is added to both numerator and denominator, the
value of the fraction is r/s.
77. Generalize Ex. 34 (p. 242), by using a letter for each
number in the example.
78. Generalize Ex. 53 (p. 245), by using a letter for each
number in the example.
79. Make up and work three examples similar to such of
the examples in this Exercise as you think are most interesting
or instructive.
248 SCHOOL ALGEBRA
143. ITtilities in Algebra.
1. Brevity of expressions which represent numbers. Brevity
means a saving of time and energy.
Thus, for instance, for " number of feet in the length of a rect-
angle," we may use a single letter as x.
2. The saving of space also opens the way for the use of
auxiliary quantity of various kinds.
See, for instance, the process of factoring o^ + a^"^ + 6^, p. 153.
3. By using a letter to represent any number (of a given
class), we are able to discover and prove general laws of numbers.
Thus, (a +by = a^ + 2ab + b^ is true for any numbers whatever.
As an example of the discovery of new and useful laws of number,
we may take the case where we know half the sum and half the dif-
ference of two numbers and desire to find the numbers themselves.
In the above description of the known facts, there is nothing to
suggest a method of obtaining the desired end. But if we express
the given facts in the algebraic form, thus, ^ and ^ , it is
at once suggested that half the difference added to half the sum will
give a, the greater of the two numbers, and subtracted will give b, th(?
smaller.
It may be well to notice that one source of this discovery is that
in the algebraic expression we used separate symbols, a and b, of
nearly equal size for the two numbers considered.
4. Combination of several rules into one formula.
Thus, the single formula p = br combines three cases (and rules)
used in arithmetic in treating percentage. Similarly, the formula
i = prt covers all cases used in treating interest in arithmetic.
This advantage comes (1) from the fact that a letter may be
used to represent any number. See 3 above.
(2) From the fact that an equation can be solved for any letter
in the equation.
(3) From the approximately uniform size of the letters employed,
which suggests that we treat all the letters alike and give each the
leadership in turn.
5. The use of letters to represent unknown numbers often
enables us to begin in the middle of a complex problem and work
I
SIMULTANEOUS EQUATIONS 249
in several directions and thus solve problems which otherwise
we could not analyze. See the examples on pp. 242-243.
6. We should also remember constantly that the symbols
used in algebra (and the advantages coming from their use)
are but a part, or detail, of the more general subject of sym-
bolism as a whole and of its utilities; and that a training in
algebra should give a better grasp of the whole subject of symbols
and their uses.
EXERCISE 76
1. Abbreviate the following as much as you can by use of
the letter x:
I a certain number + J the number = 25. How much
shorter is your expression than the given expression?
2. Make up and work an example similar to Ex. 1.
3. Why does a knowledge of algebra suggest to us that a
number like 27001 can be factored and also the method of
doing this, while a knowledge of arithmetic does not do the
same? (See Ex. 29, p. 127.)
4. Is a railroad ticket a symbol or representative of the
money paid for it? What are the advantages in the use of
the ticket? The disadvantages?
5. Discuss in the same way a check drawn on a bank and
used in paying a bill.
6. In canceling a railroad ticket, what are the advantages
in punching the ticket as compared with crossing it with a
pencil mark? With burning it?
7. What is a newspaper (or a book) a symbol or represen-
tative of? What are the advantages in its use? The disad*
vantages?
250 SCHOOL ALGEBRA
' 8. A certain firm occupied a building running from 10 to
20 Barclay St. in a certain city as their place of business. In
advertising in one magazine they gave their address as 10
Barclay St. ; in another they gave their address as 12 Barclay
St.; in another as 14 Barclay St. What was the advantage
in doing this? By this means what double use was made of
the symbols 10, 12, 14, etc.
9. If a teacher has a set of papers from each of several
classes, what is the advantage in arranging them at different
angles when piling one set upon another?
10. Can you give another instance where difference of
position is utilized as a symbol?
11. What are the advantages in using a flag as a symbol
or representative of a nation?
12. What are the advantages and disadvantages of read-
ing a book of travels as compared with traveling?
13. Why does a policeman in a large city have a number
as well as a name? Name other classes of men which have
numbers as well as names.
14. What are the advantages to a person in having a name?
15. Let each pupil make up (or collect) and work as many
examples as possible similar to the examples in this Exercise.
SuG. This work is of such a nature that it may readily be ex-
tended in various directions at the option of the teacher.
CHAPTER XIII
GRAPHS
144. A variable is a quantity which has an indefinite
number of different values.
A function is a variable which depends on another variable
for its value.
Thus, the area of a circle is a function of the radius of the circle;
the wages which a laborer receives is a function of the time that the
man works.
A graph is a diagram representing the relation between a
function and the variable on which the function depends for
its value.
A function may depend for its value on more than one variable;
thus, the area of a rectangle depends on two quantities — the length
of the rectangle and the breadth. The present treatment of graphs,
however, is limited to functions which depend on a single variable.
In algebra we study only those functions which have a definite
value for each definite value of the variable.
145. Uses of Graphs. A graph is useful in showing at a
glance the place where the function represented has the
greatest or least value and where it is changing its value most
rapidly, and in making clear similar properties of the function.
Graphs of algebraic equations are useful in making clear
certain properties of such equations which are otherwise
difficult to understand. A graph also often furnishes a rapid
method of determining the root (or roots) of an equation.
251
252
SCHOOL ALGEBRA
^
146. Framework of Reference. Axes are two straight
lines perpendicular to each other which are used as an auxil-
iary framework in constructing graphs; as XX' and YY',
The X-axis, or axis of abscissas, is the horizontal axis; as
XX', The y-axis, or axis of ordinates, is the vertical axis;
as YY',
The origin is the point
in which the axes inter-
sect; as the point 0,
The ordinate of a point
is the line drawn from
the point parallel to the
i/-axis and terminated by
the ar-axis. The abscissa
of a point is the part of
the ic-axis intercepted between the origin and the foot of the
ordinate. Thus, the ordinate of the point P is AP, and the
abscissa is OA.
i7- ; +.
The ordinate is
sometimes termed
the "2/" of a
point, and the ab-
scissa, the " x " of
a point.
(-3,2)
X'\ — I — ^-
i--+-
B
(-2,-2)
Ordinates
above the a:-axis
are taken as plus;
those below, as
minus. Abscissas
to the right of
the origin are plus; those to the left are minus.
The co-ordinates of a point are the abscissa and the ordi-
nate taken together. They are usually written together
J,0
-» — I — I — I — I — I — \x
r
I
GRAPHS 253
in parenthesis with the abscissa first and a comma
between.
Thus, the point (2, 4) is the point whose abscissa is 2 and ordi-
nate 4, or the point P of the figure. Similarly, the point ( — 3, 2) is
Q; (-2, -2) is R; and (1, -4) is S.
The quadrants are the four parts into which the axes di-
vide a plane. Thus, the points P, Q, R, and S lie in the firsty
second, third and fourth quadrants, respectively.
EXERCISE 77
Draw axes and locate each of the following points:
1. (3,2), (-1,3), (-2, -4), (4,-1).
2. (2,§), (-3, -IJ), (5, -f), (-2,1).
3. (2, 0)_, (-3, 0),_(0, 4), (0, -1)^(0, 0).
4. (1,V2), (l,-V2),(V3,0),(V5,-3), (-iV5,2V2).
5. Construct the triangle whose vertices are (1,1), (2,-2)
(3, 2).
6. Construct the quadrilateral whose vertices are (2, — 1),
(-4, -3), (-3, 5), (3, 4).
7. Plot the points (0, 0), (1, 0), (2, 0), (5, 0), (-1, 0),
(-3, 0).
8. Also (0, 0), (0, 1), (0, 2), (0, 3), (0, 5), (0, -1), (0, -3).
9. All points on the x-axis have what ordinate?
10. All points on the ?/-axis have what abscissa?
11. Plot the following pairs of points and find the distance
between each pair of points:
{1) (6, 5), (2, 8) (3) (3, -6), (-2, 6)
(^) (3, 0), (0, 6) U) (0, 0), (-3, 5)
12. Construct the rectangle whose vertices are (1, 3),
re, 3), (1, -2), (6, -2), and find its area.
254
SCHOOL ALGEBRA
13. Construct the rectangle whose vertices are ( — 3, 4),
(4, 4), (-3, -2), (4, -2), and find its area.
14. Construct the triangle whose vertices are (-3, -4),
(-1, 3), (2, -4), and find its area.
15. In which quadrant are the abscissa and ordinate both
plus? Both minus? In which quadrant is the aKscissa minus
and the ordinate plus? In which is the abscissa plus and the
ordinate minus?
16. Practice oral work with small fractions as in Exercise
58 (p. 190).
Graphs of Equations of the First Degree
147. To Construct the Graph of an Equation of the First
Degree Containing Two Unknown Quantities, as x and y.
Let X hive a series of convenient values y as 0, 1, 2, 3, etc.,
-1, -2, -3, etc.;
r
OJ
GRAPHS
255
Find the corresponding values of y;
Locate the points thus determined, and draw a line through
these points.
Ex. Construct the graph of the equation y = 2x —1.
Construct the points (0, -1), (1, 1), (2, 3), (3, 5),
( — 1, —3), (—2, —5), etc., and draw a line through
them. The straight Une AB is thus found to be the
graph oi y = 2x — I.
X
y
0
-1
1
1
2
3
3
5
etc.
etc.
-1
-3
-2
-5
etc.
etc.
148. Linear Equations. It will always be
found that the graph of an equation of the
first degree containing not more than two
unknown quantities is a straight line. Hence,
A linear equation is an equation of the first degree.
149. Abbreviated Method of Constructing the Graph of
a Linear Equation. Since a straight line is determined by
two points, in order to construct the graph of an equation
of the first degree it is sufficient to construct any two points of
the graph and draw a straight line through them.
Ex. 1. Graph 3y - 2a; = 6.
When a; =0,?/ = 2;
when y=0,x= — 3.
Hence, the graph
passes through the
points (0, 2) and
(-3,0), or CD is the
required graph.
The greater the
distance between
the points chosen, the
more accurate the
construction will be.
It is usually advis-
able to test the
result obtained by
locating a third point and observing whether it falls upon the
graph as constructed.
256 SCHOOL ALGEBRA
If the given line does not pass through the origin, or near the
origin on both axes, it is often convenient to construct the line
', by determining the points where the Une crosses the axes.
\ Ex. 2. Graph 4x -}- 7y = 1.
\ When X = 0,y = j; when y = 0, x = I. Hence, the graph
' passes close to the origin on both axes. Hence, find two points on
the required graph at some distance from each other, as by letting
x = 0, 9, and finding y = |, —5. Let the pupil construct the figure.
EXERCISE 78
Graph the following. (It is an advantage, if possible, to
draw the graph line in red, the rest of the figure in black ink.)
1. y = x + 2 7. 4x - 5y = 1
2.y = x-2 8. ?^ = 32^
3. Sx + 2y = 6 ^
4. 3a: - 22/ = 6 9' ^ = ^(2/ " 1)
5. 3a: - 5?/ + 15 = 0 \Q. y = - x
6. y = 2x 11. 2/ = 4
12. If a: = 2, show that whatever value y has, x always
= 2. Hence the graph of a; = 2 is a line parallel to the y-axis.
13. Graph a: = 0; also y = 0.
14. Show how to determine from an inspection of a linear
equation whether its graph passes through the origin; near
the origin on one axis; near the origin on both axes.
15. Graph 5x -\- Qy = 1 ; also Qx — y = 12.
16. Obtain and state a short method of graphing a linear
equation in which the term which does not contain a: or y is
missing, as 2y — 3x = 0.
GRAPHS
257
Before graphing the following, determine the best method
of constructing each graph, and then graph:
17. x + 2y = 4: 20. ja; + ^2/ = 2 23. x - y = 5
18. 2y = X 21. a: = - 3 24. i/ + 2 = 0
19. 5x-6y=l 22. bx-\-4y = 0 25. 3x-2y + -i=0
26. Construct the triangle whose sides are the graphs of
the equations, y — 2x + \=0,3y — x — 7 = 0,y + 3a: + 11 = 0.
27. An equation of the form y = b represents a line in
what position? One of the form x = a?
28. Make up and work an example similar to Ex. 4. To
Ex. 26.
150. Graphic Solution of Simultaneous Linear Equations.
If we construct the graph of the equation x — y = 3 (the
line AB) and the graph of 3x -\- 2y = 4: (the line CD), and
c -^
^ -4.
258 SCHOOL ALGEBRA
measure the co-ordinates of their points of intersection, we
find this point to be (2, —1).
x-y = 3
If we solve the pair of simultaneous equations , ^ , ^ _ a
by the ordinary algebraic method, we find that x = 2 and
y=-l.
In general, the roots of two simultaneous linear equations cor-
respond to the co-ordinates of the point of intersection of their
graphs; for these co-ordinates are the only ones which sat-
isfy both graphs, and their values are also the only values of
X and y which satisfy both equations.
Hence, to obtain the graphic solution of two simultaneous
equations.
Draw the graphs of the given equations, and measure the co-
ordinates of the point {or points) of intersection.
Graphing two simultaneous equations forms a convenient
method of testing or checking their algebraic solution.
151. Simultaneous Linear Equations whose Graphs are
Parallel Lines. Construct the graph of x -{- 2y = 2 and also
of 3a: + 6?/ = 12.
You will find that the graphs obtained are parallel straight lineis.
Now try to solve the same equations algebraically. You will find
that when either a; or ?/ is eliminated, the other unknown quantity
is eliminated also, and that it is therefore impossible to obtain a
solution. The reason why an algebraic solution is impossible is
made clear by the fact that the graphs, being parallel fines, cannot
intersect; that is to say, there are no values of x and y which
will satisfy both of these lines, or both equations, at the same
time.
152. Graphic Solution of an Equation of the First Degree
of One Unknown Quantity. By substituting for y in the
(y — X — 3
_ r. the two equations
GRAPHS 259
reduce to a: — 3 = 0. Accordingly, the graphic solution of
an equation like x — 3 = 0 can be obtained by combining
the graphs oi y = x — 3 and y = 0. In other words, the root
of a: — 3 = 0 is represented graphically by the abscissa of
the point where the graph oi y = x — 3 crosses the a:-axis.
EXERCISE 79
Solve each pair of the following equations both graphically
and algebraically, and compare the results in each example:
^^ {2x-\-3y = 7 ^ lx + 7y + n=0
2.
x-y=^l ' \x-3y + l=0
y = 3x-^ e ly = ^
.y=-2x + l ' \9x-5y = 3
^ \2y = x 1 \y^^
^' \x + y-\-6 = 0 ' l2/ = 2a: + 3
(y = 2x ^' Solve graphically 2a:+3 = 0
' \x -{- y = 0 9. Solvegraphically 3a:— 5 = 0
10. Discover and state the relation between the coefficients
of two linear simultaneous equations whose graphs are par-
allel lines. ,
11. Solve graphically | g^ + 2^/ = 11. , ^ , _ ,
^x — oy — o.
12. Solve both algebraically and graphically \ n _ g _ r
13. Construct the quadrilateral whose sides are the
graphs of the equations, x — 2y — 4: = 0, x -\- y = 1,
3y — 5a;— 15 = 0, a: + 2?/ — 4 = 0, and find the co-
ordinates of the vertices of the quadrilateral.
14. Make up and work an example similar to Ex. 1. To
Ex.6.
15. How many examples in Exercise 26 (p. 110) can you
now work at sight?
260
SCHOOL ALGEBRA
163. Graphic Solution of Written Problems.
I. Railway Problems.
Ex. The distance between New York and Philadelphia
is 90 mi. At a given time, a train leaves each city, bound
for the other city, the train from New York going at 40 mi.
an hour and the one from Philadelphia at 30 mi. In how
many hours will they meet, and at what distance from New
York?
The train dispatcher represents the distance between the stations
by the Une AB, each space denoting 10 mi. Each space on AI rep-
/ resents 1 hour. He lo-
cates E three units to the
right of A and one unit
above AB, and F four
units to the left of B and
one unit above AB. He
produces AE and BF to
meet at C, and draws
SS§s§8gg (J J) perpendicular to AB.
He obtains the distance from A at which the trains meet, by
measuring AD to scale (and hence determines the siding at which
one train must wait for the other). He obtains the time that elapses
before the trains meet, by measuring CD to scale.
The problem may also be solved algebraically in the same way
as Exs. 57-61, p. 87.
The advantage of the graphical method is that in this solution
it is easy to make allowance for any waits which trains may make
at stations. Hence, railroad time-tables are often constructed
entirely by graphical methods.
4hr.
Shr.
2hr.
Ihr.
J
^^
c:^
^
"f .
M
T*-^
^
.^
B
II. Problems in the Mixture of Materials.
Ex. In order to obtain a mixture containing 20% of butter
fat, in what proportion should cream containing 30% of fat
be mixed with milk containing 4%?
JJ
20
16
4
10
GRAPHS 261
Graphical Solution
We construct a rectangle, and write in two adjacent corners
(here the left-hand corners) the per cents
of fat (30 and 4) in the two given fluids;
and in the middle of the rectangle we
write the per cent (20) desired in the mix-
ture. The differences between the num-
ber in the middle and the numbers in the
corners (16 and 10) are then found and placed as in the diagram.
The differences thus found show the relative amounts of the given
fluids to be used; viz. : 10 parts of milk, and 16 of cream.
Now solve this problem algebraically by the method used for
Exs. 26-30, pp. 241-242; and by an examination of this solution,
discover for yourself the reason for the above graphical solution.
EXERCISE 80
Solve the following problems graphically:
1. The distance between New York and Philadelphia is
90 mi. If a train leaves New York at noon and goes 40 mi.
an hour, and another train leaves Philadelphia at the same
time and travels 20 mi. an hour, at what time and how far
from New York will they meet?
2. Make up and work an example similar to Ex. 1.
3. The distance between New York and Buffalo is 440
mi. If a train leaves New York at 11 a. m. and travels at
the rate of 40 mi. an hour, and a train traveling 30 mi. an
hour leaves Buffalo at the same time, at what time and how
far from New York will the trains meet?
4. Make up and work an example similar to Ex. 3.
5. In order to obtain a mixture containing 22% butter
fat, in what proportion must cream containing 32% of fat
be mixed with milk containing 5%?
262 SCHOOL ALGEBRA
6. In order to obtain a mixture containing 28% of butter
fat, in what proportion must cream containing 35% of fat
be mixed with cream containing 2'b%t
7. Make up and work an example similar to Ex. 6.
8. In what proportion must oats worth 50f!^ a bushel be
mixed with corn worth 80^ a bushel in order to make a mix-
ture worth 60^;^ a bushel?
9. Make up and work a similar example concerning mixing
different grades of coffee.
10. The distance PQ is 48 mi. At 8 a. m. one boy starts
from P and walks toward Q at the uniform rate of 4 mi. an
hour. At the same time another boy starts from Q on a
bicycle and rides toward P at the rate of 8 mi. an hour but
at the end of each hour of riding rests \ an hour. By means
of a graph determine where and when the two boys will meet.
11. Make up and work an example similar to Ex. 10.
12. How many examples in Exercise 27 (p. 112) can you
now work at sight?
EXERCISE 81
Review
1. Tell the degree of each term of
5a;3 _ 4^22^2 _ \\^ — xy + xy^ — xHj + Sa;^ —7y-\- 11.
2. Factor (1) x^ + 4.
(2) m? — 2mn + n^ + 5m — 5n.
(3) a2 - n2 - m2 - 2ab + 2mn + ¥.
3. Factor 2(x^ - 1) + 7{x^ - 1).
Simplify:
' 4 4x — 5 4: -\-x 2 _ x — 5
~45~~~30~'^3 ~18"*
1 - 5a; 3a; + 5 23; - 3 '
6a;2-6"^4a;+4"^3 -3a;*
p
GRAPHS 263
6.
2
r ^
2 1
1 - a; X
(X-
1)3' {i-xr
7
m
x^ -
V^-
^)
. (-^)'
-<■*»
9.
1--
a
2)
/
a;
-3'
o
~ a + x
Solve:
10.
1^+:
x-1 1
^= 2 +3-
5
11.
'+1
-K-D-
313 V
a;+l\l
2 /r
12.
3a;-!
2 5a:+14
3-2a;_
li.
6
7x+15
4 "^
13.
X
a;4-l X-
8 a;-9
x-2
x— 1 a;-
6 a;-7
14.
Sx
2 ■
y.
3"
4
'9*
"■ '-i-^-
10- a; 2/- 10
3 4
5x Sy
4"^ 2
= 3i.
2?/+ 4
2a;+?/ a;+13
3
8 4
15.
2y
A
- x =
3^
a;
= 4a:2/.
17. (a-b]
(a; + (a + 6)?/= a +6.
y
9.
{x-y){a?-¥) = a^+b\
18. .Sx+.2y=: 1.3.
.32/ +.22 =.8.
.32+. 2a; =.9.
19.
X
y'
5
'6*
20. Solve
a'^b'ah'
X-
■y =
2
15'
X ,y 1
21. Is it allowable to divide each term in 16a; = 96 by 16?
Is it allowable to divide each term of 16a; — 96 by 16? Give
reasons.
ax+1 r , ,, a{x^- 1) ^,
when X
264 SCHOOL ALGEBRA
22. Obtain the value of
[a(x+l)- J,
23. Solve 1+^=1+^11^.
24. Why is it proper to change —x=—3 into a; = 3, and not
proper to change —x—3 into a; + 3?
o c
25. What number added to the denominators of r and -j, respec-
tively, will make the results equal?
26. Does r equal -— r? Does equal -+-? Does
a—b a 0 c c c
c c c
— — T equal -+r? Verify your statements for the special case
when a = 4, 6 = 8, and c = 2.
27. The sums of three numbers taken two and two are 20, 29,
and 27. What are the numbers?
28. Factor 8{x+ yY- {2x- yY.
29. What is the advantage of being able to add the same number
to both members of an equation? In being able to transpose a term?
To divide both members of an equation by the same number? (See
Art. 70.)
30. Solve (a + c)x - (a- c)y = 2ab.
(a + b)x — {a— h)y = 2ac.
31. Does (a^ + b^) (a + b) equal a^ + 6^? Verify your state-
ment by letting a and b have convenient numerical values. Can
you prove your statement without the use of numbers?
32. If K = ttR^ and C = 2nR, find K when C = 10 and tt = ^i^.
33. If s = ^gt"^ and v = gt, find s when g= S2 and v = 64.
34. If C = 2nR and V = UR\ find V when C = 33 and tt = ^^
(use cancellation wherever possible).
Solve:
35. 4(x+2/) + ^- = 13. 36. 3a;+--^ = 6.
x-y 2x-Sy
Si^-^y) —= -1. 4x---?— = 25.
x-y 2x — 3y
SuG. Let p = (x -\-y), q =
x-y
37. Show that
reduces to
GRAPHS 265
12t{t^ + 2)2 (21'^ - sy - 6t^ (2t^ - Sy (t^ + 2)
{t'+2y
6(3^2+40 (2^2-3)=^
(f+2y
38. Graph y=2x + b when 6=1. On the same diagram
graph y= 2x+b when 6=2. When 6 = — 1. When 6=0.
39. Graph y=ax+2 when a= 1. On the same diagram
graph y= ax+ 2 when a= 2, 3, —1, —3.
40. Graph y= 3a; + 2, and y=—ix+2on the same diagram.
41. Make up and work an example similar to Ex. 38. To Ex. 39.
42. The Fahrenheit reading at the boiling point of alcohol is
95° higher than the Centigrade reading. Find each of the readings.
43. Make up an example similar to Ex. 42, using the fact that
ether boils at 96° Fahrenheit.
44. Give the value of - -j- a, a -v- -, —-x^-r- -x^.
a a 5 5
45. What is the reciprocal of - + t ?
a 0
46. Show that elimination by comparison is a special form of
elimination by substitution.
47. Show that eUmination by addition and subtraction may also
be regarded as a form of elimination by substitution.
48. Eliminate a between the equations F = Ma and s = ^af^.
49. Given I = a+ {n — V)d and s= ^(a+ I), find s in terms of
d, n, and I.
SuG. What letter must be eliminated?
rl—a
50. Eliminate I between 1= ar"^ ^ and s
r-1
CHAPTER XIV
HISTORY OF ELEMENTARY ALGEBRA
295. Epochs in the Development of Algebra. Some knowl-
edge of the origin and development of the symbols and
processes of algebra is important to a thorough under-
standing of the subject.
The oldest known mathematical writing is a papyrus roll,
now in the British Museum, entitled *' Directions for Attain-
ing to the Knowledge of All Dark Things." It was written
by a scribe named Ahmes at least as early as 1700 b. c, and
is a copy, the writer says, of a more ancient work, dating,
say, 3000 b. c, or several centuries before the time of Moses.
This papyrus roll contains, among other things, the begin-
nings of algebra as a science. Taking the epoch indicated
by this work as the first, the principal epochs in the develop-
ment of algebra are as follows:
1. Egyptian : 3000 B. C -1500 B. C.
2. Greek (at Alexandria): 200 A.D.-400 A.D. Principal
writer, Diophantus.
3. Hindoo (in India): 500 A.D.-1200 A.D.
4. Arab : 800 A. D.-1200 A. D.
5. European: 1200 A. D.- Leonardo of Pisa, an Italian,
pubhshed in 1202 a. d. a work on the Arabic arithmetic
which contained also an account of the science of algebra as
it then existed among the Arabs. From Italy the knowledge
266
HISTORY OF ELEMENTARY ALGEBRA 267
of algebra spread to France, Germany, and England, where
its subsequent development took place.
We will consider briefly the history of
I. Algebraic Symbols.
II. Ideas of Algebraic Quantity.
III. Algebraic Processes.
I
I. History of Algebraic Symbols
296. Symbol for the Unknown Quantity.
1. Egyptians (1700 b. c): used the word hau (expressed,
of course, in hieroglyphics), meaning " heap."
2. Diophantus (Alexandria, 350 A. d.?): <5', or 9°'; plural,
99.
3. Hindoos (500 a. D.-1200 a. d.) : Sanscrit word for
" color," or first letters of words for colors (as blue, yellow,
white, etc.).
4. Arabs (800 a. D.-1200 a. d.) : Arabic word for " thing "
or '^ root " (the term root, as still used in algebra, originates
here).
5. Italians (1500 A. D.) : Radix, R, Rj.
6. BomboUi (Italy, 1572 a. d.): kij
7. Stifel (Germany, 1544): yl, B, C,
8. Stevinus (Holland, 1586) : ®
9. Vieta (France, 1591): vowels A, E, I, 0, U.
10. Descartes (France, 1637) : x, y, z, etc.
297. Symbols for Powers (of x at first) ; Exponents.
1. Diophantus: Bwafi^;, or 8" (for square of the unknown
quantity); kv^o<;, or /c" (for its cube).
2. Hindoos: initial letters of Sanscrit words for " square "
and " cube."
^(^^ SCHOOL ALGEBRA
3. Italians (1500 a.d.): "census" or " zensus " or "s"
(for x''); " cubus " or " c " (for a^).
4. Bombelli (1579): ®, ®, ®, (for x, x\ :>?),
5. Stevinus (1586): ®, (D, ®, (for x, x\ x?).
6. Vieta (1591): A, A qitadratus, A cubus (for x, o^ 3!^).
7. Harriot (England, 1631) : a, aa, aaa,
8. Herigone (France, 1634) : a, a2, a3.
9. Descartes (France, 1637) : x, x'^, a^.
Wallis (England, 1659) first justified the use of fractional
and negative exponents, though the use of fractional expo-
nents had been suggested earlier by Oresme (1350), and the
use of negative exponents by Choquet (c. 1500).
Newton (England, 1676) first used a general exponent, as
in a*", where n denotes any exponent, integral or fractional,
positive or negative.
298. Symbols for Known Quantities.
1. Diophantus: ixovahe^ (i. e. monads), or fx .
2. Regiomontanus (Germany, 1430) : letters of the alphabet.
3. Italians: d, from dragma,
4. BombelH:®.
5. Stevinus: ®.
6. Vieta: consonants, B, C, D, F, , , .".
7. Descartes: a, b, c, d.
Descartes possibly used the last letters of the alphabet, x, tj, z, to
denote unknown quantities because these letters are less used and
less familiar than a, b, c, d, . . . . , which he accordingly used to de-
note known numbers.
239. Addition Sign. The following symbols were used:
1. Egyptians: pair of legs walking forward {to the left), -A.
2. Diophantus: juxtaposition (thus, a6, meant a -i- b).
I
HISTORY OF ELEMENTARY ALGEBRA 269
3. Hindoos: juxtaposition (survives in Arabic arithmetic,
as in 2 1, which means 2+1).
4. Italians: plus, then p (or e, or <j>),
5. Germans (1489): -h, +, +.
300. Subtraction Sign.
1. Egyptians: pair of legs walking backward (to the right),
thus, A_; or a flight of arrows.
2. Diophantus: ^ (Greek letter yjr inverted).
3. Hindoos: a dot over the subtracted quantity (thus, mn
meant m — n).
4. Italians : minus , then M or m or de,
5. Germans (1489) : horizontal dash, — .
The signs + and — were first printed in Johann Widman's
Mercantile Arithmetic (1489). These signs probably originated in
German warehouses, where they were used to indicate excess or
deficiency in the weight of bales and chests of goods. Stifel (1544)
was the first to use them systematically to indicate the operations
of addition and subtraction.
301. Multiplication Sign. IMultiplication at first was
usually expressed in general language. But
1. Hindoos indicated multiplication by the syllable hha,
from bharita, meaning " product," written after the factors.
2. Oughtred and Harriot (England, 1631) invented the
present symbol, X .
3. Descartes (1637) used a dot between the factors (thus,
a-b).
302. Division Sign.
1. Hindoos indicated division by placing the divisor under
the dividend (no line between). Thus, 5 meant c -i- d.
2. Arabs, by a straight line. Thus, a — 6, or a I 6, or —
b
270 SCHOOL ALGEBRA
3. Italians expressed the operation in general language.
4. Oughtred, by a dot between the dividend and divisor,
5. Pell (England, 1630), by -.
303. Equality Sign.
1. Egyptians: Z □ (Also other more complicated sym-
bols to indicate different kinds of equality).
2. Diophantus: general language or the symbol, K
3. Hindoos: by placing one side of an equation immediately
under the other side.
4. Italians: w or a; that is, the initial letters of cequalis
(equal). This symbol was afterward modified into the form,
:» , and was much used, even by Descartes, long after the
invention of the present symbol by Recorde.
5. Recorde (England, 1540): =.
He says that he selected this symbol to denote equality because
" than two equal straight Unes no two things can be more equal."
304. Other Symbols used in Elementary Algebra.
Inequality Signs, > <, were invented by Harriot (1631).
Oughtred, at the same time, proposed ~Z1, j as signs of in-
equality, but those suggested by Harriot were manifestly superior.
Parenthesis, ( ), was invented by Girard (1629).
The Vinculum had been previously suggested by Vieta
(1591).
Radical Sign. The Hindoos used the initial syllable of
the word for square root, Ka, from Karania, to indicate
square root.
Rudolph (Germany, 1525) suggested the symbol used at
present, V, (the initial letter, r, in the script form, of the
word radix, or root) to indicate square root, /W to denote
the 4th root, and /VW to denote cube root.
r
HISTORY OF ELEMENTARY ALGEBRA 271
Girard (1633) denoted the 2d, 3d, 4th, etc., roots, as at
present, by v^, V, ^, etc.
The sign for Infinity, oo , was invented by WalHs (1649).
305. Other Algebraic Symbols have been invented in recent
times, but these do not belong to elementary algebra.
Other kinds of algebra have also been invented, employing
other systems of the symbols.
306. General Illustration of the Evolution of Algebraic Sym-
bols. The following illustration will serve to show the
principal steps in the evolution of the symbols of algebra:
At the time of Diophantus the numbers 1, 2, 3, 4, ... . were de-
noted by letters of the Greek alphabet, with a dash over the letters
used; as, a,~^,y, . . . .
In the algebra of Diophantus the coefficient occupies the last
place in a term instead of the first as at present.
Beginning with Diophantus, the algebraic expression,
x^ -{- 5x — 4, w^ould be expressed in symbols as follows:
B^'a ^oe^t^fiB (Diophantus, 350 A. D.)
Iz p.5 RmA (Italy, 1500 a. d.).
IQ + 5iV - 4 (Germany, 1575).
l^)p.5[i)mA[0] (BombelH, 1579).
1(2) + 5® - 4(0) (Stevinus, 1586).
1^^ + 5^-40 (Vieta, 1591).
laa -\- 5a — 4 (Harriot, 1631).
Ia2 + 5al - 4 (Ilerigone, 1634).
x'^ -}- 5x — 4 (Descartes, 1637).
307. Three Stages in the Development of Algebraic Sym-
bols.
1. Algebra without Symbols (called Rhetorical Algebra).
In this primitive stage, algebraic quantities and operations
were expressed altogether in words, without the use of sym-
272 SCHOOL ALGEBRA
bols. The Egyptian algebra and the earUest Hindoo, Arabian,
and ItaUan algebras were of this sort.
2. Algebra in which the Symbols are Abbreviated Words
(called Syncopated Algebra). For instance, p is used for jdus.
The algebra of Diophantus was mainly of this sort. European
algebra did not get beyond this stage till about 1600 a. d.
3. Symbolic Algebra. In its final or completed state,
algebra has a system of notation or symbols of its own, inde-
pendent of ordinary language. Its operations are performed
according to certain laws or rules, " independent of, and dis-
tinct from, the laws of grammatical construction.-"
Thus, to express addition in the three stages we have plus,
p,-\-;to express subtraction, minus, m, — ; to express equality,
cequalis, ce, =,
Along with the development of algebraic symbolism, there
was a corresponding development of ideas of algebraic quan-
tity and of algebraic processes.
XL History of Algebraic Quantity
308. The Kinds of Quantity considered in algebra are
positive and negative; particular (or numerical) and general;
integral and fractional; rational and irrational; commensur-
able and incommensurable; constant and variable; real and
imaginary.
309. Ahmes (1700 b. c.) in his treatise uses particular, pos-
itive quantity, both integral and fractional (his fractions, how-
ever, are usually limited to those which have a unity for a
numerator). That is, his algebra treats of quantities like 8
and I, but not like — 3, or — f , or V2, or — a.
310. Diophantus (350 a. d.) used negative quantity, but
only in a limited way; that is, in connection with a larger
HISTORY OF ELEMENTARY ALGEBRA 273
positive quantity. Thus, he used 7 — 5, but not 5 — 7, or
— 2. He did not use, nor apparently conceive of, negative
quantity having an independent existence.
3U. The Hindoos (500 A. D.-1200 A. D.) had a distinct
idea of independent or absolute negative quantity, and used
the minus sign both as a quaUty sign and a sign of operation.
They explained independent negative quantity much as it is
explained to-day by the illustration of debts as compared
with assets, and by the opposition in direction of two lines.
Pythagoras (Greece, 520 b. c.) discovered irrational quan-
tity, but the Hindoos were the first to use this in algebra.
312. The Arabs avoided the use of negative quantity as
far as possible. This led them to make much use of the pro-
cess of transposition in order to get rid of negative terms in
an equation. Their name for algebra was " al gebr we'l
mukabala," which means " transposition and reduction.**
The Arabs used surd quantities freely.
313. In Europe the free use of absolute negative quantity
was restored.
Vieta (1591) was principally instrumental in bringing into
use general algebraic quantity (known quantities denoted by
letters and not figures).
Cardan (Italy, 1545) first discussed imaginary quantities,
which he termed " sophistic *' quantities.
Euler (Germany, 1707-83) and Gauss (Germany, 1777-
1855) first put the use of imaginary quantities on a scientific
basis. The symbol i for V — 1 was suggested by Gauss.
Descartes (1637) introduced the systematic use of variable
quantity as distinguished from constant quantity.
274 SCHOOL ALGEBRA
III. History of Algebraic Processes
314. Solution of Equations. Ahmes solved many simple
equations of the first degree, of which the following is an ex-
ample:
" Heap its seventh, its whole equals nineteen. Find heap."
In modern symbols this is,
X
Given --\-x = 19; find x.
The correct answer, 16f , was given by Ahmes.
Hero (Alexandria, 120 b. c.) solved what is in effect the
quadratic equatioriy
where d is unknown, and s is known.
Diophantus solved simple equations of one unknown quan-
tity, and simultaneous equations of two and three unknown
quantities. He solved quadratic equations much as is done
at present, completing the square by the method given in
Art. 226. However, in order to avoid the use of negative
quantity as far as possible, he made three classes of quadratic
equations, thus,
ax^ -\-hx = c,
ax^ -\- c = hx,
ax^ = hx -\- c.
In solving quadratic equations, he rejected negative and
irrational answers.
He also solved equations of the form ax"^ = 6.r".
He was the first to investigate indeterminate equations, and
solved many such equations of the first degree with two or
three unknown quantities, and some of the second degree.
HISTORY OF ELEMENTARY ALGEBRA 275
The Hindoos first invented a general method of solving a
quadratic equation (now known as the Hindoo method, see
Art. 233). They also solved particular cases of higher de-
grees, and gave a general method of solving indeterminate
equations of the first degree.
The Arabs took a step backward, for, in order to avoid the
use of negative terms, they made six cases of quadratic equar
tions; viz.:
ao^ = bXf ax^ -\-bx = c,
ax^ = c, ax^ + c = bx,
bx = c, ax^ = bx -\- c.
Accordingly, they had no general method of solving a quad-
ratic equation.
The Arabs, however, solved equations of the form ax'^^-i-bx^
= c, and obtained a geometrical solution of cubic equations
of the form x^ + pa: + g = 0.
In Italy, Tartaglia (1500-1559) discovered the general so-
lution of the cubic equation, now known as Cardan's solution.
Ferrari, a pupil of Cardan, discovered the solution of eqvxir
tions of the fourth degree.
Vieta discovered many of the elementary properties of an
equation of any degree; as, for instance, that the number of
the roots of an equation equals the degree of the equation.
315. Other Processes. Methods for the addition, sub-
traction, and multiplication of polynomial expressions were
given by Diophantus.
Transposition was first used by Diophantus, though, as a
process, it was first brought into prominence by the Arabs.
The word algebra is an Arabic word and means " transposi-
tion " {al meaning " the," and gebr meaning " transposition").
The Greeks and Romans had a very limited knowledge of
276 SCHOOL ALGEBRA
fractions. The Hindoos seem to have been the first to reduce
fractions to a common denominator.
The square and cube root of polynomial expressions were
extracted by the Hindoos.
The methods for using radicals, including the extraction
of the square root of binomial surds and the rationalizing of
the denominators of fractions, were also invented by the
Hindoos.
The methods of using fractional and negative exponents
were determined by Wallis (1659) and Sir Isaac Newton.
The three progressions were first used by Pythagoras (569
B. C.-500 B. c.)
Permutations and combinations were investigated by Pascal
and Fermat (France, 1654).
The binomial theorem was discovered by Newton (1655),
and, as one of the most notable of his many discoveries, is
said to have been engraved on his monument in Westminster
Abbey.
Graphs of the kind treated in this book were first invented
by Descartes (France, 1637).
The fundamental laws of algebra (the Associative, Com-
mutative, and Distributive Laws; see Arts. 316-317) were
first clearly formulated by Peacock and Gregory (England,
1830-45), though, of course, the existence of these laws had
been implicitly assumed from the beginnings of the science.
Students who desire to investigate the history of algebra in more
detail should read the second part of Fine's Number System of
Algebra, Ball's Short History of Mathematics, and Cajori's History
of Elementary Mathematics.
MATERIAL FOR EXAMPLES
MATERIAL FOR EXAMPLES
Formulas
Formulas used in the following subjects may be made the '
basis for numerous examples.
I. Arithmetic
p == br
i = prt
a = p-\-prt
^ _ 4(1 + r)" - 1]
r
II. Geometry
K = ibh
K = iaVS
K = ^h(b + 60
C = 2itR
K = 7rR^
K = 7rRL
S = 47rR^
T = TrRiR + L)
T = 2itR{R +H)
V = ■irmi
V = i-TrR'H
V = |,r7J3
^~ 180
K =
V =
= Vs(s -a){s- b) (s - c)
■■ \H{B + b + VBb)
III. Physics
v = gt
r
s = vt + ^gf
278
MATERIAL FOR EXAMPLES 279
^ 2a 9-\-s
^ 4PP ' l = i + i
^ bh^m f P V'
Jl H = MC^Rt
= ir\J
g E = ^^—
^^R C = f (F - 32)
IV. Engineering
Tj p = ^fcm (horse-power in an engine)
33,000^
5 = — and /' = / + If (sag in a suspended wire)
E = iri_ (elevation of outer rail on a curve)
jy = BXD J, (^gight a beam will support)
Ju
^ ^ .0045(P-0 (T-t)C ^i^^^^Yi of hot-water pipe to heat
a house)
T = ^ (tractive force of a locomotive)
G = "^^^^^y^^ (no. gal. water delivered by a pipe)
n = V ^ (diameter of a pump to raise a given
amount of water)
7) ^ (/ K — (diameter of balloon to raise a given
^ ^ .5236(^-(?) ^
weight)
280
SCHOOL ALGEBRA
Important Numerical Facts
Areas
Sq.Mi.
Rhode Island 1250
New Jersey 7815
New York . 49,170
Texas 265,780
United States . 3,025,600
North America 6,446,000
Land surface of earth 51,238,800
Great Britain and Ireland 121,371
France 207,054
Europe 3,555,000
Astronomical Facts
Planet
Diameter
Distance from Sun
Time of Revolu-
Synodic
Period
in Days
in Miles
in Million Miles
tion about Sun
Mercury
. 3030
36
88 da.
116
Venus
7700
67.2
225 da.
584
Earth
7918
92.8
365 da.
Mars
4230
141.5
687 da.
780
Jupiter
86,500
483.3
11.86 yr.
399
Saturn
73,000
886
29.5 yr.
378
Uranus
31,900
1781
84 yr.
369
Neptune
34,800
2791
165 yr.
367
Sun's diameter 866,400 mi.
Moon's diameter 2162 mi.
Moon's distance 238,850 mi.
Distance of nearest fixed star, 21 millions of millions of
miles (or 3.6 Hght years).
I
MATERIAL FOR EXAMPLES 281
Dates (a. d. unless otherwise stated)
Rome founded . . 753 b. c.
Battle of Marathon 490 b. c.
Fall of Jerusalem . . 70
Fall of Rome .... 476
Battle of Hastings . . 1066
Printing with movable
type 1438
Fall of Constantinople 1453
Discovery of America 1492
Jamestown founded . . 1607
Declaration of Indepen-
dence 1776
Washington inaugurated 1789
Battle of Waterloo . . 1815
Telegraph invented . . 1844
First transatlantic cable
message 1858
Telephone invented . . 1876
Battle of Manila Bay . 1898
Distances
From New York to Miles
Boston 234
Buffalo 440
Chicago 912
Denver . 1930
San Francisco .... 3250
From New York to Miles
Philadelphia 90
Washington 228
New Orleans 1372
Havana 1410
London 3375
San Francisco to Manila 4850
New York to San Francisco via Panama 5240
London to Bombay via Suez 6332
Heights of Mountains
Feet
Feet
Mt. Washington .
. ^6290
Mt. Mitchell . .
. . 6711
Pike's Peak . . .
. 14,147
Mt. Whitney . .
. . 14,501
Mt. McKinley „ .
. 20,464
Mt. Blanc . . .
. . 15,744
Mt. Everest . . .
. 29,002
Acongua ....
. . 23,802
282
SCHOOL ALGEBRA
Heights (or lengths) of Structures
Bunker Hill Monument ,
Washington Monument
Singer Building (N. Y.) .
Metropolitan Life
Building 700
Eiffel Tower 984
221
Olympic ....
. 882 ft.
555
Deepest shaft . .
. 5000 ft.
612
Deepest boring .
. 6573 ft.
Simplon Tunnel .
. 12imi.
700
Panama Canal .
. 49 mi.
984
Suez Canal . . .
. 100 mi.
Hudson
Ohio 950 Rhine
Colorado 1360 Amazon
Missouri 3100 Nile .
Lengths of Rivers
Miles Miles
280 Mississippi 3160
850
3300
3400
Rainfall (mean annual)
Inches Inches
Phoenix (Ariz.) ... 7.9 New York 44.8
Denver 14 New Orieans .... 57.4
Chicago 34 Cherrapongee (Asia) . 610
Records (year 1910)
100-yard dash 9f sec.
Quarter-mile run 47 sec.
Mile run 4 m. 15f sec.
Mile walk 6 m. 291 sec.
Running high jump 6 t. 5f in.
Running broad jump .2 ft. 7J in.
Pole vault 12 ft. 10| in.
MATERIAL FOR EXAMPLES 283
100-yard swim 55f sec.
1-mile swim 23 m. I65 sec.
100-yard skate 9| sec.
1-mile skate 2 m. 36 sec.
1 mile on bicycle 1 m. 5 sec.
1 mile in automobile 27^ sec.
1 mile by running horse 1 m. 35f sec.
1 mile by trotting horse in race . 2 :03^ m.
Throw of baseball 426 ft. 6 in.
Drop kick of football 189 ft. 11 in.
Transatlantic voyage (from N. Y.) 4 da. 14 h. 38 m.
Typewriting from printed copy. . 123 words in one minute
Typewriting from ney material . 6,136 words in one hour
Shorthand 187 words in one minute
Cost 1 lb. radium $2,500,000
Corn crop per acre 255f bu.
Milk from 1 cow (1 year) .... 27,432 lb.
Butter from COW/ (1 year) .... 1164.61b.
Resources (crops, etc., year 1910)
(All these figures are approximate estimates.)
Coal lands in U. S 400,000 sq. mi.
Coal in U. S 2,500,000,000,000 tons
Iron ore in U. S 15,000,000,000 tons
Water-power of Niagara 7,000,000 H. P.
Natural water-power in U. S. . . 75,000,000 H. P.
Possible water-power in U. S. (de-
veloped by stor: ?e dams, etc.) . 200,000,000 H. P.
Reclaimable swai' p lands in U. S. . 80,000,000 acres
Lands in U. S. rt Claimable by irri-
gation 100,000,000 acres
284 SCHOOL ALGEBRA
National forest reserves of U. S. . 168,000,000 acres
Corn crop of U. S 3,000,000,000 bu.
Wheat crop of U. S 700,000,000 bu.
Cotton crop of U. S 13,000,000 bales
Temperatures (Fahrenheit)
Normal temperature of the human body 98.7°
Ether boils at 96° Temperature of arc light 5400°
Alcohol boils at 173° (approx.)
Water boils at 212° Average change of temperature
Sulphur fuses at 238° below earth's surface 1° per
Tin fuses at 442° 62 ft. (increase)
Lead fuses at 617° above earth's surface 1° per
Iron fuses at 2800° (approx.) 183 ft. (decrease)
Velocities
Wind 18 mi. per hr. (av.)
Sensation along a nerve .... 120 ft per sec. (av.)
Sound in the air 1090 ft. per sec. (av.)
Rifle bullet 2500 ft. per sec. (av.)
Message in submarine cable . . 2480 mi. per sec.
Light 186,000 mi. per sec. (approx.)
Weights
Boy 12 years old 75 lb. (av.)
Man 30 years old 150 lb. (av.)
Horse • . . . . 1000 lb. (av.)
Elephant 2^ tons (av.)
Whale 60 tons (approx.)
1 cu. ft. of air 1| oz. (approx.)
1 cu. ft. of water 62.5 lb.
MATERIAL FOR EXAMPLES 285
Specific Gravities
Air ^0 0^ Stone (average) ... 2.5
Cork 24 Aluminum 2.6
Maple wood 75 Glass 2.6-3.3
Alcohol 79 Iron (cast) 7.4
Ice 92 Iron (wrought) .... 7.8
Sea water 1.03 Lead 11.3
Water ....... 1 Gold 19.3
Clay ........ 1.2 Platinum 21.5
I
Miscellaneous
Heart beats per minute — Frog 10
Man 72 '^
Bird 120
Smallest length visible to unaided eye . . . 25T inch
Smallest length visible by aid of microscope 125,000 inch
Accuracy of work in a machine shop . . . to oT inch
Accuracy in most refined measurements . . 10.000.000 inch
Dimensions of double tennis court .... 78 X 36
Dimensions of single tennis court .... 78 X 27
Dimensions of football field 160 X 300
Standard width of railroad track 4 8j
Weights and Measures
Avoirdupois weight, 1 ton = 2000 lb. ; 1 lb. = 16 oz. = 7000 gr.
Troy weight, 1 lb. = 12 oz. = 5760 gr.; 1 oz. = 20 pwt. =
480 gr.
Long measure, 1 mi. = 1760 yd. = 5280 ft. = 63,360 in.
Square measure, 1 A. = 160 sq. rd. = 43,560 sq. ft.; 1 sq.yd.
= 9 sq. ft, = 9 X 144 sq. in.
286 SCHOOL ALGEBRA
Cubic measure y 1 cu. yd. = 27 cu. ft. = 27 X 1728 cu. in.
Dry measure, 1 bu. = 4 pk. = 32 qt. = 64 pt.
Liquid measure, 1 gal. = 4 qt. = 8 pt.; 1 pt. = 16 liquid oz.
Paper measure, 1 ream = 20 quires = 480 sheets.
Metric system, 1 meter = 39.37 in.; 1 kilometer = .621 mi.
1 liter = 1.057 liquid qt.; 1 kilogram =
2.2046 lb.
1 hectare = 2.471 A.
1 kilometer = 10 hectometers = 100 decame-
ters = 1000 meters = 10,000 decimeters =
100,000 centimeters = 1,000,000 millimeters.
At the option of the teacher, the pupil may insert on the
blank pages at the end of the book other important formulas
or numerical facts, particularly those which are important
in the locality in which the pupil lives.
ANSWEES
EXERCISE 1
1. 56, 28. 4. 24, 12 quarts.
2. Daughter, $8000; son, $4000. 6. $12.40, $6.20.
3. Man, $96.60; boy, $32.20. 9. 11,250,000 bales.
10. Tenant, $4000; owner, $2000.
11. New York, 49,200; Mass., 8,200 sq. mi.
12. 200 and 40. 15. .0036 and .0009.
13. 4.84 and 2.42. 16. $90, $30.
14. i and j\. 17. 4f and f .
19. Lowest part, 12 ft.; middle, 24 ft.; top, 96 ft.
20. $1000, $2000, $3000.
21. Hat, $7; coat, $14; suit, $21.
22. Niece, $12,000; daughter, $24,000; wife, $48,000.
23. Cement, 3,375; sand, 6,750; gravel, 16,875 cu. ft.
25. Nitrate of soda, 500; ground bone, 500; potash, 1000 lb.
26. Lime, 190i^; potash, 952^t; sand, 2857/1 lb.
27. Boy, $9.90; adult, $19.80. 30. .0062, .0124, .0186; ^^, tV, /i-
28. 20, 40, and 60. 31. 30, 30, 60, 120.
29. 20, 40, and 60. 32. $94.74-, $284.21+, $1421.05+.
33. 35^ lb.
EXERCISE 2
19. (1) 6; (2) 3; (3) h; (4) 4; (5) 8; (6) 1; (7) 9; (8) 11; (9) 10; (10) 21.
35. Walter, 25 marbles; brother, 35 marbles.
37. 9 hr. 14 min. 42. 7 and 8.
38. 15f, 12|. 43. 10, 11, and 12.
39. 35. 44. 121, 391 sq. mi.
40. 7,258. " 45. 6290 ft.
u
SCHOOL ALGEBRA
EXERCISE 3
16. (1) 40; (2) 72; (3) 324; (4) 9; (5) 18; (6) 26; (7) |; (8) 9.
17. (9) 80; (10) 125; (11) 0; (12) 4.
23. 10; 32. 28. 2K
26. 2, 4, 8, 32, 128. 29. 1200 sq. ft.
27. 81, 125. 30. $24; $14.
31. State, $360; county, $720; township, $720.
32. Pedestal, 155 ft.; statue, 151 ft.
33. Charcoal, 500 lb.; sulphur, 500 lb.; niter, 1000 lb.
34. 800,000,000 bu.
EXERCISE 4
1. 17.
16.
16.
31.
I
46.
2+\^.
2. 4.
17.
12.
32.
33.
34.
35.
36.
37.
38.
39.
I
1.
f.
I
7.
!f.
1.
24.
47.
48.
49.
50.
51.
55.
0.
3. 0.
4. 2S.
5. 7.
6. 18.
7. 2.
8. 29.
18.
19.
20.
21.
22.
^.
24.
"57
9.
37.
108.
21.
63.
When a
When a
When a
When a
X
: = 3.
; = 2.
J = 2. 3.
J = 3.
2x + l
9. 6.
1
3
10. 9.
25.
2.
40.
15.
2
5
11. 3.
26.
15.
41.
i.
3
7
12. 0.
27.
26.
42.
¥.
5
11
13. 17.
28.
i
43.
i.
i
2
14. 18.
29.
3.
44.
w.
• i
li
15. 12.
30.
3.
45.
4.
1.5
4
EXERCISE 5
1. 66; 60.32.
8.
402 ft.
2. 180; 183.976.
9.
35°; 37f .
3. 374.5;
.3456; 105.
10.
1482f.
4. $37.50;
$2052.05.
11.
78.54.
5. 314.16.
16
Daughter,
$14,400; son, $9,600.
6. 10.
17.
$31.20.
7. 257.28;
100.5.
18.
Tenant, $1890,
owner.
$2520.
ANSWERS ifl
19. Ownar, $4125; oUmt. 12475.
20. 21, 105; 30, 90.
21. .0041, .0231; 4)08, .02.
22. Towoahip, $10,800; oouoty, $5,400; itoto, $l,80a
23. GniYel, 2000 lb.; «&nd, 1000 lb.; cement, 500 lb.
IXSRCI8I6
2.ar.
7. r; -U*.
3.7*.
8. $9000.
4. 5*; -la^; 3(f.
9. -2f .
5. -4^; 7*; -16!**.
10. $50; -$25;
-$50.
15. 15 gAme0.
15. Defeated candidate, 6,105; wmniag candidate, 6,315.
W. Walter, 30; brother, 53.
21. f, V.
19. 81, 19.
22. 234 mi.
20. IXn, 3.33.
23. 555 ft.
EXERCI8I7
1. 3. 6. - 4.
11. 13.
16. -5.
2. -2. 7. -2.
12. -9.
17. -2.
3. 0. 8. 6.
13. 5.
18. 62*.
4.-2. 9.-1.
14. -5.
19. +5.
5. -6. 10. -1.
15. -2.5.
22. Man, 240; boy, 80.
24. Youafer, $8.20;
older, $16.40.
23. $1880; $1340.
25. $3.84 and $8.84.
26. Ztne, 400; tin, 800; copper, 3400 lb.
27. 3100 mi. J». 22661 lb.
29. State, $4000; county, $8000; townihip, $12000.
30. $3000, $500a $6000. 31. 612 ft.
EXERCISE 9
1. -5.
5. 5z.
12. -6a5«.
18. 2:r.
2. -6.
6. -5o
14. 4(aH-fe).
20. 3o» - 2a!».
3. 2z.
' 7. 72*,
16. Va+x.
21. a-^b.
4. -4j.
11. 4eu:.
17. f^.
22. 22^-llv».
IV SCHOOL ALGEBRA
^ 23. 2hy^. 35. 2.
24. 7x2 ^ 22/2. 36. x" + a:2 - 7a; + 2.
25. w? -mn- 2n«. 41. $3000; $6000; $9000.
29. 2x21/ + 4xt/2. 43. 11, 12, 13.
30. - 2x - 4y + 2z. 44. 25, 26, 27, 28.
31. -xy + 2ax + y^- 3x«. 46. 716,555 sq. mi.
EXERCISE 10
1. 4a6. 4. 8x. 7. 2(a + 6). 11. x'- 5x.
2. -4x. 5. x2. 9. V^Tx. 13. 3x3-7.
14. 6x2 -}- 7a; - 8. 22. 1 - 2x - 2x3 + x^ + x«.
15. a + 46 - 4c + d. 23. m - 3d - x + 3c.
16. - 8 + X + 7x2. 24. - 3x4 -f 3x3 + 43^2 _ e^..
19. -l-2x + 2x2 + x3 + 3x4. 26. -2x"» + 4x» - x2 + 2.
20. 12x?/2 - x22/2 _ 9a:2y. . 27. Gy.
28. 3x; -X + y; - 3a2 + 2ab - ¥.
31. 4x3 - 2x - 2. 35. 3 in.
32. 2x3 4. 6x2 _ 2x - 4. 36. $10.10, $14.70.
33. - 2x3 - 2x2 _ 8x - 2. 37. $6.20, $18.60.
34. 4x3 - 4x2 + 8x + 4. 38. $1300, $1600, and $2100.
39. $1200, $1500, and $2300.
40. $3529tV, $17641?, and $705H
41. $857f, $17141, and $3428f.
EXERCISE 11
1. 5a - 6.
9. 4.
17. 2c - 6 - cf.
2. X + 1.
10. 4x - 1.
18. 3x - 2x3.
3. 1 - X.
11. 0.
19. -7x3 + x2-
-2x-l,
4.-1.
12. a - 1.
20. 2.
5. 2x + 1.
13. 0.
21. -2y.
6. -x + Zy.
14. 2 - 2x.
22. -3x.
7. 1 - 2x.
15. X + 1. ,/
25. i
27. 3.
8. 9x - 1.
16. 6.
26. 5.
28. H.
ANSWERS V
EXERCISE 12
1. 3^- (3x« - 3x + 1). 5. x^ - (- 4a; + rc« + 4).
2. a - (b - c - d). 6. a^b' - {2cd + c^ + d^).
3. 1 - ( - 2a + a2 + 1). 7. 4x* - (9x2 _ i2xy + 4y^).
4. 1 - (a2 + 2a6 + ¥). 8. x* - 4x3 4. 43^2 _ ( _ 4^ ^ 4 _|. ^.s)
9. (m + 2)x - (n + 4)2/ + (3 + n)^.
10. (1 - a - 6)x - (1 - 6 + a)2/ - (2 + a - c)z.
11. ( - 7 - 2a + 26)x + (12 - c - Qd)y - (10 - 36)z.
12. (5 - Scd)y - (3ac+ 4a6 - 2c + 5a)x - (5cd + 4)«.
13. (3 - a - c)x3 - (2 - a + c)x^ + (1 - 2a - c)x - 5.
14. ( - a + 1 - 26)x3 - (1 - 6 + a)x2 - (1 + a + 36)x + 3a.
15. (- ¥ - a3)x3 + (a2 - 26^ - c)x2 - (a - 3b - 3c)x - a - c.
EXERCISE 13
1. 3.
6. 2.74
11. -.3. 17. 3i SI
2. 5.
7. 1.
12. - I. 18. 4, 9.
3. 3.
8. 1.
14. f 19. 3^ 81
4. 3.
9. h
15. 12. 20. 8.9, 7.5.
5. 4.
10. fi
16. 6. 21. 8.9, 7.5.
22. 8.9, 7.5.
24. 13, 15, 17; 5, 7, 9, 11, 13.
25. 18,
20,
22;
8, 10, 12, 14, 16.
EXERCISE 14
1. 8.
6. 3V3 - 3V2 - 1.
2. 12.
7. 5x3 + x2 - 5x + 2.
3. 297.28.
8. - 2x» + 3x^2/ + 7x2/« - 22/».
4. 36; 144.
9. 12x - 3.
5. 2x*-3x' + 2x2-7x-
-2.
10. 12x.
11. (1 + a - 2c)x5 - (3 -h c)x4 - (1 + a - 3c)x» - (2a + 5)x2 + 2.
12. 1 + 2a - (1 + 2a + 36 - c)x - (1 - 2a - 36)x2 - (1+ 2a - 36)x».
13. 6. 14. 3. 15. - i 16. i-
17. - 8^2 4. 2oj:2 4- -Sac + 2a« - a*.
VI
SCHOOL ALGEBRA
23. Ihx^ - llx.
24. fa:2 - Ix - I
25. .95a2 + .45a + .3.
26. 1.23a2-h2.12a + .6.
27. Six + y) + {y + z).
18. 79.
19. x^-^Sax + 5a\
20. 6x2 - 4ax - Qa\
21. -4.
22. 6,405,000 sq. mi.
28. 3a3 - lOab + 3a^¥.
29. First, x^ - x^ + x - 1; second, - x"^ -{- 4:X - 1; third, - x^ -h x
+ 14; fourth, -2:2 + x-l.
30. First, - a^ + 2x^ - 3x + 1; second, a:^ - 3x - 9; tMrd, 2x2-32;
-6; fourth, 2x^ -2x -a.
31. 2x3 _ 2a;2z/ + 6x2/2 + 52/». 33. - 2x3 + 2x2/2 + 3i/'.
32. 4x3 _ 2x^y + 2x?/2 + 72/3. 34^ 6^ _ 2^22/ - 3y>.
36. 17,480; 15,064 votes.
37. Lowlands, 44 mi.; Culebra Cut, 5 mi.
EXERCISE 16
1.
-20.
10.
20a26cc?2.
21. 2».
40.
0.
2.
6a.
11.
- lS(^d\
23. x^+\
41.
0.
3.
- 15a6.
12.
16x32/3^4.
24. o2x»+«.
42.
16.
4.
- 30x22/2.
13.
.8x5.
25. -a3x«+».
43.
4.
5.
-8x2.
14.
ia3x«.
27. -a3x2»-«.
44.
1.
6.
15x2.
15.
.015x2.
29. 10(a + 6)s.
45.
3.
7.
- 12a2x2.
17.
Y^.
32. 21(a + 6)»'-H.
46.
iS.
8.
42x22/4.
18.
fx».
33. 2"-i.
47.
2.
9.
- 21a2xy.
19.
2»»+i.
35. 27. 39. 0.
48.
1.
EXERCISE 16
1.
6a2x + 9ax2.
8. 2x"+3 - 3x"+-2.
2.
- 15x22/ + 10x2/2.
9. - 12x"+2 - 28x«+i.
3.
8x32/2 - 2x22/3.
10. 3x"+i + 3x".
4.
- 21a2fex2^ + 12a62a:2/2.
11. 3x9«+5x7«.
5.
40a2c2n - 15am^nl
12. -4a7'^ + 14a5^
6.
— 7m3n + 7m
*n + 21m5n.
13. x3 - 1.48x2 + ,
.204
X.
7.
24x32/2 - 15x22/3 _
3X2/*.
14. fx2 - 1x3 - |x
.
ANSWERS Vii
15. - i^a»x3 + i^aV + ia'x.
16. ^x* + lx' + iafi- ix\
18. 10(a + by - 6(a + b)^ - 10(a + b).
19. 21 (x - yy + 6(x - yy - 18(x - yy.
22. 56. 24. 5.567. 26. 56. 28. 80.
23. 18. 25. 40. 27. 30. 29. $238.25.
30. 60. 31. 169.9. 32. 69.75.
33. Daughter, $10,500; son, $5500.
34. Iron, 460 lb.; aluminum, 158 lb.
35. 19. 37. - 9. 38. 3.
EXERCISE 17
1. 2x2 - 7x - 4. 7. 32a5c - 2a¥(^.
2. 3x2 _ 7a. _ 6. 8 33^^ + x^i/^ _ 143.^^
3. 2x2 - 9x - 35. 9. a' + 63.
4. 12x2 - 25x2/ + 12?/2. 10. x* -y.
5. 28X4 + X22/2 - 15?/4. 11. 8x4 - 2x3 + x2 - I
6. 30x2i/2 -\-xy- 42. 12. 6x3 _ 19^,22, + 21x1/2 _ iq^^^
13. 2x6 - 5x4 - 2x3 + 9^2 _ 7a; + 3.
14. 3x*y - 10x3i/2 + 4x22/3 _|_ 63.^4 _^ ^^
15. x6 - 5x4?/ + iQ^y2 _ iOa:22^ 4. 5a:y4 _ yS^
16. 4x« + 9x4 - 16^:3 _j_ 22x2 - 21x + 6.
17. x« - x* - 7x4 + 3x3 + 173-2 -5x- 20.
18. x« - 6x4?/ + 9x22/2 - 2/«.
19. a* + a'b^ + 64.
20. 16x4 + 36x22/2 + 817/4,
21. X^ - 9x^2/2 + 7x42/3 _|_ 13a^^ _ 19^.2^5 4. 8^.^ _ y7,
22. - x^ + 2ax4 + 8o2x3 _ 160^3^.2 _ 16^4^. ^ 3206.
23. a3 + 63 + x3 + 3a62 + 3a26. 26. Ax3 - 17fx + 2.
24. 0252 + c2d2 _ a2c2 - 62(^2. 27. .Ia2 - .23a6 + .126*.
25. ia2 - 162. 28. 4.5x3 - 7.1x2 - .4x + .24.
29. x"+i - x«-i - 6x"-2 - 2x + 4.
30. X2"+1 - X2" - 2x2"- 1 + 3x2«-2 - 10x2"-3.
31. x"-4 + x»-3 - x"-2 + x"-i - x" + 7x«+i + lOx'H-*.
Vlll SCHOOL ALGEBRA
34. 2x< + 5x3 - 8x2 _|. iia; _ 20.
35. 12x* - x3 - 27x2 - 3x + 10.
36. 6x« + 5x*y - m:f?y^ + Uxh/ - 6xy* + y^.
38. 35. 40. 50. 42. 60. 44. 60.
39. 35, 145. 41. 45, 50. 43. 24, 46. 45. $200.
46. 315, 85. 47. $780, $220.
EXERCISE 18
1. - a«. 5. - X - 2. 8. 10x2 -h 7x - 12.
3. (x + y)2 (x - yy. 6. x* - x - 2. 9. - 5a + 24.
4. 16, - 1. 7. 17x - 12. 10. 24a + 20.
11. x2. 21. 2x3 + 4x2 _ 2x.
12. 4x<- 12x3 ^ 13^2 _ 6x + 1. 22. x2 + lOx - 16.
13. 40a - 24a6. 23. 0.
14. - 20x. 24. x2 - 5x + 8.
15. - 6x2 + I3x - 4. 25. 3x2 _ lOxy + j/».
16. - 2x2 - 3x + 6. 26. x2 - z^.
17. x« - 7x -f 6. 27. 4a2 -ax + bx + my + cy.
18. 2a6 + 862. 28. 0.
19. 2/2 - 4x2 + 2yz + z*. 29. 5a3 + 2a2x - llax2 + 10x».
20. 2x + 1. 30. &xy.
32. -12. 35.- 12. 38. -1. 41. 5.
33. 0 36.- 18. 39. -26. 42. 29.
34. 12. 37. 16. 40. 1. 43. 8a«.
44. 76p2. 52. U, i.
45. - 2a262 + I4a6 - 5. 53. $45, $55.
46. 27, 9. 54. $10, $45, $45.
47. - 6a + 296. 55. $13^, $28f , $28f , $28^.
48. 50. 56. 80.
49. $100. 57. 22, 11, 17.
50. $920, $80. 58. 13, 14, 15, 16, 17.
51. .0012, .0003. 59. 15, 9.
60. Daughter, $940; each son, $1780.
61. 25, 11. 62. 21, 22. 63. $26, $37, $35.
ANSWERS ix
EXERCISE 19
1. - 3. 4. 5xy. 12. - 16a. 15. .5a.
2. - 3^2. 5. - X. 13. - 2iy. 17. ^x.
3. -2a. 6. -72/'. 14. 40m. 19. 2r2,47r,4r.
21. ^ ; m; 2mt;. 22. - 5(x + i/)'; - 10(:c + y).
4 23. .2(a - 6)2; .7(a - 6).
24. a*"; a^"; - a^^.
25. - 3a2; - 6a; - 6a'; 2a4; - 6a-»+».
26. a«-i-<; a'; a"+». 29. 2«-2; 2"-^
33. 0. 34. 0. 37. 0.
EXERCISE 20
1. -x^ + Sx. 3. - 26 + 3ac. 6. 1 + m - m« + m».
2. 5a; - 2?/. 5. 3x2 _ 2xy - y^. 8. 3x^ + 2x - 5.
9. - 2a;2 + Ax - 30. 14. - 2x + 5a:2 - 3x2-».
10. .04a2 - .08a6 - 1.662. 15. x^ - 2:i^ -h Sx^ -{■ x.
11. -fa;2+a; + ¥. 16. - 2x^ ^- ^2? + x^ + Zx.
12. - fa'6 + ia262 + f a6'. 17. 3x"-i - 2x" + 4x"+i _ a;n+2,
13. - 3a:2» + 2a;" - 4. 18. - 5(a + 6) + 4.
19. (x - 2/)2 - .3(x - y).
20. X - !/. 24. 60. 27. 300. 30. $600.
22. x + l. 25. 120. 28. $300. 31. 125 nickels.
23. 90. 26. 84. 29. 600. 32. $37.50.
33. 14 quarters, 7 bills. 34. 15 each. 35. 17 each.
36. 2. 37. 9. 38. 2. 39. - 2^.
EXERCISE 21
1. 3x + 1. 4. 3x + 7. 7. - 5a; + 8.
2. 2a; + 1. 5. 3a; - by. 8. 4a; + y.
3. 4a; - by. 6. 3a + 4c. 9. a + 26.
10. a;2 + a;?/ 4- 2/*. 14. 4a2x2 - 2axy'^ + j/*.
11. 9a;2 - 6a; + 4. 15. a;2 - 3a; + 1.
. 12. 3x - 7. 16. 7a;2 + 8x + 1.
13. 25 + 20a; + 16a;2. 17. 3a2 - 4aa; + xK
SCHOOL ALGEBRA
18. 2y^ -4y^ + y- 1. 21. 2:i^ - x + 1.
19. c* + cH^ + X*. 22. 3x3 _^ 4^2^, _^ 5^y2 _|_ 2^3^
20. 2a;» - Sx^ + 4x - 5. 23. 2x< - 3x22/ - 2y\
24. x3 + 2x^y + 4x2/2 + 8^/3.
25. x^ - 2x3?/ + 4x22/2 - 8x7/3 _f. ig?/*.
26. x^ — x^y + ^y^ — ^^y^ + ^y^ — y^'
27. 64x6 _j_ iQ^yi ^ 4^.2^ 4. 2^,
28. 2x2 -5x-l. 39. 2x3 - 3x2 + x - 5,
29. 3x2 - X - 5. 40. la - ift.
30. 2x3 - 4x2 - X + 3. 41. la; + i^.
31. 2x3 _|_ sx^y - 4x1/2 _|_ j^. 42. ia2 + ^ab + ^62.
32. 3a3 - 4a26 + 3a52 - 263. 43. ^x2 + ^x - I
33. x^ + y^— z^— xz + xy— yz. 44. .4x — .5y.
34. c2 + d2 + n^-cd-cn-dn. 45. 2.4x - 3.
35. y*-\-2y^ + Sy^ + 2y + 1. 46. 2x" - Sx'^K
36. 2x*- 4x3 + 3x2- 2x + 1. 47. 4x3" + 3x2" - x«.
37. 2x2/ - 2x2 - 3yz. 48. 4x"+i - 3x" + x"-».
38. x2 - 3x + 1. 49. 3x"-i + 2x"-2 - 3x»-^.
52. X* + x32/ + x22/2 + x^ + 2/* + ^^
X*
x-y
53. 1 + X + x2 + x3 + ^— -. 56. X + 5, X + a, X + 2/.
54. 1 + ax + a2x2 + 58. 6 hr.
55. 15, 3a, 3x, 3(x + 2). 60. 4^ hr., 36 mi.
61. 108 mi., 126 mi.
63. 5f hr. after second boy starts.
64. 2 hr. 56 min. after second train starts.
66. 6 hr.
68. 4 mi., 8 mi.
67. 8 hr.
69. 5 mi., 10 mi.
72. A, 24
mi.;
B, 21 mi.
EXERCISE 22
1. 3.2x2 - 2.42x2/ - -242/2.
8. 6.
2. - 7.15a2 - 1.5a6 - I.862.
9. -18.
3. - 3.8p2 - .5p + 3.85.
10. 0.
4. 2.6x2 - .5x + 2.
11. 8x^-18x3- 13x2 + 9x
5. 45.
12. 1 + 2x - 3x2 - ^^
7. -22X + 54.
13. 10, 11, 12.
ANSWERS XI
14. Cement, 400 lb.; sand, 800 lb.; gravel, 1600 lb.
15. 9f sec.
25. 5a3 (x
-yY-
17. a:« + X - 1.
27. -5.
18. x2 - 3.
29. (1) 512; (2) ,
512; (3) 64.
19. 38.
30. 6x3 _
22x2 _|. loa; + 10.
20. a + h] 3a-56 + 4c + 7d
32. ix<-i^jax3+|a2x2 + fa*.
21. -7-Sx + 2y-.
3, etc.
33. x2" + ;
rV +
y^\
22. -2a + 36;
-5.
34. 2-3x + 3x2-
-.3x3 + 3x*..,
35.
X2-
1.
36.
4.8x3
- 17.95x2|/ + 18.45x2/2
- 6.32/3
37.
x^-
x3 + 2x2 ■
- 3x + 5 + . ,
. . .
38. Qx-^y- i
42. -
3.
39. 1.6x2 - 2xy + 2AyK
43. 2z
— X.
40. a2 + 62 + c2-
-ah-
ac — he.
44. 0.
41. 2.
48. 54
■■; 2; ^.
EXERCISE 23
1. 3.
10.
17 in.
19. 3.
28. 3.
2. 4.
11.
-2.
20. 1.
41. 3.
3. 10.
12.
11.
21. 1.
42. 2.
4. 4.
13.
0.
22. 14.
43. 2|.
5. 2.
14.
-I
23. -3.
44. -4.
6. 3.
15.
10.
24. 100.
45. 2i
7. -5.
16.
6.
25. 4.
46. -1.
8. 4.
17.
-f.
26. 1.
47. 5.
9. 10 ft.
18.
i
27. 3.
48. 7.8 ft.
50.
8.4.
52.
2yr. 3
mo.
EXERCISE 26
1. $63, $21.
5. A, $61
;B,$39.
2. $48, $36.
6. Owner
•, $45; i
3ther, $22.50.
3. $48, $24, $12
1.
7. 23.22.
4. 36, 12.
8. 38.
9. Wife, $9000; each daughter, $3000.
10. Wife, $12,200; each daughter, $2200.
11. Alcohol, 50; water, 62.5 lb.
Xll SCHOOL ALGEBRA
12. 20, 21, 22. 15. 91 sec. 18. $26, $37, $35.
13. 150. 16. 11, 24. 19. 73, 19.
14. 21, 22. 17. John, 36; WilUam, 60. 20. 22, 11, 17.
21. Horse, $67; cow, $27.
23. Limestone, 50; coke, 350; iron ore, 400. 25. 12 lb.
24. 87 yd., 174 yd., 99 yd. 26. 234 mi.
27. Passengers, $1410; freight, $1675.
28. Niece, $12,000; daughter, $16,000; wife, $36,000.
29. 15. 33. 8 ft. 37. T X 12'.
30. 4' X 9'. 34. 5 yd. 38. 8 in. X 12 in.
31. 4 yd. X 5 yd. 35. 20' X 40'. 40. 36' X 78'.
32. 6 in. 36. 20' X 60'. 41. 160' X 300'.
42. Boy, 15 yr. ; brother, 5 yr. 43. Man, 20 yr. ; brother, 10 yr.
44. 8 lb. 51. 8 mm. 56. 6.
45. 8J lb. 52. 26' X 34'. 67. 4, 5, 6, 7, 8.
47. 13, 14, 15, 16, 17. 53. 30. 58. 8, 9.
48. 19, 21, 23. 54. |. 59. 8 yr., 12 yr.
49. 21 words. 55. 43. 60. 26|, 6i
61. 27' X 78'; 36' X 78'.
62. Mon., 52; Tues., 104; Wed., 57; Thurs., 97.
63. $4. 65. 8 hr. after second boy starts.
64. 3 hr. after second boy starts. 66. 38 da.
EXERCISE 26
1. 7*2 + 2ny + y'^. 4. 9x2 _ \2xy + 4!/».
2. c2 - 2cx + x^. 8. 1 - 14?/3 _|_ 49^^.
3. 4x2 - 4x2/ + y^- 9. 9x8 ^ 30^7 -f 25x«.
10. 36x*l/2 - 132x21/32^ + 1212/426.
11. 25x2" - 30x"y"2"' + 92/2"22m^
12. 16xV°2*'' + 72x3?/3"+522n _^ gl^^n^
13. ix8 + 1x^2/ + I2/2. 18. 2.25m2 - .06m + .0004.
15. .04x2 + .12x2/ + .092/2. 19. a^ + 2ah + fe2 + ga + 86+16.
16. .09a2 + .024a62 + .001664. 20. a2 + 2a6 + 62 - 6a - 66 + 9.
23. 9 + 6a + 66 + a2 + 2a6 + 62.
25. 4a4 - 4a26 + 8a2c -j- 62 - 46c + 4c2.
26. x2 + 2x2/ + J'^- 2ax - 2ay - 26x - Ihy + a« + 2a6 + 62.
ANSWERS XIU
28. 998,001. 31. 2,601. 34. 992,016.
29. 994,009. 32. 1,006,009. 35. 99,940,009.
30. 99,960,004. 33. 9,409. 36. 9,840.64.
EXERCISE 27
1. x^ - zK 5. X* -4:. 9. la^ -^b".
2. 2/2 - 9. 6. a^x* - ¥y\ 12. .0025a* - .096«.
3. 9X2 _ y2^ 8, 4a.2n _ 257/2"». 13. ^zx^ - .492/2.
14. a2«+2 - 162^-2 27. a2 - 62 + 66x - 9x2.
15. a? + 2a6 + 62-9. 28. x^ + x'^y'^ + y^.
16. x2 + 2xy + 2/2 - a2, 29. a* + a2 + 1.
18. 16 - a:2 - 2x - 1. 30. 4x* - 29^2 + 25.
19. 4x2 _ 9^2 + 301/ - 25. 31. 4x4 _ 2<dx'^y^ + 2/*.
20. a2 + 2a6 + 62 - 9. 32. x* - 3x2^/2 + 2/^.
22. 16 - x2 - 2x - 1. 33. a? + 2a6 + 62 - c2 + 2c - 1.
23. 4x2 - 92/2 + 302/ - 25. 34. x* + 2/* - xV - 1-
25. x4 + 6x3+9x2 -4. 35. x2 + 2x2/ + 2/2 -22 -22 - 1.
40. 8096. 43. 999,975. 46. 9996 sq. ft. 61. 996,004.
41. 9991. 44. 1200. 47. $8.96. 62. 999,996.
42. 9975. 45. 292.40. 48. $48.91. 63. 9409.
64. 9991.
EXERCISE 28
1. 4x2 + ^2 _^ 1 _|. 4^.^ + 4x + 2y.
2. x2 + 4^2 _|_ 4^2 _ ^xy + 4x2 — 82/2.
3. 9x2 _|. 4y2 -I- 25 - 12x2/ - 30x + 2O2/.
4. 4a2 + 62 + 9c2 - 4a6 + 12ac - 66c.
5. x2 + 42/2 + 922 _ ^xy - 6x2 + Vlyz.
6. 16x2 + 92/2 + 1 + 24x2/ - 8x - 62/.
7. x2 - 2x3 + 3x2 _ 2x + 1.
8. 4a* + 20a3 + 13a2 - 30a + 9.
9. x« + 2/2 + 2* + 1 - 2x2/ + 2x2 - 2x - 2yz + 2y - 2z.
10. 4x2 4. Qy2 4. 1622 + 25 + 12x2/ - 16x2 - 20x- 24yz -30y + 40z,
11. 9x« - 24x« + 22x* - 20x3 + 17x2 - 4x + 4.
12. ix* - 1x3 + ^^a;2 - ^x + 25.
13. fx« - fx« + ^x* + ^x' - 11x2 + 9x + 36.
14. .04a2 + .0962 4. .25c2 + .12a6 - .2ac - .36c.
29. .0004x* - .012x3 _}_ .ii3;2 _ ,3^ _j_ 25.
Xiv SCHOOL ALGEBRA
EXERCISE 29
1. x^ + 7x + 10. 15. a2 + ia - i
2. x» - 8x + 15. 16. a262 + ^ahx + Sx^.
3. a:» - 3x - 28. 17. a'b^ - 2ahx - 3x^.
4. x^ + 4x - 32. 18. xV - 4a;y22 _ 2I3*.
5. x2 - 6a: - 7. 19. a:^" - 25.
6. a:< - 5^2 + 6. 20. (x + ?/)2 + 8(x + y) + 15.
7. x^ + 4x2 + 3. 21. {x + ?/)2 + 2(x + y) - 15.
8. a2 - 7ax - 30x2. 22. (x + 2/)^ + 2(x + 2/) - 15.
9. x2 - 6x1/ - 7i/2. 23. (a + 26)2 _|_ 8(a + 26) + 15
10. x2 + .7x + .1. 24. (2x+3?/)2-2a(2x+3!/) -15(
11. x2 + fx + h 25. x2 - a2 - 2a6 - 6*.
12. x2 + 5.02X + .1. 26. 4x2 - a^ - 6a6 - 962.
13. a2 + .52o + .01. 27. 4x2 - a^ + 6a6 - 962.
14. x' - ix - i. 30. 381 ft.
EXERCISE 30
1. 2x2 _j_ 11a; _|_ 12. 7. 5x2 _|_ 34a. _ 7.
2. 2x2 _ iix + 12. 8. 3x2 +xy- 24y«.
3. 2x2 -5x- 12. 9. 12a< - lla26 - 56«.
4. 2x2 + 5x - 12. 10. |x2 .+ |x + i.
5. 6a2 + 19a + 15. 11. 2a2 + .la6 - .0662.
6. 6a2 - a - 15. 12. fx2 + 2ax - la\
EXERCISE 31
19. a« + 2a6 + 62 - x2 - 2xy - yK
20. a* + a2x2 + x<. 36. 8a6. 40. 3xy.
22. 16a* + 4a2 + 1. 37. 8x2- gx + 2. 41. - 169x<.
29. a2 - 62.
38. -
8x.
42.
c2 - 2c + 2a.
35. 2a2 + 862.
39. -
6a + 5.
43.
x^y^-x^y-y^+y
46. 72i.
51.
99001.
56.
38,025.
61. 23.04.
47. 3801.
52.
56.25.
57.
990,025.
62. 9604.
48. 39,800i.
53.
380.25.
58.
94.09.
67. -4^.
49. 2401.
54.
9900.25
59.
96.04.
68. 2.
50. 2450i.
55.
5625.
60.
92.16.
69. 9991 sq. rd
70. 9604 sq. rd.
71
. $35.96.
72. $80.75.
ANSWERS XV
EXERCISE 32
1. a+x. 11. .2x - ,Sy. 28. 54, 46.
2. 3 + 2x. 15. a; + 1 - a. 29. 53, 47.
4. 5x + Qy\ 16. a + fe - 2c. 30. 109, 91.
5. 4x^ - 7y\ 18. a - 6 - c + 1. 31. 10.9, 9.1.
7. a?>2 - 6c3d<. 19. l-a-h+c. 32. 89, 71.
8. \x + fy. 27. 54, 46. 34. 12.
10. .5a + .46.
EXERCISE 33
1. a2 - 2a + 4. 10. W -iab^ + iVfe*.
2. a;2 + X + 1. 14. c2 - c + ex + 1 - 2x + xK
3. 9x2 _|_ 12a; + 16. 15. 4 + 2x + 2?/ + x^ + 2x2/ +2/«.
5. 25 + 5x3 _}_ a^. 16. 9x* - 15xV 4. 252/6.
6. 9a4 - 3aV + V^- 17. a^ - 2a + 1 + ax2 - x^ + x«.
8. .04x2 + 2xy + 2/2. 18. x^ - 2x3 + 2x2 + 2x + 1.
9. ix2 + \xy + ly^. 19. 4x2 _ gxi/ + 4i/2 + 2x2 - 2yz + ^2.
29. 23. 30. 23. 31. 23. 32. 17.
33. 53. 34. 47. 36. 90.
EXERCISE 34
1. a* — a'x + a2x2 — ax^ + x*.
2. a< + a^x + a2x2 + a7? + x^.
3. 6« - l^y + W - 632/3 + 62|/< - 52/6 4. 2/J.
4. 66 + 65y _|_ 5Y + 632/3 + 52^ 4. 5^/5 + 2/6.
5. a^ - 2a3 + 4a2 - 8a + 16.
6. a^ + 2a6 + 4a4 + 8a3 + 16a2 + 32a + 64.
7. X* + x3 + x2 + X + 1.
8. x4-x3+x2 -x + 1.
9. 16x4 + 8X32/ + 4x22/2 + 2x2/3 + 2/4,
10. a}^ - aH + a8x2 - aV + aV-a^x^ + a4x«-a3x'+a2x' -ax«+x".
11. X8 + X«2/3 + a^^^ + a;2^9 + ^12.
12. 81 + 27a + 9a2 + 3a3 + a*.
23. 11. ^ 24. 13. 25. 12.
26. 7. 28. 210.
XVI SCHOOL ALGEBRA
EXERCISE 36
31. x^+ax + a^+x + a. 42. 10.
32. x^ -\- ax + a^ -{- 5x + 5a. 43. 6.
35. a;2 - 2ax + a^ + 5x + 5a. 44. 1^.
37. (a2 - 62)2. 45. _ 4.
38. (a2 - 462)2. 46. qq
39. (9x2 _ 4^/2)2. 47 180, 540, 280.
40. 3x2 + 14^ _ 21. 48. 12i mi.
41. - 11x2 4- 29x - 13. 49. 617°.
50. 120 ft. per sec. Latter is liV times as great.
EXERCISE 36
1. x2(2x + 5). 6. 3a2x2(a - 5x). 9. a^x{l - 2x).
2. x(x2 - 2). 7. 9x4(2x - Sy). 10. ix2(2x + 1).
3. x(x + 1). 8. x2(l -X- x2). 11. lah{3a - 206).
12. ix3(5a - 12x). 22. (a + b) (7x + 5y).
13. .2x2(x + 2a). 23. xy{a + 6) [7x + 5(a + h)yl
14. .02ax2(l - 20a). 24. 7(a: - ym - 2{x - y)].
15. .6m(2n - m). 25. 3(2x - a)3I3 - 4(2x - a)2].
16. 3(a2 - 2ax + 3x2). 26. 1694.
17. 2x(l + 2x - 3x2). 27. 938.25.
20. 2xV(2/'- 4x"2/ 4- 3x2"). 28. 58,190.
21. a"^63c2"(l + lie). 29. 314f.
33. }? ' 36. ^ "^ ^
a + 6 H- c 3(x2 - 2)
10 1 15 "^
31. -^, ^- 34. V" . ^ • 37.
a + 6 2 "20-6 + 30 *"' ' 2(2x - 3)
^^•a + 6 '^^•a + 26 ^^- 2(2p2g2 _ i)
EXERCISE 37
1. (2x + y)K 3. (5x - 1)2. 5. c(7 + 26c)*.
2. (4a - 32/)2. 4. (x - lOy)K 6. 0^(6 + 2)\
7. x(2/ + 1)2. 12. 2y{2a - 5x)\
10. x(2x + 112/2)2. 13. 2x2(x - 2)«.
11. a6(9a + 76)*. 16. (x" + j/)*.
ANSWERS
19. (8a ■
20. (5x ■
24. (.2a - .36)2.
25. (5a - 3x)2.
26. a - 36.
-12
-5y
-6)«.
- 12xy)K
27. a + 36.
28. 1 - 2a.
29. 2a + 36.
EXERCISE 38
XVll
17. (a - 6 - c)«.
18. {3x + Sy + 2zy.
21. (a + 6 + c)«.
22. {ix + Sy)\
23. ihx + ly)K
1. (x + 3) (x - 3). 9. x(x + 3a) (x - 3a).
2. (5 + 4a) (5 - 4a). 10. x^ix + 3a) (x - 3a).
3. (2a + 76) (2a - 76). • 11. m(l + 8a) (1 - 8a).
4. (x + 2y) (x - 2y). 12. 2(11 + x) (11 - x),
7. (1 + ^m) (1 - 8m). 15. (a^ + x^) (a + x) (a - x).
8. 3(a: + 2y) (a; - 2y). 16. (a2 + 96^) (a + 36) (a - 36).
17. {7^ + y') (a:2 + t/^) (x + y){x- y).
18. x(x2 -f 1) (x + 1) {x - 1).
20. x{a^ + 1) (a2 + 1) (« + 1) (a - 1).
21. (15x« + y) (15a:" - y). 27. (.9x + .056) (.9a; - .056).
22. (fa: + ly) (fa; - fi/). 28. (a:^ ^ y^) ^^ _ ^)^
23. (fa + 36) (fa - 36). 29. (x^" + y^'z^) (x^" - i/V).
24. (.3x + Ay) (.3x - .4i/). 30. {x + y ■\-\) {x + y - 1).
25. (.la + .26) (.la - .26). 31. {x + y + l){x-y- 1).
26. (.52/ + \h) {.by - \h). 32. (x - 2/ + 3) (x - 2/ - 3).
33. (2x - 21/ + 5) (2x - 2y - 5).
34. (1 + 6x + I2y) (1 - 6x - I2y).
35. (4x+22/ + l)(22/-2x-l).
36. (11a - 86) (9a - 26).
37. 2/(xV + 2^) {x^y^ + 2*) i^y"^ - 2*).
38. (9x« + 4^2) (3x3 + 2?/) (3x3 _ 2y).
39. x(x2 4- Vly^) (x2 - 121/23).
40. (a - 6 + 2c + 2) (a - 6 - 2c - 2).
41. (10x2 - lOx - 9) (- 10x2 + lOx + 11).
42. a - 26. 43. a + 26. 44. 3 + a. 45. 1+6.
EXERCISE 39
1. (X f 'i) (x + 2). 3. (x + 3) (x - 2).
2. (x + 2) (x - 3). 4. (x + 11) (x - 4),
XVIU SCHOOL ALGEBRA
7. (x + 8y) (x - 2y). 20. {x - 8a) {x + 3a).
8. {x - 8y) {x + 2y). 2L (x^ - S) (x + 1) {x - 1).
11. (x - 9) (x + 4). 24. a:(a; + 4) (x + 3) (x- 4) {x- 8).
12. (x2 + 4) (a: + 3) (x-3). 25. (x" - 8) (a:» + 7).
15. {x - 12) (a: + 4). 26. (a6 - 13c^) (ah + 2c2).
16. (x + 16) (x - 3). 27. (x -i-a){x + 6).
17. (x - 24) (a: + 2). 28. (a: + 2a) (x - 36).
18. (x - 12) (x + 8). 29. {x + a-{-b){x + b + c).
19. (xy - 12) (xi/ - 11). 30. {x + a-c)ix + b + c).
31. (x-y-6)ix-y + 3).
1. (2a: + 1) (a: + 1).
2. (3a: - 2) (a: - 4).
3. (2x + 1) (a: + 2).
4. (3a: + 1) (a: + 3).
5. (3a: - 5) (2a: + 1).
6. (x + 3) (2a: - 1).
7. 2x(a: + 4) (3a: - 2).
19. (2a: + 3) (a: + 1) (2x - 3) (a: - 1).
20. (3a: + 2) (a: + 4) (3a: - 2) (x - 4).
21. (4a: + 3^) (3a: - 4z).
22. 2a:(6a:-t/2)(2a: + 92/2).
23. (5a + 46) (5a - 46) (a^ + 6^).
24. (8a:2 - 9y^) {2x^ + y^).
25. iSx^ + y){x^-Sy).
26. (5a + 46) (a + 6) (5a - 46) (a - 6).
29. (a + fe + 8) (a + 6 - 3).
30. (3a: - Sy - 2z) (x - y + Sz).
31. (3a:2 + 6a: + 4) (a: + 3) (x - 1).
32. 4a:(a: + 4) (a: + 2) (a: + 1) (a: - 1).
33. 2(1+ 3a:) (2 -a:).
EXERCISE 41
1. (m - n) (m2 + mn + n^). 4. (a + 26c) (a2-2a6c + 462c2).
2. (c + 2d) (c^-2cd + 4(^2). 7. (ah + 1) (a262 - a6 + 1).
3. (3 - a:) (9 + 3x + a:^). 8. (1 - IQx) (1 + 10a: + lOOx^).
EXERCISE 40
10.
(2a: + 5) (a: -
•2).
11.
(4a: + 1) (3a: ■
-2).
12.
(4a:-
- 1) (X + 3).
13.
(5a:-
- 1) (:^ + 5).
14.
3a:(a:
- 2) (3a:
+ 1).
15.
2y{Sx
^ + 2) (a:
-1).
18.
(8a-
- 96) (4a
+ 56)
ANSWERS XIX
9. xiSx + a) (9x^ - Sax + a^).
10. {8x - 2/2) (64x2 + Sxy^ -\-y*).
11. a(l +7a)(l -7a + 49a2).
12. (a + x)(a- x) (a^ - ax + x^) (a2 -{- ax + x"^).
13. (x2 + y) (x2 - 2/) (x^ - a:2y + y') (x* + x2|/ + y^).
14. (a + 2n2) (a - 2^2) (a2 + 2an^ + 4n*) (a* - 2an2 + An').
15. 2x(5 -x2)(25 + 5x2 + x4).
16. (2x2 _|. y) (4^4 _ 2x'y + y').
17. (a + & + 1) (a=^ + 2a6 + 62 - a - 6 + 1).
18. (5 + 26 - a) (25 - 106 + 5a + 462 _ 4^5 -f a').
19. (2 - c - d) (4 + 2c + 2d + c2 + 2cd + d^).
20. (-2x - 2/) (13x2 - 5xy + 2/2).
2L 2x(2x2/2 - 32) (4x2?/4 4. Q^yi^ + 922).
22. (x + y)(x'-3^y + xY - xy' + y').
23. (x - y) (x« + x^y + x^y^ + s^y^ + xY + xy^ + t)^
24. (a2 + m2) (a* - a2m2 + m^).
25. (x4 + ^) (x8 - x^?/^ + 7/«).
26. (a - 26) (a6 + 2a% + 4a^62 + 8a363 + 16a26< + 32a65 + 646«).
27. (a + x) (a" - aH + a8x2 - aV + a^x^ - a^x^ + a^x* - a^x^ + a2x«-
- ax9 + a:^").
31. (3 - x) (81 + 27x + 9x2 + 3^:3+ ^).
32. (4 - o + 6) (16 + 4a - 46 + a2 - 2a6 + 62).
33. (2x - 41/ + 1) (4x2 _ 16^^ + iQyi _ 2a; + 4i/ + 1).
36. (2x - a2) (16x4 + 8a2x3 + 4a4x2 + 2a6x + a*).
37. {a^^-y'){a'-aY+'!f)'
38. (2X4 _|_ yi) (4a;8 _ 2x^'i^ + ^).
39. [8x - (a + hy] [64x2 + 8x(a + 6)2 + (a + 6)4].
EXERCISE 42
1. (a + 6) (x + y).
9. {y' + 1) (2/ + 1).
2. (x - a) (x + c).
10. (ax - 1) (x - 2a).
3. (52/ - 3) (x - 2).
11. (X -y){x- 3).
4. (m - 2y) (3a - 4n).
12. (2 - 1)2 {z + 1).
5. x(a + 3) (a + c).
13. (6 - 1) (a - y).
6. (3a - 5n) (a + 6).
14. (x - 1) (x2 + 2) (x2 - 2).
7. x(x2 4-2).(x + l).
15. (x ^y){a^ 6).
8. 2x(x + a)(x-'a)(x
-1).
16. (xH-4)(x + 2)«.
XX SCHOOL ALGEBRA
17. {a + 3) (a» - 3). 19. {x - 1) {2x - 1)«.
18. (x - y) {2x + 2y- 1). 20. (x - 1) {x^ + 3x + 3).
21. (x - 2/) (x + 2/ + x2 + xy + 2/').
22. {x-y){x'-{-xy + y^ + l),
23. (x - 2/) (2J^ + a:i/ + 2/2 - X - 2/).
24. (x - y) (x2 + X2/ + 2/' - X + 2/).
25. (x + 2) (1 + a) (x - 2) (1 - a).
26. (x - y) (x2 + X2/ + 2/=^ + X + 2/ + 1).
27. 4o(x-l)(x2+2).
28. (3a - x) (3a + 2x) (a- x). 32. (2x - 3) (4x - 3) (x - 2),
29. (x - 2) (x + 3) (x - 1). 33. (x + 2) (x + 1) (x - 3).
30. (x + 3) (2x - 5) (2x - 1). 34. (x + 3) (x - 2) (x - 4).
31. (2x + 1) (4x - 3) (x + 1). 35. (x - 2) (x - 1) (x - 5).
EXERCISE 43
1. {a + b + x){a + b- x). 11. {x + a-h)(x-a + b).
2. (a- b + 2x) (a- b- 2x). 12. {x - a + y) (x - a - y).
3. (a + X + 2/) (a - X — y). 13. (a + 2/ + x) (a + 2/ - x).
4. (3a + a; + 2y) {U-x-2y). 14. (a* + x^ + y) (a^- x»- 2/)
5. (4a + X - y) (4a - x + 2/)- 15. {x + y + l){x-y- 1).
6. (m + X + 2/) (m - X - 2/). 16. {I + x - y) {I - x + y).
7. (o + 6 H- 2x) (a + 6 - 2x). 17. {c -\- a -b) {c - a -\-b).
8. (a + 6 + 2x) (a + 6 - 2x). 18. (a - 6 + c) (a - 6 - c).
9. (a + 6 + 2x) (a + 6 - 2x). 19. (a6 + 1+ x) (a6 + 1 ^ x).
10. (x + o + 6) (x - a - 6). 20. 2{z - I + 2«) (z - 1 - 2>).
21. (x + 22/-5«)(x-22/ + 52).
22. (a + 6 + c + d) (a + 6 - c- d).
23. (x - 22/ + 32 + 1) (x - 22/ - 3z - 1).
24. (3a - 26 + 5x + 1) (3a - 26 - 5x - 1).
25. (a - 56 + 36x - 1) (a - 56 - 36x + 1).
EXERCISE 44
1. {(? + cx-\- x«) (c2 - ex + x«).
2. (x» + X + 1) (x* - X + 1).
3. (2x» + 3x - 1) (2x« - 3x - 1).
4. (2a' - 3a6 - 362) (2a2 + 3a6 - 36«).
ANSWERS XXI
5. (3a:« + Sxy + 2y^) (Sx^ - 2xy + 2y^).
6. (7c« + 9cd + 5^2) (7c2 - 9cd + Sd^).
7. (4x2 + X - 1) (4x2 _ a; _ 1).
8. (10x2 + X - 3) (10x2 - X - 3).
9. (15a262-f Safe + 2) (15a2&2- 8a6 + 2),
10. 2(4a2 + 6a6 + 62) (4a2 _ 6a6 + 62).
11. (a2 + 2a6 + 262) (a2 - 2a6 + 262).
12. (1 + 4x + 8x2) (1 _ 4^ _|_ 8^.2) .
13. (x22/2 + 6x2/ + 18) (x22/2 - 6x?/ + 18).
EXERCISE 45
10. 3x(x2 + X + 1) (x2 - X + 1) (x + 1) (a: - 1).
11. (2a + 1) (a + 1) (2a - 1) (a - 1).
12. 2(x2 + 2x + 2) (x2 - 2x + 2) (x2 + 2) (x2 - 2).
13. (x + 9) (x - 5).
14. (2x + a - 1) (2x - a + 1).
15. 5a(x« + x3 + 1) (x2 + X + 1) (x - ]).
16. 3x(3x + 4) (2x - 3).
17. (X2 +XZ+ 2^2) (x2 - X2 + 2^2).
18. (x =fc 1) (a =fc 3). 25. (a2 + 2) (a - 1).
19. (11 4- x) (10 - x). 26. 2x(3x + 1) (a: - 1).
20. (3x - 52/) (x + 62/). 27. (1 =*= 5z + z^).
21. 7a(l ± a62): 28. 2(4 - y) (16 + 4^/ + y^).
22. 2(3x + 4) (x + 1). 29. (1 + o + 6) (1 - a - 6).
23. (x + 2) (x- 1) (x2-x + 2). 30. (3a - 5) (7a + 6).
24. 3a(l + a) (1 - a + a2).' 31. (x^ -\. y^) {x^ - 3^ + y^),
32. (2x + 9z') (4x2 - 18x23 + 8l2«).
33. 45x4(3^/2 =t 1).
34. (a3 + 5)(a + 2)(a-2).
35. (c + d - 1) (c2 + 2cd + (i2 + c + d + 1).
36. (x-2/)(x-2/ + 2).
37. (8x - 92/) (3x + 42/).
38. (x + 2y) (x - 2y)K
39. (a + 3) (a + 2) (a - 3) (a - 2).
40. (z2 ± z + 1).
41. (a + 6 + ^) (a + 6 - c) (a - 6 + c) (a - 6 - c).
XXli SCHOOL ALGEBRA
42. {7x + ^y){^x-7y).
43. (2 + n) (16 - 8n + 4n2 - 2^3 + n^).
44. 5x(x2 + 2/=) (a:^ - a:^ + y^).
45. (m + n) (m^— m^/i + m^n^— m?n^ + m^/i* — mn^ + n*).
46. ia(2a; + 2/) (4x2 - 2xy + y^). 49. (2a ± 1) (a =t 3).
47. (1 + x2) (1 + xy (1 - a;). 50. {x ^ 2) (x^ ± 2a: + 4).
48. 6(x - 3) (4 - x). 5L {x + l){x^ 2) (x - 3).
52. y{2x - 02) (16a:4 _|_ 33^22 + 4^224 + 2xz^ + 28).
53. (x4 + 6a:2^2 + 2/4) (^^ + y^ (^^ _ 2^)2.
54. (x2 + 2/2 + 22) (re + 2) (x - 2).
55. (aa:-2/)(a:-l)(a:2+x + l).
56. (a + &)(a-76).
' 57. (a-6+ a:-2/) ia^-2ab + 62-aa: + &a; + a2/-^2/ + x^-2xy+ y^).
58. a262(o - 6)2. 60. {x + 2/) (2a:2 + xy + 2y^).
59. (a:< + a3)(a;8-a3a^_|_a6). gi. (^-6) (a: + 2/) (a -6 + a: + 2/).
62. (a - 6 + 2a: + 22/)2.
63. (a2 + 1) (a4 - a2 + 1) (a =fc 1) (a2 ± a + 1).
64. (2a - 36) (2a + 36 + 2). 68. (a =t 52) (1 - a:) (1 + x + a:2).
65. (2a + 36 + 1) (2a - 36 - 1). 69. (3 + a) {x ^ 3).
66. {x + 2/)2 (a: - y)\ 70. (a + 6)^.
67. {x - 1)2 (a: + 2)2. 71. (a - x)\
72. (a6c — mnp) {ax — my).
73. (a: + 2 + 2a - n) (a: + 2 - 2a + n).
74. (a: - 2) (a: + 5) (2a: + 1). 76. ^x-'y^x + yY {x - y)\
75. (a2 + 262) (a2 -262 + 1). 77. 2 (9a: - y) {x + 3y).
78. (1 + a: - a:2) (1 + 2a: + 2a:2 + ar» + x*).
79. (1 - x)2 (1 - 2a: - a:2). 82. (a - 36 + 3) (a - 36 - 3).
80. (a: + 1) (ax - c). 83. (a:2 + 4) {x + 2)^ (ar - 2)2.
81. (a:2 =t 9a: + 1). 84. 9(1 ^ x) {7x* + 6a:2 + 3),
85. (a:2 - 3 + 7y) {x^ - 3 - 7y).
86. (a:22/2 -2+2x-y) (x^y^ ^2-2x + y).
87. {ax — bmy) {an + cmz).
EXERCISE 46
1. 2, 3.
3. 3, 4. 5. 4, - 3.
7. =^=3.
2. 2,-1.
4. 3, - 2. 6. =t 4.
8. 0, ±
ANSWERS
XXIH
9. 0, =fc 5. 13. 0, 3, - 2. 17. - 1, ± 2.
10. 0, =t 3. 14. 0, - 2. 18. - 1, ± 3.
11. 1, h 15. 0, - a. 19. =t 1, =t 2.
12. 2, - f . 16. 0, =t o. 20. 1, =t 2.
25. h 1. 26. 2, 2.
28. x2 - 7x + 12 = 0.
29. x2 + 3a: - 10 = 0. .
30. x^ + 10a: + 21 = 0.
34. - 5, 9. 36. 3, - 4.
35. 4, - 10. 37. 0, 9.
42. 50' X 150'. 43. 617°.
31. x^
32. x^
21. - 1, f.
22. ± 1.
23. 0, =t 3.
24. 1, 2.
27. =«= 2, ± 3.
- 4a: + 4 = 0.
2a: = 0.
33. r' -5x^-hQx = 0.
38. 0, 25. 40. 8, 9.
39. 0, ± 5. 41. I, - 2.
44. 49,170 sq. mi.
1. 2ab.
2. 5x%
3. 8a2x2.
6. a: - 3.
7. a:(2a: + 3).
8. a — x.
9. a: - 1.
10. a(2a + 1).
EXERCISE 47
11. a: + l.
12. 4aa:(a — x).
13. x{x - 1).
14. 2a: - y.
15. a:(a: - 2).
16. 6(1 - a2).
17. 1+a + aK
20. 0262(a _ 5)8.
21. 3a:2(a: - y^.
22. a: - ?/.
23. (a: - y)^.
24. 2 - X.
25. a - 3.
26. a;(a: - a).
EXERCISE 48
1. 6a262.
2. SQa^xY-
3. 12a6c.
4. 12a262c2.
7. 2a:(a:2 - 1).
8. 6ah{a + 6).
9. 14a:2(a: - 3).
10. (a:3 - 1) (x + 1).
11. (a:2 - 2/2) (a: - 2y).
12. 6a:(a: + 1) (a: - 1)2.
13. 15 abx^yix -\- y) (x - y)^.
14. a:(a: + 5)(a:-8)(a:--l).
15. a« - ¥.
16. 6a:2(a: + 1) (a; - 1).
17. 12a6(a + 6) (a - b).
18. (2a: + l)(a: + l)(2x-.l).
19. 6a:(a:3 _ i) (-^ _ i).
20. 6a:(3a: + 10) (2a:-7) (a:-3).
21. 2a:(l + x^) (1 + a:) (1 - x).
22. uxY(x + ly {x - ly.
23. Q^^iSx + 1) (a; - 1) (3a: - 1)*.
24. {x - 1)2 (x + 1)2 {x + 3)2 (x-3).
27. a'¥(a + by (a - by.
28. lSa^¥(^{c ^d)(a- dy.
29. a*bHa + 6)2 (a - by.
30. 36a:4(a, 4. 2^)2 (3. _ y)3.
31. a-b.
32. 36(a - by.
Xxiv SCHOOL ALGEBRA
33. (a + b)(a- b) (x - y). 35. {x + 1) (x + 2) {x - 2)K
34. (a + 6) (a - 6) (x - 2/)3. 36. x'ia + x) {a - x)\
37. 2, 3.
39. Irrigable land, 100,000,000 A.; swamp land, 78,000,000 A„
40. 2 da. 14 hr. 40 min. 41. 866,400 mi.
42. Son, $3600; daughter, $5600; wife, $10,800.
43. 2162 mi. 44. 39.37 in.
EXERCISE 49
^ a + 6 +c . 6c
A' 7i • O.
3 "• o
I
43,560'
3 -^—. 8 -^^-t^.
4.
2a
Zx
5.
4x
by'
6.
X
2-3ax
7.
Sxz
4^2
8.
3a
462*
9.
1
2a -1
10.
1
2a*
11.
1,
a
12.
x-y^
1
EXERCISE 60
1. I 13. 3^
2a
3a;*
14 2(a; + l)
15.
16.
17.
3
5
2ix - y)
a-hh
2(a - b) '
2
3x — 4y
-i
19.
20.
21.
. 22.
x+y x-y
2x + 3y
7x-hSy
2x2
x2 + 3a; 4- 9
a;-3
1
^u.
2a; + 3y
24.
2x + y
2x-y
25.
a + b — e
a — b — c
26.
1+a-x
a; + a-l
27.
3a; + 4a
x-\- a
28.
x-2
y*
29.
a; + 3
a; + 2
30.
x^-y^.
31.
x-y^
a + b
oo
x-y-z-2
ANSWERS XXV
EXERCISE 61
2.^. ^.-Z^. 10.-5^.
3. y-z^. 7. - 1. 11. ^ + %+'-
y + 2x X -3
A 3+m -1 IP 2 + a + b
4— m o + 0 4- c 2 — a + o
18. f 19. 6. 20. |. 21. 4. 22. 20. 23. 5.
EXERCISE 52
1. 6f. 2. 13f. 3. lOJrf
4.x-2 + ^- 14.x-l-^2.
5. 2x^ + 3 -A. 15. 3a+ ^"^
2a; ' 3a2 - 2b
6. 2a2x2 + l -y^- 16. x2-2a; + 2+ ^
5ax ' X + 3
7. x2 - 4x + 5 T-r- 17. 2a - 26 +
x+1 ' a+6
n.^_, + 2-4^^^. 20.1 -. + 2x^-3^-2-'
X^+X-1 ' ' 1+X-X2
12. x»+:»'H-2x + 3+ !"" "^ V 21. 4-2x + 3x2 So;' - 3a:*
X2-X-1 ' 2 + x-a:2
EXERCISE 63
1. ^. ^ a:^-2re _ a:» - g
2. i^. X — 1 x^ +x + 1
, a'-a + l 7 8x2 -y, o' + 1
4- ^^ ^' 2x + l ^^' a-2'
7? Q q2 + Q6 . x»4-xy
X — 1 a + 2o X -t a
XXyi SCHOOL ALGEBRA
,„ 2hc-¥+c'-a* .. (a - b)^ .a ^ - I
i-o. ^rr • 10. -: • lo. ; — —
26c 4 X + 1
14.(2 + 30)!. 17.^. 1, ^
4 l+x -' x-1
20. 3248 mi. 2L 27^ sec.
EXERCISE 64
1. if, if. - _a 46_ 2ab_
2. tiH,n. • 2026=^' 20=^62' 2a262-
o 4a; 15x ,,1 2a^ - 2a 3a
"• To» "To" •
18' 18 a2 - a a^ - a ' o^ - a
jQ x» + x2 + a; X + 1
(x + 1) (x' - D' (X + 1) (x» - 1)
jj X x(2x-3) 4x2-9
15.
x(4x2 - 9)' x(4x2 - 9)' x(4x2 - 9)
2x + 4 15x - 30 18
6(x2-4)'6(x2-4)'6(x2-4)*
EXERCISE 56
1 1?. 7 3x + l j^ 3w2+l
6x 24 ^" (m+ 1) (m - 1)2
8x - 9 + 12a 8. 1. ^^ {a - 36)«
12ax * 9 4x • 4(a - 6)2*
_ 15b - 4c - 6a ' 1 - f 16. ^' .
3- 6^6^ 10.^^^. ^^-4
, , , , . 3x2 2x2 + 3x - 1
4. ?^. jj 25a -20b 17- x(x2-l) '
^^^ ^^- 12 18. 0.
g 9 + 10ax2 y8-3x2g»-6y02 1
12ax2
^^' 6x22/22 • ^^- 8x2-2
6.^^. 13.-J?£-. 20.^.
o2 — o2 x* — 4 x2 — 1
01 x2 + 5x + 10 04 5^'y - ^y"
^"^ (x + 1) (x + 2) (x + 3) ' ^^- x(x2 - t) ' 25. 0.
5x(x + 3) x2 + 4x - 13
^ • (2x + l)(2x-l)(x + l)* 2(x2-l)
23 ^' 27 ^-^^
(a + 6)» X3 + 64
ANSWERS XXVn
28. ^^ + ?Q^-l.. 29. -1
6(x2 -9){x- 3) (x - 3) (x - 4)
Qn ^'^ + 2x - 1 ^^ o-3b 36. 0.
^^- x^-1 ^^' ¥^^^' 13
31. 0. 3xy *"'• 8(1 -a«)
K 42/2 _a.2-
32. r-^- „. ^ 38.
X
1 - x2 35. 0. 1 - x2
39. . ...^-1. ^. 43. ^^ - ^^^ - 1^
40.
{x - 2) (x - 3) (x - 5) (x2 - 9) (x - 1)
5-46 44. 0.
(a - 3) (a - 2) (6 - 2)* ITx^ - 42x + 39
41. 0. 15(x2 - 9)
42..^ 7^, .. 46. ^^-4^-22
12x(x + 1) (x - 2) (x - 3) (x - 5)
47. 0. 49. 0. ^^ 1
48. 1. 50. 1. 51. 0. ^i^ + 1)
EXERCISE 66
263x a + x 20. 1.
Sacy x^{a — x)
21. 1.
2. ^^. , lo g' + g + l ..
4x2 3. 1. 12. 22. -— .
a
a
4- --V" IM fa+1) (2x-l) „„ ,
^- ;t(2x - 1)' ,, 2 24. " + ''"'?•
„6 ^*-^TT- «-- + !
2^^' 15 X 25. 1.
»« + 2x - 3 '3^-xy + y^' g'c + g^ + 6c»
X jg^ a4-64-c
8 -^LzJ_. 1 27. 1.
''• a(x + 1) 17. -\.
2a; + 3 28. i-
^- 3(3:c-l)' 18- 1-
10. ^£ + l|. 19. -^. 29. 2^i^.
(x + 1)2 x + y ^mn
^^XVlll SCHOOL ALGEBRA
EXERCISE 57
- 2(2 - ar) i o ^2 . m
I. -^ -' 12. — i— . OQ 2 - q2 + 6^
^ 2x2 - 1 ^'^' 2
^^ + 1 « + 1 2a + 26
^*'^"^^* 14 e^_±^^_Z_^. 25. -1.
• . 6 * ac 4- 6c + a6
., x-a + 1 26.^^±1±^,
5 o-J^ 1^- ^T^^- 26c
' * a + 1 16. 0. 27. i.
6 ^^-2. 17 a6-cd + l ' ""'
18 a6-cd-l* 28. 2x.
8. - 4±1-. 19 <^ + 1) 30. - 1.
9. 1. 20. ?^^. ^^- rT2i-
^ a
10. a+a:. 21. a - 1. ^2. 14.
11 L±y. 22 -^ ^^- ~ ^^•
""-y '^-x 34.102.
EXERCISE 69
1. 3a + 56 - 4c.
2. -5-a-2x + y-^z;0-a~2x + y + z.
3. 9a2 + 62 + c2 + 4c^ - 6a6 - 6ac + 12a(i +26c - 46./ - 4cd.
7. -8a2 + 20a+9. 19. - ^.
8- 35. o^ a + 1
9.
23a 2^- "3^
^ 21. 5 + 2a - 3a2.
^^- 2' ^' 4x3 _ 26x
12. 32, 12, lOf. 28. ^^,_^^^^,_^y
^^' ^^'^^'''- OQ 4x3- 26x2- 26x + 144
14 (J^SI^, ^^' ' (x2-4)(x2-9) '
18. f, -A. 30.-1^,.
ANSWERS
X
31 ^(2x-
^^- 2(3:. -
3)
1)'
38.
1+rr*
32. 0.
39.
40.
1
1.
31 ^
**• s(2 - x) (x
-3)
41.
2x
3'
3«-(l-.)(..
4
- 2) (X - 3)
.
42.
43.
5a:.
_1
X
37. L
48.
5240]
oai.
EXERCISE 60
1. 2.
17. 3.
32. ^.
47.
6f.
2. 3.
18. 10.
33. -5.
48.
12.
3. 2.
19. ff.
34. -7.
49.
.5.
4. 2.
20. A.
35. -3.
50.
.2.
5. -1.
21. 5.
36. -4.
51.
15,200.
6. 13f.
22. 4.
37. 2.
52.
.05.
7. 5.
23. 13.
38. Y.
53.
.05.
8. 1.
24. ih
39. 3.
54.
3 yr. 6 mo.
9. ¥.
25. -7.
40. -h
55.
122, 212.
10. -2.
26. 4.
41. 3.
56.
20.
11. 5.
27. 2.
42. 8.
57.
14^.
12. -f.
28. I.
43. 1.
58.
19i^.
13. -2.
29. -i
44. 73.
59.
36f.
14. f.
30. -H.
45. 5.
60.
10.
15. 0.
31. 0.
46. 30.
61.
12.
EXERCISE 61
1. -i
5. -9.
9. -f.
13.
5.
2. -3.
6. -I
10. -23.
14.
7.
3. -H.
7. -I
11. 8.
15.
0.
4. 12.
8. -2.
12. 1.
16.
-3.
17
'. i
20. 183 ft.
XXX SCHOOL ALGEBRA
EXERCISE 62
8. 2, - 1*. 9. 5, - 1*. 10. 2, -2*
11. 1*, -J^. 12. 3,-1*.
EXERCISE 63
1- 3a. a 17. 17a. ]
h h'
2. -• * ^ 18. 19a\
_« n.S—t^ 20. ^
a-6^
0-&
2c
3-6
5 -2a
36 + 2d
2a -c
a6
a + 6 ' 2a + b
12. ^. 21 ^^(^-^) .
2 a' - 2a62- fta
13. ^. 22 a^
14. 0.
23. 0.
7- -^^— • 15. 3 24. "^ + «
3a2-l
0-6' : ■'''• ab + bc + ac' 26. ^1 - 2a - o»),
8. -^. 16. ^^^
EXERCISE 66
1- 4- 4. 2q + 6 6. 5. 10. 24.
2-12 ^ 7. 59. 11. 30.
,* * 5^. 8-60. 12. ^;f;A.
3- - 2. > 9. 247. 13. 5V. 14. 45.
EXERCISE 66
1. 120. 3. 336. 4. 120.
6. Tin, 37A; zinc, 75A; copper, 301i^ lb. 6. 27, 28.
7 1^ A 1156
• lOO' 20' 100 *
8. Owner, $2800; other, $2000.
9. State, $4000; county, $8000; township, $6000.
10, $350, $420; $330, $440. 11. 33, 42.
ANSWERS XXXI
12. 15,000,000,000 tons.
13. India, 234,375,000; China, 421,875,000.
14. First, $30,000; second, $22,500.
15. 177^ cu. ft. 22. 11. 29. 100 lb. 34. U da.
16. 96. 23. 30 gal. 30. 80 lb. 35. 6 da.
17. 860 million. 24. li^ gal. 31. | gal. 36. 4 da.
18.10. 26. 4flb. 32 1, i. 37. 28| min.
19.8. 27. 9itlb. ' 9' x' 38. 36 min.
21. 60. 28. 150 lb. 33. 5f da. 39. 169iV min.
4j I 54t^ min. past 4. 42 I ^^^^ ^^' P^^* ^'
1 38 A min. past 1. ' 1 54tt min. past 10.
40 I ^iT a^d 38t\ min. past 4.
1 21 A and 54iV min. past 7.
44. 779 - da. 46. 686 da. 48. 24 hr. 50. 6 hr.
45. 398 + da. 47. 15 hr. 49. 108 min. 51. 19ii mi.
f 1st, each 12 hr. 63. 36 lb.
g2 I A, 42 mi.; B, 40 mi. 64. $20,000.
2d, each 492 hr. g^ 22^ min.
^A, 1722 mi.; B, 1640 mi. ^, , , , „„ « .
gg g^^' ' ' 66. 5A and 38A mm. past 10.
54. 1100 ft. ^^- ^•
55. 2357+ ft. 68. 29|yr.
56. gold, 21 lb.; snver, 18f lb. 69. 12 da.
67. Aluminum, 35 lb.; iron, 45 lb. 70. U gal.
58. Copper, 51^ lb.; tin, 48| lb. 71. 6.
^g r 46f bu. oats. 72. 15.
*' ^ ^^3 bu. corn. 73 ^p + 2, 2p + 3.
f $3250 at 4%. ^5, .
^"- I $1800 at 5%. 74. ^-q— mi.
61. 10; 14; 6; 24. _abc_ ..
62. 26, 27, 28. ^^' b-a
EXERCISE 67
15. 5i yr. 17. (1) 122°; (2) 32°; (3) 4892*^.
18. 4
20. 60 lb.
16. 4^%. 18. 442 + °; 617°; 1995 + °; 2804*.
Xxxii SCHOOL ALGEBRA
EXERCISE 68
1. 1, 1. 5. 2, - L 9. I |. 13. 15, 10.
2. 1, - 1. 6. 1, - h 10. - I 2. 14. 3, - 4.
3. h -h 7. - 3, U. 11. 3, - 7. 15. 10, - 10.
4. 2, 3. 8. i - I 12. 8, 9. 16. 12, 18.
17. 7, 5. 18. 12, 21. 21. Sugar, 6^; rice, 5^
EXERCISE 69
2. 1, 2. 4. 3, - 2. 6. 3, - J. 8. - 4, 3.
3. - 1, - 1. 5. 2,-1. . 7. 5, 4. 9. 2, 6.
EXERCISE 70
2. 3, - 2. 4. 3, - 2. 6. 1, 3. 8. 5, 1.
3. - 3f, - ^. 5. 3, 6. 7. - 2, 1. 9. ff, 4.
10. - 61, - 3i 11. ii, ¥.
EXERCISE 71
1. 5, 12. 5. 3, 1. 9. 2, 4. 13. 18, 12.
2. 5, 2. 6. - 1, 4. 10. -.2, .6. 14. 9, - 1.
3. i, i. 7. - h, h 11- .015, .01. 15. 17, 6.
4. -5, - 1. 8. 1, - 1. 12. 2, - 3. 16. 2, - 1.
17. -2, -3.
EXERCISE 72
1. 2a, — a. o a 6
8. r> -•
2. -6, 2a. «> "
¥ -h a -a' 9 1, 1.
'^' ab' -a'b' ah' -a'h ' <^ ^
^. m-\-n,m-n. 10. n-m,n+m.
^ 2b + r a-2 11.3, ?^.
6 a 0
6. a + 26, 2a- 6. 12. a, - 6. 13. a + 6, a - 6.
' cn — bd ad — cm -. a — d a — h
' on — bm an — bm ' b — d b — d
ANSWERS XXxiil
17.^
1 1 2
15. a,b. -- a + 2b a - 2b
1 1
16.
a + 6 + c a + 6 + c 18. - a, b. 19. b, a.
20. a + 1, 6 - 1.
EXERCISE 73
1. 1, 2, 3. 4. 2, 2, 2. 7. - 3, 3^, - 2„
2. 2, 3, - 4. 5. U, 1|, U. 8. 2, 3, 1, 4.
3. 3, 4, 7. 6. 2, 3i - 4. 9. 12, 18, - 24
10. a + 6, a - 6, 2a. 11. 6, 40, 20.
12. X = -a+b + c. , , 3a - 26
, , 14. X = ^
y = a — b -{- c. 6
2=a + 6-c. 2/= 2a + 36
13. X = a - 6 + 1. 6 '
?/=— a + 6 + 1. _aH-6
2 = a + 6 - 1. "^ 6
.,_ 6 + c + c? — a a + 6 + c? — c
15. a; =— —
y
4 4
o + c + c? — 6 a + 6 + c — d
EXERCISE 74
1. - 1, 1.
2. i - i
3. i iV.
4. i - i.
10.
1,1.
11.
12.
13.
14.
h - 1
1, - h h
2, - h 1.
1, i I.
5.
6.
7.
2n
2n .
8. i, 1.
1+^2'
a, -a.
1 -w2
9. ^, ^.
6 a
ire -2&C
- 2ac. - 2a6
a-{- c a + 6
lb. T~, '
26 2c
I -\-m . I -]-n m -{-n
1,1.
EXERCISE 75
1. 9, 14. 2. 9, 12. 3. 2, 8.
4. Flour, 3f^; sugar, 5^. 6. 57 pear trees; 43 apple trees.
5. Man, $3; boy, $2. 7. Silk, $1.80: mtiu. $1.50.
XXXiv SCHOOL ALGEBRA
9. Iron, 480 lb.; lead, 700 lb. 13. $4600; $5400.
10. First, 3 points; second, 1 point. 14. 67| and 172f .
11. First, 5 points; second, 3 15. 50 — 3.
points; third, 1 point. 17. Wheat, $7; potatoes, $50.
18. Wheat, $8; corn, $20; potatoes, $24.
19. First-class, $1.52; second-class, $1.36.
20. Fourth-class, $1.02; fifth-class, $.81; sixth-class, $.70.
21. Corn, 2,772,000,000 bu.; wheat, 737,000,000 bu.; oats, 1,007,-
000,000 bu.
22. Copper, 550 lb.; iron, 480 lb.; aluminum, 156 lb.
23. .2875 in., .5025 in., .3625 in.
24. Eiffel Tower, 984 ft.; M. L. B'ld'g, 700 ft.; Wash. Mon., 555 ft
25. Nitrogen, 15^; potash, 5^; phosphate, 5?f.
26. Oats, 461 bu.; corn, 53^ bu.
27. 20 lb. of 20j!f coffee; 40 lb. of 32^ coffee.
29. 40 lb. of 7H tea; 60 lb. of 50^ tea.
30. Cream, llf gal.; milk, ^^ gal.
31. $3250 at 4%; $1800 at 5%.
32. $2000 at 5%; $10,000 at 4%.
34. /' X 5''.
35. 15' X 6'.
36. 12 boys; $60.
37. 90 mi.
39. 13 played, 8 won.
40. 68 cases, 50 successful.
41. f.
42. -i^.
52. A, in 24 da.; B, in 48 da.
53. A, 14 A da.; B, 18tV da.; C, 34f da.
57. 49. 58. 23. 59. 64. 60. 151.
61. Oarsman, 4 mi.; stream, 2 mi. per hr.
62. Oarsman, 5^ mi.; stream, If mi. per hr.
63. Oarsman, 6 mi.; stream, 1^ mi. per hr.
64. Cast iron, 450 lb.; wrought iron, 480 lb.
65. From earth, 93,000,000 mi.; from Mars, 141,000,000 mi.
44. x\.
45. ¥. i.
46. 1.
47. 16, 81.
48. 21, 79.
49. 14, 54.
50. 32, 18.
51. 51 hr.;
17 hr.
ANSWERS XXXV
66. Tea, 50ff; coffee, 30^.
67. 24 bu. from 1st; 16 bu. from 2d.
68. A, $70; B, $110.
69. 80 lb. of 25f^ spice; 120 lb. of 50ff spice.
70. 480 mi. 71. 11, 36.
' 2*2* ' p-q' q -p *
EXERCISE 77
11. (1) 5; (2) V45; (3) 13; (4) Vu.
12. 25 sq. spaces. 13. 42 sq. spaces. 14. 171 ftsj. spaces.
1,0.
EXERCISE 79
1. 2, 1.
3. - 4, - 2. 5. - 4, - 1.
7. -
2. 1, - 1.
4. 0, 0. 6. 2, 3.
8. -
9. f. 11. 2.9 -h,
-3.3-.
EXERCISE 80
1. 60 mi.; 1:30 p. m. 6. 3:7,
3. 251f mi.; 5:17^p. M. 8.2:1.
5. 17: 10. 10. At end of 5 hr. 20 mi. from 1*.
EXERCISE 81
4. tV q a'+x^ 16. 7, 10.
5.^. ' "^ ' 17.-^,^.
J 10,-2, a-b a + b
^•^^Ay' 11. -f. 18.3,2,1.
12. - 3, 19. I, f .
13. 4. __ a' -a h'-b
^0.
x{x-]
ly
3(4x-
15)
5(2x-
-3)
x^ + l
x^-l
23. a'
-bK
25. '-^
-ad
14 8 i ' a'b—ab' ab' —a'b
15. ii i. 22. - a - 1.
27. 11, 9, 18.
28. 9^(4x2 + 2xy + 2/2).
a — c 30. c — a + b, — a — 6 — c.
XXXvi SCHOOL ALGEBRA
32- 7H. 48. a = 1^.
33. 64. 2M
36. 4, 3. _ ar" —a
42. 174 - ° F.; 79 - ° C. '^"' ^ " ^ _ 1 *
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