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GIFT OF

MATHEMATICAL TEXTS

FOR SCHOOLS

EDITED BT

PERCEY F. SMITH, Ph.D.

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD

SCIENTIFIC SCHOOL OF YALE UNIVERSITY

FIRST COURSE IN ALGEBRA

BY

HERBERT E. HAWKES, Ph.D.

PROFESSOR OF MATHEMATICS IN COLUMBIA UNIVERSITY

WILLIAM A. LUBY, A.B.

HEAD OF THE DEPARTMENT OF MATHEMATICS

KANSAS CITY POLYTECHNIC INSTITUTE

AND

FRANK C. TOUTOH, A.M. ' '

FORMERLY PRINCIPAL OF CENTRAL Hit*H SCEOOL

ST. JOSEPH, MISSOURI '

REVISED EDITION

GINN AND COMPANY

BOSTON â€¢ NEW YORK â€¢CHICAGO â€¢ LONDON

ATLANTA . DALLAS -COLUMBUS IAN FRANCISCO

1911

ENTERED AT STATIONERS' HALL

COPYRIGHT, 1909, 1910, 1917, BY

HERBERT E. HAWKES, WILLIAM A. LUBT

; , .': '. AND FRANK C. TOUTON

ALit BIGHTS RESERVED

' ' " A 321.7

GINN AND COMPANY â€¢ PRO-

PRIETORS â€¢ BOSTON â€¢ U.S.A.

O-C

PREFACE

In this revision of their ''First Course in Algebra" the authors

have in general followed the plan of that text in the order of

topics treated and in the method of their presentation.

The most important modification of the order of topics is

found in the transference of the work on Eatio and Propor-

tion to the last chapter in the book and the omission of the

chapter on the Highest Common Factor and Lowest Common

Multiple. The latter topic is treated in connection with the

related material on fractions, while the former is placed among

the Supplementary Topics at the end of the book.

Material for which there is no strong demand from teachers

has been omitted, and the entire work has been rewritten in

the interest of greater simplicity and directness of appeal. The

collections of exercises and problems arÂ© for the most part new

and contain a larger proportion of easy exereises with simple

results than the first edition.

A striking feature of the revision is the inclusion of a large

number of oral exercises in connection with the introduction of

each new idea or operation. It is the object of these exercises

to present the new concept in complete isolation from any com-

plication of notation or technique so that the student becomes

familiar with its content and bearing before he is asked to

make use of it in written work. These oral exercises may well

be taken up when the advance lesson is assigned, so that the

pupil may be certain that he understands the idea involved

in the new work before he leaves his instructor.

Another feature scarcely less important is the character and

position of the examples and hints. The aim has been to

459907

vi FIEST COUESE IN ALGEBRA

help the student at the exact point where he needs it and

to avoid the insertion of lengthy and difficult solutions before

they can be completely understood.

The definitions and axioms have been expressed in the sim-

plest language which is consistent with scientific accuracy.

Many definitions which are usually found in elementary texts

but which do not contribute to the clearness of the subject

are omitted.

The first presentation of the subject of graphs has been limited

to the study of the straight line and a few exercises of a com-

mercial or scientific character. These exercises not only have

a very definite human interest apart from their mathematical

value but also serve to familiarize the student with the kind

of graphs he will meet in his ordinary reading.

The first consideration in the treatment of radicals has been

the needs of the student for his later study of the quadratic

equation and for his work in geometry.

Frequently a student's knowledge of algebra is limited to a

greater or less facility in the use of the rules of operation â€”

to mere technique. To obviate this result the development of

the problem work in this text has received full and careful

attention.

The authors have received suggestions of great value from

many teachers in all parts of the country, for which they

extend their thanks. They are under especial obligation to

Mr. E. L. Brown, of Denver, Colorado, Professor H. E. Cobb,

of Chicago, Illinois, and to Mr. L. A. Pultz, of Eochester,

New York, for helpful criticism.

CONTENTS

CHAPTER PAGE

I. Introduction (Sects. 1-11) 1

XL Positive and Negative Numbers (Sects. 12-19) . 18

III. Addition (Sects. 20-25) 33 ^

IV. Simple Equations (Sects. 26-28) 39

V. Subtraction (Sects. 29-30) 49 "^

VI. Identities and Equations of Condition (Sects.

31-34) 54

Vll. Parentheses (Sects. 35-36) 64

Vlll. Multiplication (Sects. 37-44) - - ^O - '

IX. Parentheses in Equations (Sects. 45-46) ... 79

X. Division (Sects. 47-49) 87 -^

XI. Equations and Problems (Sects. 50-51) .... 95

XII. Important Special Products (Sects. 52-55) . . 105

XIII. Factoring (Sects. 56-66) 113

XIV. Solution of Equations by Factoring (Sects.

67-70) 137

XV. Fractions (Sects. 71-81) 148

XVI. Equations containing Fractions (Sects. 82-88) . 175 '

XVII. Graphical Representation (Sects. 89-94) ... 200

XVIII. Linear Systems (Sects. 95-100) 217

XIX. Square Root (Sects. 101-102) 240

XX. Radicals (Sects. 103-114) 250

XXI. Quadratic Equations (Sects. 115-117) .... 270

XXII. Ratio and Proportion (Sects. 118-125) .... 282

Supplementary Topics (Sects. 126-130) 293

Index 807

vii

ILLUSTRATIONS

PAGE

JOHN WALLIS 48

SIR WILLIAM ROWAN HAMILTON 70

SIR ISAAC NEWTON 100

JOHN NAPIER 186

REN:^ DESCARTES 210

FRANCOIS VIETA 268

I3f

FIRST COUESE IN ALGEBRA

CHAPTER I

INTRODUCTION

1. The numbers and symbols of arithmetic. The simple

operation of counting employs the numbers we call in-

tegers. To represent these integers and the other numbers

with which it deals, arithmetic uses the nunierals 0, 1,

2, 3, 4, 5, 6, 7, 8, and 9. Operations on the numbers of

arithmetic are indicated by the symbols +, â€” , X, and -^.

The operation of division applied to integers gives rise

to fractions. With these two kinds of numbers, integers

and fractions, the student's work in arithmetic is mainly

carried on.

2. Symbols of algebra. Symbols are employed far more

extensively in algebra than in arithmetic, and many new

ideas arise in connection with their meaning and their use.

Some symbols represent numbers, others indicate opera-

tions upon them, others represent relations between them,

and still others represent kinds of numbers with which

arithmetic does not deal. Letters as well as arable numer-

als are used to represent numbers. The following symbols

of operation, -f ? â€” ? X, and -r-, have the same meaning as in

arithmetic. The sign of multiplication is usually replaced

by a dot or omitted.

For example, 3 x a is written 3 â€¢ a, or 3 a, and 2 x a x b is

written 2 ab. Also a -^ 6 is often written - .

b

2 FIEST COURSE IK ALGEBRA

The sign = is read equals^ or is equal to. As the need

for them arises, other symbgls will be introduced.

3. The use of letters to represent numbers. The use of

the letters of the alphabet to represent numbers is the

most striking difference between arithmetic and algebra.

In arithmetic we speak thus : If the sides of a triangle

are 6, 7, and 9 inches respectively, its perimeter is 6 + 7 + 9,

or 22 mches. The corresponding statement in algebra is:

If the sides of a triangle are* a, ^, and c inches respectively,

and its perimeter is p inches, then p = a-\-h-{- c. Here the

second statement is true for every triangle, while the first

is not true for every triangle.

Similarly: If a rectangle is 8 inches by 12 inches, its

perimeter is 8+12 + 8+12, or 40 inches. And if a rec-

tangle is I inches long and w inches wide and if p denotes

its perimeter in inches, then p â€” l-{-W'\-l-\-w^ or 2/+2?^^.

Here, again, the arithmetical statement is particular and

applies to one rectangle only, while the algebraic state-

ment is general; that is, it is true for all rectangles.

The gain in power which the general symbolic language

of algebra affords over the particular numerical language

of arithmetic constitutes one of the most important advan-

tages of the algebraic method. As the student progresses he

will meet with many illustrations of this feature of algebra.

The purpose of the following exercises is to familiarize

the student with the use of letters in the place of numbers.

ORAL EXERCISES

1. What numerical value has 5 a when a is 3 ? when a is 5 ?

when a is 10 ?

2. What numerical value has 6Â«^ + 2& when a is 2 and 5 is 4?

3. Express h-\-?nn in seconds if h and m stand. for the

number of seconds in an hour and in a minute respectively.

INTKODUCTION 3

4. Express lOy + 4/ in inches if y and / stand for the

number of inches in a yard and in a foot respectively.

5. If g' and d represent the number of cents in a quarter

and in a dime respectively, express 4 5' + 6 c? in cents.

6. If ^ and h represent the number of pounds in one ton

and in one hundredweight respectively, express 4 i^ + 6 /z, in

pounds.

7. 3 ir + 5 ic = how many a; ? 9. 5 â€¢ 2^ + 10 â€¢ ?^ = (?) ?^.

8. ^x-\-hx = {^)x. 10. 5a^ + 3cc + 6a: = ?

11. 3x-2aj + 7a;-5^ = ?

12. 7 books + 3 chairs + 2 books + 5 chairs = (?) books +

(?) chairs.

13. 8 books + 4 chairs â€” 2 chairs + 4 books = (?) books +

(?) chairs.

14. 6 books + 7 chairs â€” 3 books â€” 2 chairs = (?) books +

(?) chairs.

15. 55 + 4c + 8Z>-2c = (?)5+(?)c.

16. 6^ + 3ic + 4^> + 8a^ = (?)^Â» + (?)a^.

17. ^x-\-2h^-Zx-h-\-x = (^.)h^(^,)x,

18. 2a^ + 2 + 3cc + 4 = (?)a^ + ?

19. 4cc + 2 + 3x + 2-^ + 8 = ?

20. iÂ«+:r + 2 + cc + cc + 2 = ? 22. 5a+18-3a-7=?

21. 7i4-7z + l + 7i + 2= ? 23. 8;r-3+18-5ir=?

24. 4i^;- 8 + 3t^ + 20 = ?

25. What value has 4 aj + 3 when a:; = 2 ? when a? = 7 ?

26. What value has 3 a; â€” 4 when aj = 3 ? when x = 21

27. The side of a square is 5 inches long. What is its area?

its perimeter ?

4 FIRST COURSE IN ALGEBRA

28. The side of a square is s inches long. What represents

its perimeter ? its area ?

29. The base of a rectangle is 12 feet, and its altitude is

4 feet. What is its perimeter ? its area ?

30. If ^ represents the number of feet in the base of a rec-

tangle and a the number of feet in its altitude, what is its

perimeter ? its area ?

31. A rectangle is twice as long as it is wide. Let w repre-

sent the number of inches in its width. Then express, in terms

of w, (a) the length ; (h) the perimeter ; (c) the area.

32. A man is three times as old as his son. If s denotes

the number of years in the son's age, what will represent the

father's age ?

33. A father is 28 years older than his son. If s represents

the son's age in years, what will represent the father's age ?

34. A rectangle is 24 inches longer than it is wide. Let b

represent the width in feet. Then represent the length and the

perimeter in terms of b and numbers.

35. A rectangle is 16 feet narrower than it is long. If w

represents the width in feet, what will conveniently represent

the length ? the perimeter ?

36. A rectangle is 4 feet longer than twice its width. Express

the width, the length, and the perimeter in terms of a letter, or

a letter and numbers.

Origin of symbols. Many of the symbols that are in common use

in algebra at the present time have histories which not only are

interesting in themselves, but which also serve to indicate the slow

and uncertain development of the subject. It is often found that

symbols which seem without meaning represent some abbreviation

or suggestion long since forgotten, and that operations and methods

which we find hard to master have sometimes recLuired hundreds of

years to perfect.

INTEODUCTION 6

In the early centuries there were practically no algebraic symbols

in common use ; one wrote out in full the words plus, minus, equals,

and the like. But in the sixteenth century several Italian mathema-

ticians used the initial letters p and m for + and â€” . Some think

that our modern symbol â€” came into use through writing the initial

m so rapidly that the curves of the letter gradually flattened out,

leaving finally a straight line. The symbol + may have originated

similarly in the rapid writing of the letter p. But in the opinion of

others these symbols were first used in the German warehouses of

the fifteenth century to mark the weights of boxes of goods. If a

lot of boxes, each supposed to weigh 100 pounds, came to the ware-

house, the weight would be checked, and if a certain box exceeded

the standard weight by 5 pounds, it was marked 100 + 5 ; if it

lacked 5 pounds, it was marked 100 â€” 5. Though the first book to

use these symbols was published in 1489, it was not until about

1630 that they could be said to be in common use.

Both of the symbols for multiplication given in the text were

first used about 1630. The cross was used by two Englishmen,

Oughtred and Harriot, and was probably an adaptation of the letter

Xy which is found some years earlier. The dot is first found in the

writings of the Frenchman Descartes. It is interesting to note that

Harriot was sent to America in 1585 by Sir Walter Raleigh, and

returned to England with a report of observations. He made the

first survey of Virginia and North Carolina, and constructed maps

of those regions.

It is strange that the line was used to denote division long before

any of the other symbols here mentioned were in use. This is, in

fact, one of the oldest signs of operation that we have. The Arabs,

as early as 1000 a.d., used both - and a/h to denote the quotient of

a and h. The symbol -^ did not occur until about 1630.

Equality has been denoted in a variety of ways. The word equals

was usually written out in full until about the year 1600, though

the two sides of an equation were written one over the other by the

Hindus as early as the twelfth century. The modern sign = was

probably introduced by the Englishman Recorde, in 1557, because,

he says, " Xoe. 2. thynges can be moare equalle " than two parallel

lines. This symbol was not generally accepted at first, and in its

place the symbols II, oc, and go are frequently met during the next

fifty years.

6 FIEST COUKSE IN ALGEBRA

4. The usefulness of symbols. Symbols enable one to

abbreviate ordinary language in the solution of problems.

For example : Three times a certain number is equal

to 20 diminished by 5. What is the number?

If n represents the number, the preceding statement and

' question can be written in symbols, thus :

3 ^ = 20 - 5.

^ = ?

The symbolic statement 3 n = 20 â€” 5 is called an equa-

tion and n the unknown number.

If 3^ = 20-5,

then 3?^ = 15,

and n = 5.

While the preceding example is very simple, it illus-

trates the algebraic method of stating and solving the

problem. The method is brief and direct, and its advan-

tages will become more apparent as the student progresses.

ORAL EXERCISES

Find the numerical value of x in the following equations :

1. So; = 18. 4. 7x = 42. 7. 6ir = 17 + 13

2. 4x = 28. 5. 4aj = 12 + 8. S. 4:X+3x = 35.

3. 5x = 30. 6. 3x = 4 + ll. 9. 6a:+2x = 32.

10. 5 a; + 4 cc = 45. 14. 4 a? â€” a^ = 15 â€” 6.

11. 4aj + 3x = 56. 15. 5x + 4aj - 2x = 10 + 4.

12. 7x + 2a; = 15 + 3. 16. 4:X-\-Sx-x = SS-S.

13. 9a;-3aj = 18-hl2. 17. 6x - x + 3x = ^2 -^ 6.

18. If one number is represented by x, what will represent

a number three times as great ?

INTEODUCTION 7

19. James had 3x cents. His brother had four times as

many. Eepresent the number of cents the brother had.

20. Paul's weight is 2 a? pounds, and his father weighs three

times as much. What will represent the father's weight ? the

weight of the two together ?

21. The area of a circle is 6 y. Eepresent the area of a

circle three times as large.

22. One number is twice a second, and the second is four

times the third. If x represents the third, what will represent

the second ? the first ?

23. One newsboy has three times as many papers as a

second, and the two together have as many as a third.

Represent in terms of x the number of papers each has.

EXAMPLE

The sum of two numbers is 112. The greater is three times

the less. What are the numbers ?

Solution. By the conditions of the problem,

greater number + less number = 112. (1)

Let I = the less number.

Then 3 I = the greater number.

Substituting these symbols in (1), we have

3 Z + Z = 112.

Collecting, 4 I = 112.Â»

Whence Z = i p = 28,

and 3 / = 3 X 28 = 84.

Therefore the greater number is 84 and the less 28.

We may verify the result by substituting 84 and 28 in the problem.

Thus 84 + 28 = 112,

and 84 = 3 . 28.

8 FIEST COUESE IN ALGEBEA

PROBLEMS

1. The sum of two numbers is 120. The greater is foui

times the less. Find each.

2. A certain number plus seven times itself equals 216.

Find the number.

3. One number is eight times another. Their sum is 72.

Find each.

4. The first of three numbers is twice the third, and the

second is four times the third. The sum of the three numbers

is 252. Find the numbers.

Hint. Let x = the third number. Then 2x = the first, and 4 x = the

Second.

5. The sum of three numbers is 117. The second is twice

the first, and the third is three times the second. Find each.

6. There are three numbers whose sum is 192. The first

is twice the second, and the third equals the sum of the other

two. Find the numbers.

7. The sum of three numbers is 312. The second is five

times the first, and the third is four times the second. Find

the numbers.

8. The sum of three numbers is 208. The second is three

times the first, and the third is the sum of the other two.

Find the numbers.

9. A man is three times as old as his son. The sum of

their ages is 44 years. Find the age of each.

10. The perimeter of a certain square is 160 feet. Find

the length of each side.

11. The perimeter of a certain rectangle is 216 feet. It is

three times as long as it is wide. Find its dimensions.

12. The perimeter of the rectangle formed by placing two

equal squares side by side is 258 inches. Find the side and

the perimeter of each square.

INTEODUCTION 9

5. Literal notation. In algebra numbers are represented

by one or more arable numerals, or by letters, or by both

combined.

Thus 3, 25, a, 2 b, 4 xi/, and 2 a: + 3 are algebraic symbols for

numbers.

Precisely what numbers 4 xy and 2x-\- S represent is

not known until the numbers for which x and y stand

are known. In one problem these letters may have values

quite different from those they have in another. To de-

vise methods of determining these values in the various

problems which arise is the principal aim of algebra.

6. Factors. A factor of a product is any one of the

numbers which multiplied together form the product.

Thus 3 ab means 3 times a times b. Here 3, a, and b are each fac-

tors of 3 ab. Similarly, the expression 4 (a + 6) means 4 times the

sum of a and b. Here 4 and a + 6 are factors of 4 (a + 6).

ORAL EXERCISES

1. Name the factors in 3 â€¢ 4 â€¢ 6, 2 xy, 3 abx, 4 abc.

In Exercises 2-5, replace a by 3 and ^ by 4 and find the

value of the resulting expression.

2. ab. 3. Sab. 4. 2ab. 5. 5ab.

7. Exponents. An exponent is an integer written at the

right of and above another number to show how many

times the latter is to be taken as a factor.

(Later this definition will be modified so as to include

fractions and other numbers as exponents.)

Thus 32 = 3 . 3 ; 5^ = 5 â€¢ 5 â€¢ 5. Also a* = ^ â€¢ a â€¢ a â€¢ a, and 4 x?/^ =

4: ' X ' y ' y â€¢ y. In a*, & is the exponent of a. If a is 4 and b is 3,

aft = 4^ = 4 â€¢ 4 â€¢ 4. The exponent 1 is usually not written.

10 riEST COUESE IK ALGEBEA

ORAL EXERCISES

1. What are the exponents in 2aG^? 3 a^c ? 5 aV ?

2. What is meant by x"? a^? b^? b' ?

3. 42 = ? 4. 52 = ? 5.2^ = ? 6. 3 . 52 = ?

In Exercises 7-14, replace a by 3 and 5 by 2 and find the

value of the result.

7. a^ 9. a\ 11. a'^b, 13. 2a\

8. a^-j-b\ 10. a^+^Â»l 12. a'^b^ 14. 5 a^^'-^.

8. Coefficients. If a number is the product of two

factors, either of these factors is called the coefficient of

the other in that product.

Thus in 4 x'^y, 4 is the coefficient of x^i/, y is the coefficient of 4 x^,

and 4 ?/ is the coefficient of x'^. The numerical coefficient 1 is usually

omitted. If a numerical coefficient other than 1 occurs, it is usually

written first. For instance, we write 5 x, not x 5.

The following examples illustrate the difference in mean-

ing between a coefficient and an exponent respectively:

a? = X ' X - X,

If a; =5 in each case, 3^ stands for the number 15,

while a? stands for 125. If x = 10 in each case, 8:2; =30,

while 0.^ = 1000.

ORAL EXERCISES

1 . What are the numerical coefficients in 4 a; ? 5a?? Sax?

4.ac? Sabc?

2. What is meant hj Sa? 4:X? 5c?

3. In 4 a^xt/y name the coefficient of a^xT/, xy, y, a'^x, and a^y.

9. Use of parentheses and radical signs. If two or more

numbers connected by signs of operation are inclosed in

parentheses, the entire expression is treated as a symbol

for a single number.

INTEODUCTION 11

Thus 3(6+4) means 3 â– 10, or 30; (17 - 2) -4- (8 - 3) means

15 -^ 5, or 3 ; (5 + 7)^ means 12^ or 144: and Q(x i- y) means six

times the sum of x and y.

As in arithmetic, the symbol for square root is V , and

the symbol for cube root is V .

The name radical sign is applied to all symbols like the

following: V , V , V , etc. The small figure in a radical

sign, like the 3 in v , is often called the index.

ORAL EXERCISES

Find the value of :

1. 2(3 + 4). 6. V9 + 7.

2. 4(7-2). 7. V32 + 42.

3. (4 + 3) (5 -2). 8. â– v^4(7-5).

4. (7 - 2) (8 -f- 3). 9. â– V(5 + 3)(6 + 2).

5. -y/S. 10. Ve^ + 81

Note. There has been a considerable variety in the symbols for

the roots of numbers. The symbol V was introduced in 1544 by the

German Stifel, and is a corruption of the initial letter of the Latin

word radix, which means **root." Before his time square root was

denoted by the symbol B, used nowadays by physicians on prescrip-

tions as an abbreviation for the word recipe. Thus a/5 would have

been denoted by B;^5. Some early writers used a dot to indicate

square root, and expressed V2 by â€¢ 2. The Arabs denoted the root

of a number by an arable letter placed directly over the number.

ORAL EXERCISES

1. What are the numerical coefficients in 2 a?? Sa^? Axy?

2 ah? 3Va?

2. What are the exponents in 3^^^? 4^^^^? 5aV? 5xhjz^?

3. What is the difference in meaning between the 4 in 4 a;

and that in x^?

4. What is meant hy 2x? 5a? Sa?

12 FIRST COUESE IN ALGEBEA

5. What is meant by 2 a^ ? ^x"?

6. What is meant by 3 (8 + 6)? 2(9-4)? (7 + 3)(8~2)?

(7 + 3)2? V3 + 6? V9-fl6? VlOO - 64 ? -v^35 - 8 ?

â– ^100 - 36 ?

7. What is the numerical value of each expression in

Exercise 6 ?

8. Read Exercises 1-16 on pages 15-16.

9. 3.52=? 13. (5-l)(8 + 3)=?

10. (8 + 2)^=? 14. 3(7-2)(5-3)=?

11. 7(6-1)=? 15. V52+122 = ?

12. (4 + 3)(5-f4)=? 16. V(10 + 8)(10-8) = ?

In Exercises 17-33, replace a by 3 and 5 by 4 and find

the value of the resulting expressions.

17. ^a''-2h. 21. 2a\ 25. ^o'bK 29. V3&2+16.

18. 3 ^2 + 8 a. 22. 2a%. 26. 4.ah^-h''. 30. â– \/4.a'-^lh,

19. -Vb. 23. 4a^l 27. V3^. 31. "V^.

20. R 24. 2a%\ 28. Va^ + b\ 32. "V^a^^ +15.

33. â€¢\/(^2_^52-17.

EXERCISES

Write, using algebraic symbols :

1. The sum of three times a and four times h.

2. Three times a subtracted from four times h. ,

3. The square of a subtracted from the square of b.

4. The cube of b subtracted from the square of a.

5. Two times a squared subtracted from three times a

squared.

6. The quotient of a and b.

7. The product of four times a squared and b.

INTEODUCTION 13

8. The sum of a and b divided by their product.

9. The product of a and 2h â€” c.

10. The product of a and the sum of h and c.

11. The result of subtracting a â€” h from Ix.

12. The sum of the square root of 5 a and the cube root

of 7^. .

13. The product oi x â€” y and the square root of Ix.

14. The square of the sum of a and h.

15. The square of h subtracted from a.

16. The quotient of three times a multiplied by the square

of h, and four times c multiplied by the cube of a.

17. The sum of the quotients of a and 3x, and 4y and c.

10. Order of fundamental arithmetical operations. If we

read the expression 6 + 4.9 â€” 12-^3 from left to right,

and perform each indicated operation as we come to its

symbol, we obtain successively 10, 90, 78, and a final

result of 26. If we perform the multiplication and

division first, the expression becomes 6 + 36â€”4, which

equals 38. These results show that the value of the

expression is determined largely by the order in which

the operations are performed. It is customary to ob-

$B 3Db MSB

A ikX^\j iÂ«^ ^LÂ»^ X \jr \

Mtsm-aiij^itifif Mt'IkifLiiVO

nm'KES-vjm-TO'urQH

/

J-^

GIFT OF

MATHEMATICAL TEXTS

FOR SCHOOLS

EDITED BT

PERCEY F. SMITH, Ph.D.

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD

SCIENTIFIC SCHOOL OF YALE UNIVERSITY

FIRST COURSE IN ALGEBRA

BY

HERBERT E. HAWKES, Ph.D.

PROFESSOR OF MATHEMATICS IN COLUMBIA UNIVERSITY

WILLIAM A. LUBY, A.B.

HEAD OF THE DEPARTMENT OF MATHEMATICS

KANSAS CITY POLYTECHNIC INSTITUTE

AND

FRANK C. TOUTOH, A.M. ' '

FORMERLY PRINCIPAL OF CENTRAL Hit*H SCEOOL

ST. JOSEPH, MISSOURI '

REVISED EDITION

GINN AND COMPANY

BOSTON â€¢ NEW YORK â€¢CHICAGO â€¢ LONDON

ATLANTA . DALLAS -COLUMBUS IAN FRANCISCO

1911

ENTERED AT STATIONERS' HALL

COPYRIGHT, 1909, 1910, 1917, BY

HERBERT E. HAWKES, WILLIAM A. LUBT

; , .': '. AND FRANK C. TOUTON

ALit BIGHTS RESERVED

' ' " A 321.7

GINN AND COMPANY â€¢ PRO-

PRIETORS â€¢ BOSTON â€¢ U.S.A.

O-C

PREFACE

In this revision of their ''First Course in Algebra" the authors

have in general followed the plan of that text in the order of

topics treated and in the method of their presentation.

The most important modification of the order of topics is

found in the transference of the work on Eatio and Propor-

tion to the last chapter in the book and the omission of the

chapter on the Highest Common Factor and Lowest Common

Multiple. The latter topic is treated in connection with the

related material on fractions, while the former is placed among

the Supplementary Topics at the end of the book.

Material for which there is no strong demand from teachers

has been omitted, and the entire work has been rewritten in

the interest of greater simplicity and directness of appeal. The

collections of exercises and problems arÂ© for the most part new

and contain a larger proportion of easy exereises with simple

results than the first edition.

A striking feature of the revision is the inclusion of a large

number of oral exercises in connection with the introduction of

each new idea or operation. It is the object of these exercises

to present the new concept in complete isolation from any com-

plication of notation or technique so that the student becomes

familiar with its content and bearing before he is asked to

make use of it in written work. These oral exercises may well

be taken up when the advance lesson is assigned, so that the

pupil may be certain that he understands the idea involved

in the new work before he leaves his instructor.

Another feature scarcely less important is the character and

position of the examples and hints. The aim has been to

459907

vi FIEST COUESE IN ALGEBRA

help the student at the exact point where he needs it and

to avoid the insertion of lengthy and difficult solutions before

they can be completely understood.

The definitions and axioms have been expressed in the sim-

plest language which is consistent with scientific accuracy.

Many definitions which are usually found in elementary texts

but which do not contribute to the clearness of the subject

are omitted.

The first presentation of the subject of graphs has been limited

to the study of the straight line and a few exercises of a com-

mercial or scientific character. These exercises not only have

a very definite human interest apart from their mathematical

value but also serve to familiarize the student with the kind

of graphs he will meet in his ordinary reading.

The first consideration in the treatment of radicals has been

the needs of the student for his later study of the quadratic

equation and for his work in geometry.

Frequently a student's knowledge of algebra is limited to a

greater or less facility in the use of the rules of operation â€”

to mere technique. To obviate this result the development of

the problem work in this text has received full and careful

attention.

The authors have received suggestions of great value from

many teachers in all parts of the country, for which they

extend their thanks. They are under especial obligation to

Mr. E. L. Brown, of Denver, Colorado, Professor H. E. Cobb,

of Chicago, Illinois, and to Mr. L. A. Pultz, of Eochester,

New York, for helpful criticism.

CONTENTS

CHAPTER PAGE

I. Introduction (Sects. 1-11) 1

XL Positive and Negative Numbers (Sects. 12-19) . 18

III. Addition (Sects. 20-25) 33 ^

IV. Simple Equations (Sects. 26-28) 39

V. Subtraction (Sects. 29-30) 49 "^

VI. Identities and Equations of Condition (Sects.

31-34) 54

Vll. Parentheses (Sects. 35-36) 64

Vlll. Multiplication (Sects. 37-44) - - ^O - '

IX. Parentheses in Equations (Sects. 45-46) ... 79

X. Division (Sects. 47-49) 87 -^

XI. Equations and Problems (Sects. 50-51) .... 95

XII. Important Special Products (Sects. 52-55) . . 105

XIII. Factoring (Sects. 56-66) 113

XIV. Solution of Equations by Factoring (Sects.

67-70) 137

XV. Fractions (Sects. 71-81) 148

XVI. Equations containing Fractions (Sects. 82-88) . 175 '

XVII. Graphical Representation (Sects. 89-94) ... 200

XVIII. Linear Systems (Sects. 95-100) 217

XIX. Square Root (Sects. 101-102) 240

XX. Radicals (Sects. 103-114) 250

XXI. Quadratic Equations (Sects. 115-117) .... 270

XXII. Ratio and Proportion (Sects. 118-125) .... 282

Supplementary Topics (Sects. 126-130) 293

Index 807

vii

ILLUSTRATIONS

PAGE

JOHN WALLIS 48

SIR WILLIAM ROWAN HAMILTON 70

SIR ISAAC NEWTON 100

JOHN NAPIER 186

REN:^ DESCARTES 210

FRANCOIS VIETA 268

I3f

FIRST COUESE IN ALGEBRA

CHAPTER I

INTRODUCTION

1. The numbers and symbols of arithmetic. The simple

operation of counting employs the numbers we call in-

tegers. To represent these integers and the other numbers

with which it deals, arithmetic uses the nunierals 0, 1,

2, 3, 4, 5, 6, 7, 8, and 9. Operations on the numbers of

arithmetic are indicated by the symbols +, â€” , X, and -^.

The operation of division applied to integers gives rise

to fractions. With these two kinds of numbers, integers

and fractions, the student's work in arithmetic is mainly

carried on.

2. Symbols of algebra. Symbols are employed far more

extensively in algebra than in arithmetic, and many new

ideas arise in connection with their meaning and their use.

Some symbols represent numbers, others indicate opera-

tions upon them, others represent relations between them,

and still others represent kinds of numbers with which

arithmetic does not deal. Letters as well as arable numer-

als are used to represent numbers. The following symbols

of operation, -f ? â€” ? X, and -r-, have the same meaning as in

arithmetic. The sign of multiplication is usually replaced

by a dot or omitted.

For example, 3 x a is written 3 â€¢ a, or 3 a, and 2 x a x b is

written 2 ab. Also a -^ 6 is often written - .

b

2 FIEST COURSE IK ALGEBRA

The sign = is read equals^ or is equal to. As the need

for them arises, other symbgls will be introduced.

3. The use of letters to represent numbers. The use of

the letters of the alphabet to represent numbers is the

most striking difference between arithmetic and algebra.

In arithmetic we speak thus : If the sides of a triangle

are 6, 7, and 9 inches respectively, its perimeter is 6 + 7 + 9,

or 22 mches. The corresponding statement in algebra is:

If the sides of a triangle are* a, ^, and c inches respectively,

and its perimeter is p inches, then p = a-\-h-{- c. Here the

second statement is true for every triangle, while the first

is not true for every triangle.

Similarly: If a rectangle is 8 inches by 12 inches, its

perimeter is 8+12 + 8+12, or 40 inches. And if a rec-

tangle is I inches long and w inches wide and if p denotes

its perimeter in inches, then p â€” l-{-W'\-l-\-w^ or 2/+2?^^.

Here, again, the arithmetical statement is particular and

applies to one rectangle only, while the algebraic state-

ment is general; that is, it is true for all rectangles.

The gain in power which the general symbolic language

of algebra affords over the particular numerical language

of arithmetic constitutes one of the most important advan-

tages of the algebraic method. As the student progresses he

will meet with many illustrations of this feature of algebra.

The purpose of the following exercises is to familiarize

the student with the use of letters in the place of numbers.

ORAL EXERCISES

1. What numerical value has 5 a when a is 3 ? when a is 5 ?

when a is 10 ?

2. What numerical value has 6Â«^ + 2& when a is 2 and 5 is 4?

3. Express h-\-?nn in seconds if h and m stand. for the

number of seconds in an hour and in a minute respectively.

INTKODUCTION 3

4. Express lOy + 4/ in inches if y and / stand for the

number of inches in a yard and in a foot respectively.

5. If g' and d represent the number of cents in a quarter

and in a dime respectively, express 4 5' + 6 c? in cents.

6. If ^ and h represent the number of pounds in one ton

and in one hundredweight respectively, express 4 i^ + 6 /z, in

pounds.

7. 3 ir + 5 ic = how many a; ? 9. 5 â€¢ 2^ + 10 â€¢ ?^ = (?) ?^.

8. ^x-\-hx = {^)x. 10. 5a^ + 3cc + 6a: = ?

11. 3x-2aj + 7a;-5^ = ?

12. 7 books + 3 chairs + 2 books + 5 chairs = (?) books +

(?) chairs.

13. 8 books + 4 chairs â€” 2 chairs + 4 books = (?) books +

(?) chairs.

14. 6 books + 7 chairs â€” 3 books â€” 2 chairs = (?) books +

(?) chairs.

15. 55 + 4c + 8Z>-2c = (?)5+(?)c.

16. 6^ + 3ic + 4^> + 8a^ = (?)^Â» + (?)a^.

17. ^x-\-2h^-Zx-h-\-x = (^.)h^(^,)x,

18. 2a^ + 2 + 3cc + 4 = (?)a^ + ?

19. 4cc + 2 + 3x + 2-^ + 8 = ?

20. iÂ«+:r + 2 + cc + cc + 2 = ? 22. 5a+18-3a-7=?

21. 7i4-7z + l + 7i + 2= ? 23. 8;r-3+18-5ir=?

24. 4i^;- 8 + 3t^ + 20 = ?

25. What value has 4 aj + 3 when a:; = 2 ? when a? = 7 ?

26. What value has 3 a; â€” 4 when aj = 3 ? when x = 21

27. The side of a square is 5 inches long. What is its area?

its perimeter ?

4 FIRST COURSE IN ALGEBRA

28. The side of a square is s inches long. What represents

its perimeter ? its area ?

29. The base of a rectangle is 12 feet, and its altitude is

4 feet. What is its perimeter ? its area ?

30. If ^ represents the number of feet in the base of a rec-

tangle and a the number of feet in its altitude, what is its

perimeter ? its area ?

31. A rectangle is twice as long as it is wide. Let w repre-

sent the number of inches in its width. Then express, in terms

of w, (a) the length ; (h) the perimeter ; (c) the area.

32. A man is three times as old as his son. If s denotes

the number of years in the son's age, what will represent the

father's age ?

33. A father is 28 years older than his son. If s represents

the son's age in years, what will represent the father's age ?

34. A rectangle is 24 inches longer than it is wide. Let b

represent the width in feet. Then represent the length and the

perimeter in terms of b and numbers.

35. A rectangle is 16 feet narrower than it is long. If w

represents the width in feet, what will conveniently represent

the length ? the perimeter ?

36. A rectangle is 4 feet longer than twice its width. Express

the width, the length, and the perimeter in terms of a letter, or

a letter and numbers.

Origin of symbols. Many of the symbols that are in common use

in algebra at the present time have histories which not only are

interesting in themselves, but which also serve to indicate the slow

and uncertain development of the subject. It is often found that

symbols which seem without meaning represent some abbreviation

or suggestion long since forgotten, and that operations and methods

which we find hard to master have sometimes recLuired hundreds of

years to perfect.

INTEODUCTION 6

In the early centuries there were practically no algebraic symbols

in common use ; one wrote out in full the words plus, minus, equals,

and the like. But in the sixteenth century several Italian mathema-

ticians used the initial letters p and m for + and â€” . Some think

that our modern symbol â€” came into use through writing the initial

m so rapidly that the curves of the letter gradually flattened out,

leaving finally a straight line. The symbol + may have originated

similarly in the rapid writing of the letter p. But in the opinion of

others these symbols were first used in the German warehouses of

the fifteenth century to mark the weights of boxes of goods. If a

lot of boxes, each supposed to weigh 100 pounds, came to the ware-

house, the weight would be checked, and if a certain box exceeded

the standard weight by 5 pounds, it was marked 100 + 5 ; if it

lacked 5 pounds, it was marked 100 â€” 5. Though the first book to

use these symbols was published in 1489, it was not until about

1630 that they could be said to be in common use.

Both of the symbols for multiplication given in the text were

first used about 1630. The cross was used by two Englishmen,

Oughtred and Harriot, and was probably an adaptation of the letter

Xy which is found some years earlier. The dot is first found in the

writings of the Frenchman Descartes. It is interesting to note that

Harriot was sent to America in 1585 by Sir Walter Raleigh, and

returned to England with a report of observations. He made the

first survey of Virginia and North Carolina, and constructed maps

of those regions.

It is strange that the line was used to denote division long before

any of the other symbols here mentioned were in use. This is, in

fact, one of the oldest signs of operation that we have. The Arabs,

as early as 1000 a.d., used both - and a/h to denote the quotient of

a and h. The symbol -^ did not occur until about 1630.

Equality has been denoted in a variety of ways. The word equals

was usually written out in full until about the year 1600, though

the two sides of an equation were written one over the other by the

Hindus as early as the twelfth century. The modern sign = was

probably introduced by the Englishman Recorde, in 1557, because,

he says, " Xoe. 2. thynges can be moare equalle " than two parallel

lines. This symbol was not generally accepted at first, and in its

place the symbols II, oc, and go are frequently met during the next

fifty years.

6 FIEST COUKSE IN ALGEBRA

4. The usefulness of symbols. Symbols enable one to

abbreviate ordinary language in the solution of problems.

For example : Three times a certain number is equal

to 20 diminished by 5. What is the number?

If n represents the number, the preceding statement and

' question can be written in symbols, thus :

3 ^ = 20 - 5.

^ = ?

The symbolic statement 3 n = 20 â€” 5 is called an equa-

tion and n the unknown number.

If 3^ = 20-5,

then 3?^ = 15,

and n = 5.

While the preceding example is very simple, it illus-

trates the algebraic method of stating and solving the

problem. The method is brief and direct, and its advan-

tages will become more apparent as the student progresses.

ORAL EXERCISES

Find the numerical value of x in the following equations :

1. So; = 18. 4. 7x = 42. 7. 6ir = 17 + 13

2. 4x = 28. 5. 4aj = 12 + 8. S. 4:X+3x = 35.

3. 5x = 30. 6. 3x = 4 + ll. 9. 6a:+2x = 32.

10. 5 a; + 4 cc = 45. 14. 4 a? â€” a^ = 15 â€” 6.

11. 4aj + 3x = 56. 15. 5x + 4aj - 2x = 10 + 4.

12. 7x + 2a; = 15 + 3. 16. 4:X-\-Sx-x = SS-S.

13. 9a;-3aj = 18-hl2. 17. 6x - x + 3x = ^2 -^ 6.

18. If one number is represented by x, what will represent

a number three times as great ?

INTEODUCTION 7

19. James had 3x cents. His brother had four times as

many. Eepresent the number of cents the brother had.

20. Paul's weight is 2 a? pounds, and his father weighs three

times as much. What will represent the father's weight ? the

weight of the two together ?

21. The area of a circle is 6 y. Eepresent the area of a

circle three times as large.

22. One number is twice a second, and the second is four

times the third. If x represents the third, what will represent

the second ? the first ?

23. One newsboy has three times as many papers as a

second, and the two together have as many as a third.

Represent in terms of x the number of papers each has.

EXAMPLE

The sum of two numbers is 112. The greater is three times

the less. What are the numbers ?

Solution. By the conditions of the problem,

greater number + less number = 112. (1)

Let I = the less number.

Then 3 I = the greater number.

Substituting these symbols in (1), we have

3 Z + Z = 112.

Collecting, 4 I = 112.Â»

Whence Z = i p = 28,

and 3 / = 3 X 28 = 84.

Therefore the greater number is 84 and the less 28.

We may verify the result by substituting 84 and 28 in the problem.

Thus 84 + 28 = 112,

and 84 = 3 . 28.

8 FIEST COUESE IN ALGEBEA

PROBLEMS

1. The sum of two numbers is 120. The greater is foui

times the less. Find each.

2. A certain number plus seven times itself equals 216.

Find the number.

3. One number is eight times another. Their sum is 72.

Find each.

4. The first of three numbers is twice the third, and the

second is four times the third. The sum of the three numbers

is 252. Find the numbers.

Hint. Let x = the third number. Then 2x = the first, and 4 x = the

Second.

5. The sum of three numbers is 117. The second is twice

the first, and the third is three times the second. Find each.

6. There are three numbers whose sum is 192. The first

is twice the second, and the third equals the sum of the other

two. Find the numbers.

7. The sum of three numbers is 312. The second is five

times the first, and the third is four times the second. Find

the numbers.

8. The sum of three numbers is 208. The second is three

times the first, and the third is the sum of the other two.

Find the numbers.

9. A man is three times as old as his son. The sum of

their ages is 44 years. Find the age of each.

10. The perimeter of a certain square is 160 feet. Find

the length of each side.

11. The perimeter of a certain rectangle is 216 feet. It is

three times as long as it is wide. Find its dimensions.

12. The perimeter of the rectangle formed by placing two

equal squares side by side is 258 inches. Find the side and

the perimeter of each square.

INTEODUCTION 9

5. Literal notation. In algebra numbers are represented

by one or more arable numerals, or by letters, or by both

combined.

Thus 3, 25, a, 2 b, 4 xi/, and 2 a: + 3 are algebraic symbols for

numbers.

Precisely what numbers 4 xy and 2x-\- S represent is

not known until the numbers for which x and y stand

are known. In one problem these letters may have values

quite different from those they have in another. To de-

vise methods of determining these values in the various

problems which arise is the principal aim of algebra.

6. Factors. A factor of a product is any one of the

numbers which multiplied together form the product.

Thus 3 ab means 3 times a times b. Here 3, a, and b are each fac-

tors of 3 ab. Similarly, the expression 4 (a + 6) means 4 times the

sum of a and b. Here 4 and a + 6 are factors of 4 (a + 6).

ORAL EXERCISES

1. Name the factors in 3 â€¢ 4 â€¢ 6, 2 xy, 3 abx, 4 abc.

In Exercises 2-5, replace a by 3 and ^ by 4 and find the

value of the resulting expression.

2. ab. 3. Sab. 4. 2ab. 5. 5ab.

7. Exponents. An exponent is an integer written at the

right of and above another number to show how many

times the latter is to be taken as a factor.

(Later this definition will be modified so as to include

fractions and other numbers as exponents.)

Thus 32 = 3 . 3 ; 5^ = 5 â€¢ 5 â€¢ 5. Also a* = ^ â€¢ a â€¢ a â€¢ a, and 4 x?/^ =

4: ' X ' y ' y â€¢ y. In a*, & is the exponent of a. If a is 4 and b is 3,

aft = 4^ = 4 â€¢ 4 â€¢ 4. The exponent 1 is usually not written.

10 riEST COUESE IK ALGEBEA

ORAL EXERCISES

1. What are the exponents in 2aG^? 3 a^c ? 5 aV ?

2. What is meant by x"? a^? b^? b' ?

3. 42 = ? 4. 52 = ? 5.2^ = ? 6. 3 . 52 = ?

In Exercises 7-14, replace a by 3 and 5 by 2 and find the

value of the result.

7. a^ 9. a\ 11. a'^b, 13. 2a\

8. a^-j-b\ 10. a^+^Â»l 12. a'^b^ 14. 5 a^^'-^.

8. Coefficients. If a number is the product of two

factors, either of these factors is called the coefficient of

the other in that product.

Thus in 4 x'^y, 4 is the coefficient of x^i/, y is the coefficient of 4 x^,

and 4 ?/ is the coefficient of x'^. The numerical coefficient 1 is usually

omitted. If a numerical coefficient other than 1 occurs, it is usually

written first. For instance, we write 5 x, not x 5.

The following examples illustrate the difference in mean-

ing between a coefficient and an exponent respectively:

a? = X ' X - X,

If a; =5 in each case, 3^ stands for the number 15,

while a? stands for 125. If x = 10 in each case, 8:2; =30,

while 0.^ = 1000.

ORAL EXERCISES

1 . What are the numerical coefficients in 4 a; ? 5a?? Sax?

4.ac? Sabc?

2. What is meant hj Sa? 4:X? 5c?

3. In 4 a^xt/y name the coefficient of a^xT/, xy, y, a'^x, and a^y.

9. Use of parentheses and radical signs. If two or more

numbers connected by signs of operation are inclosed in

parentheses, the entire expression is treated as a symbol

for a single number.

INTEODUCTION 11

Thus 3(6+4) means 3 â– 10, or 30; (17 - 2) -4- (8 - 3) means

15 -^ 5, or 3 ; (5 + 7)^ means 12^ or 144: and Q(x i- y) means six

times the sum of x and y.

As in arithmetic, the symbol for square root is V , and

the symbol for cube root is V .

The name radical sign is applied to all symbols like the

following: V , V , V , etc. The small figure in a radical

sign, like the 3 in v , is often called the index.

ORAL EXERCISES

Find the value of :

1. 2(3 + 4). 6. V9 + 7.

2. 4(7-2). 7. V32 + 42.

3. (4 + 3) (5 -2). 8. â– v^4(7-5).

4. (7 - 2) (8 -f- 3). 9. â– V(5 + 3)(6 + 2).

5. -y/S. 10. Ve^ + 81

Note. There has been a considerable variety in the symbols for

the roots of numbers. The symbol V was introduced in 1544 by the

German Stifel, and is a corruption of the initial letter of the Latin

word radix, which means **root." Before his time square root was

denoted by the symbol B, used nowadays by physicians on prescrip-

tions as an abbreviation for the word recipe. Thus a/5 would have

been denoted by B;^5. Some early writers used a dot to indicate

square root, and expressed V2 by â€¢ 2. The Arabs denoted the root

of a number by an arable letter placed directly over the number.

ORAL EXERCISES

1. What are the numerical coefficients in 2 a?? Sa^? Axy?

2 ah? 3Va?

2. What are the exponents in 3^^^? 4^^^^? 5aV? 5xhjz^?

3. What is the difference in meaning between the 4 in 4 a;

and that in x^?

4. What is meant hy 2x? 5a? Sa?

12 FIRST COUESE IN ALGEBEA

5. What is meant by 2 a^ ? ^x"?

6. What is meant by 3 (8 + 6)? 2(9-4)? (7 + 3)(8~2)?

(7 + 3)2? V3 + 6? V9-fl6? VlOO - 64 ? -v^35 - 8 ?

â– ^100 - 36 ?

7. What is the numerical value of each expression in

Exercise 6 ?

8. Read Exercises 1-16 on pages 15-16.

9. 3.52=? 13. (5-l)(8 + 3)=?

10. (8 + 2)^=? 14. 3(7-2)(5-3)=?

11. 7(6-1)=? 15. V52+122 = ?

12. (4 + 3)(5-f4)=? 16. V(10 + 8)(10-8) = ?

In Exercises 17-33, replace a by 3 and 5 by 4 and find

the value of the resulting expressions.

17. ^a''-2h. 21. 2a\ 25. ^o'bK 29. V3&2+16.

18. 3 ^2 + 8 a. 22. 2a%. 26. 4.ah^-h''. 30. â– \/4.a'-^lh,

19. -Vb. 23. 4a^l 27. V3^. 31. "V^.

20. R 24. 2a%\ 28. Va^ + b\ 32. "V^a^^ +15.

33. â€¢\/(^2_^52-17.

EXERCISES

Write, using algebraic symbols :

1. The sum of three times a and four times h.

2. Three times a subtracted from four times h. ,

3. The square of a subtracted from the square of b.

4. The cube of b subtracted from the square of a.

5. Two times a squared subtracted from three times a

squared.

6. The quotient of a and b.

7. The product of four times a squared and b.

INTEODUCTION 13

8. The sum of a and b divided by their product.

9. The product of a and 2h â€” c.

10. The product of a and the sum of h and c.

11. The result of subtracting a â€” h from Ix.

12. The sum of the square root of 5 a and the cube root

of 7^. .

13. The product oi x â€” y and the square root of Ix.

14. The square of the sum of a and h.

15. The square of h subtracted from a.

16. The quotient of three times a multiplied by the square

of h, and four times c multiplied by the cube of a.

17. The sum of the quotients of a and 3x, and 4y and c.

10. Order of fundamental arithmetical operations. If we

read the expression 6 + 4.9 â€” 12-^3 from left to right,

and perform each indicated operation as we come to its

symbol, we obtain successively 10, 90, 78, and a final

result of 26. If we perform the multiplication and

division first, the expression becomes 6 + 36â€”4, which

equals 38. These results show that the value of the

expression is determined largely by the order in which

the operations are performed. It is customary to ob-

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