Herbert E. (Herbert Edwin) Hawkes.

First course in algebra online

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Online LibraryHerbert E. (Herbert Edwin) HawkesFirst course in algebra → online text (page 1 of 18)
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COPYRIGHT, 1909, 1910, 1917, BY

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

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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 ar© 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



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 —
to mere technique. To obviate this result the development of
the problem work in this text has received full and careful

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.



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














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 +, — , 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 ? — ? 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 • a, or 3 a, and 2 x a x b is

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



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 — 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.


1. What numerical value has 5 a when a is 3 ? when a is 5 ?
when a is 10 ?

2. What numerical value has 6«^ + 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.


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

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

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 — 2 chairs + 4 books = (?) books +
(?) chairs.

14. 6 books + 7 chairs — 3 books — 2 chairs = (?) books +
(?) chairs.

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

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

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

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

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

20. i«+: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; — 4 when aj = 3 ? when x = 21

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


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.


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 — . Some think
that our modern symbol — 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 — 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.


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 — 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.


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? — a^ = 15 — 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 ?


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.


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.»

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.



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

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.


5. Literal notation. In algebra numbers are represented
by one or more arable numerals, or by letters, or by both

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

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).


1. Name the factors in 3 • 4 • 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 • 5 • 5. Also a* = ^ • a • a • a, and 4 x?/^ =
4: ' X ' y ' y • y. In a*, & is the exponent of a. If a is 4 and b is 3,
aft = 4^ = 4 • 4 • 4. The exponent 1 is usually not written.



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^+^»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.


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


Thus 3(6+4) means 3 ■ 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.


Find the value of :

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

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

3. (4 + 3) (5 -2). 8. ■v^4(7-5).

4. (7 - 2) (8 -f- 3). 9. ■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 • 2. The Arabs denoted the root
of a number by an arable letter placed directly over the number.


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?


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 ?
■^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. ■\/4.a'-^lh,

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

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

33. •\/(^2_^52-17.


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

6. The quotient of a and b.

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


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

9. The product of a and 2h — c.

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

11. The result of subtracting a — h from Ix.

12. The sum of the square root of 5 a and the cube root
of 7^. .

13. The product oi x — 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 — 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 + 36—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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Online LibraryHerbert E. (Herbert Edwin) HawkesFirst course in algebra → online text (page 1 of 18)