21. Check each of the preceding exercises by performing the multiplication first, then the addition or subtraction. Which is the shorter way?

Find the following sums and differences :

67. bh+b'h.

68. r-.lr-.01r.

40. 4r-r.

41.

28. 18n-2n.

29. 52−2.

30. 19y-y.

x+x+x.

42. 9.13+13. 17.

43. a+a-a.

44. 456+18b+b-10b.

45. 3.8a-2.1a+.04a.

46.

07.05.7.

57. cn-xn. 62. y+ry.

58. ar- -na.
59. ay-2y.
60. 9m+mx.

63. ry+r.

64. c+rc.

65. c-rc.

61. ad-ca.

66. p+prt.

69. 2mp-3p.

71. rx+xy-xz.

70. ac-bcnc. 72. ap-bpp.

In expressions such as 3a+5+7a+8 the terms do not all contain a common factor. Such expressions are simplified by combining the terms which have a common factor. Thus, 3a+5+7a+8=10a +13.

The expression ax+bcd+a't is composed of three terms. The first term is ax, the second is bcd, and the third is a2t.

If a term is separated into two factors either of the two factors may be called the coefficient of the other. Thus the term bcd may be separated into the two factors b and cd. Then b is the coefficient of cd, and cd is the coefficient of b. Like terms are terms having a common factor. Thus 3x and 5x are like terms; also ax and bx.

Rule. To add like terms, multiply their common factor by the sum of its coefficients.

EXAMPLE 1. Add 4·7+7·6+7.

SOLUTION. The common factor of these three terms is 7. Its coefficient in the first term is 4, in the second term is 6, and in the third term is 1. The sum of the coefficients is 11.

Exercise 7

1. Read the terms of the expression 4x2+7xy+3yx. What is the coefficient of x2 in the first term? Of xy in the second term? Of x in the third term? Of x in the second term?

2. In the expression 2x+3x what are the factors of the first term? Of the second term? What is the common factor of the two terms? What is its coefficient in the first term? In the second term? Find the sum of the coefficients. Multiply this sum by the common factor.

3. Answer the questions of the preceding exercise for the expression xa+cx.

Combine the terms in the following:

7. Equations. The statement that two expressions are equal is called an equation. The two expressions are called the members of the equation. Thus in the equation 3x+6= 2x+8, 3x+6 is the left member and 2x+8 is the right member.

The formulas that we have used are examples of equations.

8. Questions asked by equations. Many questions of elementary arithmetic may be stated in the form of equations. Thus the question, What number added to 4 gives 11? may be written, 4+?=11, or 4+n=11, where n stands for the unknown number.

Exercise 8

State the following as equations and give the answers:

1. What number added to 9 gives 16?

2. What number subtracted from 19 gives 6?

3. Forty-four is 8 more than what number?

4. Six times what number is 45?

5. One-half of what number is 19?

6. Thirteen is one-eighth of what number?

7. What must be subtracted from 42 to get 12?

8. What must 8 be multiplied by to get 92?

In the following the number to be found is represented by x.

Many of the equations with which the pupil will have to deal should be thought of as questions. The equation, 2n+1 =7, asks the question, For what value of n is 2n+1 equal to 7? The answer is 3. We test the truth of this answer by substituting 3 for n in the equation. This gives 6+1=7, or 7=7, which shows that the answer is correct.

This testing the answer by substituting is called checking. If, in this example, we say that the answer is 4 instead of 3, and then substitute 4 for n in the equation, we get 8+1=7, or 9=7, which is not true.

The number 3 is said to satisfy the equation 2n+1=7.

A number that satisfies an equation is called a root of the equation.

The process of finding the root of an equation is called solving the equation.

We shall learn how to solve certain kinds of equations. The letter in an equation whose value we wish to find is called the unknown. In the equation 2n+1=7, n is the

unknown.

When an equation is solved for an unknown the unknown will stand by itself in one member of the equation, and its value will be given by the other member of the equation. Thus, when the equation 2n+1=7 is solved for n, we have the equation n=3. The pupil should note that when an equation is solved for an unknown, this unknown is found in only one member.

9. The equation as a balance. The equation may be thought of as a balance. Thus, in the equation x+2=14,

x+2

FIG. 6

14

x+2 and 14 may be thought of as weights which balance when x+2 is placed in one scale pan and 14 in the other. (See Figure 6.)

When one set of weights balances another set, if a weight is added to or taken from one scale pan, a weight of equal size must be added to or taken from the other pan if the resulting weights are still to balance. Thus, if we subtract 2 from x+2 we must subtract 2 from 14 if we wish the resulting weights to balance each other. This means that x= = 12. Suppose that 3x pounds balance 15 pounds, that is, 3x=15. Then of 3x pounds will balance of 15 pounds. That is, x pounds will balance 5 pounds, or x=5.

Suppose that x pounds balance 7 pounds, that is, x=7. Then 2 times pounds will balance 2×7 pounds. That is, x pounds will balance 14 pounds, or x=14.