NOTE 1.

66

66

In Step 2, S15" means subtract 15 from both sides of the previous equation." In Step 3, "C. T." means "collect terms in the previous equation."

NOTE 2. - Step 2 of the foregoing solution may be performed mentally as in Steps 2 and 3 of the following solution.

Example 2.

Solution. - 1.

2. S4.

3. S2t.

4. D3.

Check.

Substitute 7 for t in the original equation.

Does 5x7+4=2x7+25? Does 35+4=14+25?
Yes. Therefore, 7 is the correct root of the equation.

In the following examples you will need to use also the

112. More complicated problems can now be solved by means of equations.

Example. -The length of a certain rectangle is 5 in. more than 3 times the width. The perimeter of the rectangle is 42 in. What are the length and width of the rectangle?

Solution. 1. Let w the number of inches in the width.

Then

=

3w+5=the number of inches in the length.

2. Then w+3w+5+w+3w+5=the perimeter,

or

3.

4. Sto

5. Dg.

and

Check.

Yes.

8w+10=the perimeter.

:. 8w+10=42, since the perimeter is 42 in. 8w=32.

w=4 in., the width,

3w+5=17 in., the length.

Does 4+17+4+17=42?

Therefore the solution is correct.

NOTE 1. Your letter must always equal a number or a number of something.

NOTE 2. Always answer the questions asked in the problem. In this problem you were asked to find the length and the width. Do not stop when you have "found the unknown number," but go back to your problem to see whether you have found all the numbers you were told to find.

EXERCISE 93

1. Twice a certain number added to 17 gives 31. What is the number?

2. Three times a certain number added to 5 gives 23. What is the number?

3. If 11 be subtracted from three times a certain number, the result is 25. What is the number?

4. If 27 be subtracted from six times a certain number, the result is 27. What is the number?

5. The length of a certain rectangle is three times its width. The perimeter is 64 in. What are the length and the width?

6. The length of a certain parallelogram is two times its width. If 6 is added to the perimeter, the result is 60 in. What are the length and the width?

7. The length of a certain rectangle is 14 in. more than its width. The perimeter is 48 in. What are the length and the width?

8. There are two numbers whose sum is 70. The larger is four times as large as the smaller. What are the numbers?

9. The larger of two numbers is 11 times the smaller. The sum of the numbers is 156. What are the numbers?

10. The larger of two numbers is 8 times the smaller. The difference between the numbers is 91. What are the numbers?

11. A triangle has three unequal sides. The second side is 8 in. more than the shortest side; the third side is 15 in. more than the shortest side. The perimeter is 50 in. What is the length of each side?

12. One of two line segments is five times as long as the other. If 20 inches be added to the shorter, and 8 inches be taken from the longer, the two results are equal. Find the lengths of the two segments.

13. Five times a certain number diminished by 7 is 23. What is the number?

14. Eight times a certain number exceeds four times the same number by 17. What is the number?

15. The lower base of a trapezoid is 3 times as long as the upper base. If 5 in. be subtracted from the lower base and 7 added to the upper base, the two results are equal. How long are the bases?

XIII. POSITIVE AND NEGATIVE NUMBERS

MULTIPLICATION AND DIVISION

113. Multiplication of positive and negative numbers. The multiplier is the number by which you multiply. The multiplicand is the number which you multiply. The product is the result.

In the following examples, the sign X is to be read "times."

Thus, 6×(-3) means 6 times (−3).

Example 1. How much is (+4) X(+3)?

Solution.

(+4)X(+3)=(+3)+(+3)+(+3)+(+3)=+12.

:: (+4)X(+3)=+12.

Example 2. How much is (+4)X(-3)?

Example 3.

Solution.

(+4)×(−3) = (−3)+(−3)+(−3)+(−3) = −12. (+4)X(-3)=-12.

How much is (-4)×(+3)?

- In arithmetic, 4×3 and 3×4 are the same.

That

is, the multiplier and multiplicand can be interchanged. If we assume that we may do this also with positive and negative numbers, then

But
So

(-4)X(+3) should be the same as (+3) ×(−4).
(+3) ×(−4) =(−4)+(−4)+(−4) = −12.
(−4) × (+3) = −12.

NOTE. When multiplying by a negative number (-4), it appears that the result (-12) can be obtained by multiplying as if the multiplier were positive (+4), and then changing the sign of the product.

Example 4. How much is (-4)×(−3)?

Solution. Using the suggestion in the note above, first multiply −3 by (+4); this gives 12. Then change the sign of the result; this gives +12.

.. (-4)X(-3)=+12.

Collecting the results of Examples 1, 2, 3, 4:

1. (+4)X(+3) = 12.

2. (+4)X(-3)=-12.

3. (-4)X(+3)=-12.

4. (-4)X(-3)= +12.

Rule. To multiply one signed number by another:

1. Find the product of their absolute values. (See all four above.)

2. Make the product positive if the multiplicand and multiplier have like signs. (See 1 and 4 above.)

3. Make the product negative if the multiplicand and multiplier have unlike signs. (See 2 and 3 above.)

EXERCISE 94

1. Multiply the following numbers by +4:
+5, +6, +8, +11, +9.

2. Multiply the numbers in Example 1 by -2.
3. Multiply the numbers in Example 1 by -5.
4. Multiply the following numbers by +3:

-6, -4, -12, -7, -20.

5. Multiply the numbers in Example 4 by +6.
6. Multiply the numbers in Example 4 by -5.
7. Multiply the numbers in Example 4 by -8.
8. Multiply the following numbers by +3:
+9, -7, -11, +10, -12.

9. Multiply the numbers in Example 8 by -4. 10. Multiply the numbers in Example 8 by -7. 11. Multiply the numbers in Example 8 by +10. 12. Multiply the following numbers by 24: