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4. Subtract 4 x + 3y-2 from 5x, and add x − 3y+2 to the result.

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6. What expression must be added to 4 a − 3 m +x to produce 7 a-5 m + 3x ?

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7. What is the value of (-3) - (+2) − (−3)+(−5) -(-2)?

8. Collect 3a-b+4c, 2c-3d-x, a -4x-d, 5c+2 a +3x, and 4 db-7c.

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9. Simplify x-1-{x-1-[x-1- x − 1 − x]}.

10. Subtract 4x-3y+11 from unity, and add 5x-3y+12 to the result.

11. Combine the m-terms, the n-terms, and the x-terms in the following, inclosing the resulting coefficients in parentheses: am + 3 bn + x + m + n· ax + 2 n − cx.

12. A given minuend is 7x+12 y - 7, and the corresponding difference, 4x-y+2. Find the subtrahend.

13. To what expression must you add 2 a-3 cm to produce 5 a+7c-9m?

14. Subtract 2 a-7x+3 from the sum of 3x+2a, 4 a -10 x, 5 a-7, and -4x-11 a.

15. From (a+c) y +(m + n) z take (a — c) y — (m — n)z.

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16. Subtract x + 17 y- z from 12x+3z and add the result to 3x + (y +11 z) — 4 y.

17. Add the sum of 4x+7c and 2x+3c to the remainder that results when x +4 c is subtracted from 5c - 11 x.

18. From (a+4)x+(a+3) y subtract (a+1)x+(a+2)y. 19. Add 3x + 2 m -1 and 2x-3 m +7, and subtract the sum from 6x — m + 6.

20. Simplify a-1-a+1+[a−1-a-a — 1 — (a − 1) —

(1 − a)].

21. From 7m+3x-12 take the sum of 3x+7 and 2 m-3y-3.

22. What expression must be added to a+b+c to produce 0?

23. What expression must be subtracted from 0 to give a+b+c?

24. From what expression must 4x-7m+10 be subtracted to give a remainder of 3x+6 m −4n+2?

25. Inclose the last four terms of a in a parenthesis preceded by a minus sign.

·8 b + 4 m − 5 n + 7 x

26. Simplify and collect a+[-2m - 4a+x

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27. If x+7y-9 is subtracted from 0, what expression results?

28. Is the sum of [−7+(− 2)− (−3)]+[−{−2+(−1)}] positive or negative?

29. Prove that [3-(-2)-(-1)]+[(-2)+(-1)-(-3)]

30. Simplify and collect

+[−2−(−1)+(−5)]=0.

10-[9—8—(7—6—{5—4—3 — 2} − 1)].

31. Show that

2 − (− 3 + a − 1) − 3 + (− 2 − a + 1) = −2 a.

32. Collect the coefficients of x, y, and z in

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1 − [− { − ( − 1+1 − a) −1} −1]— a.

34. What expression must be subtracted from a give (a+[ -x-(-2a-x + 1) −3]) ?

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CHAPTER IV

MULTIPLICATION

51. Multiplication is an abbreviated form of addition.

Thus,

3 x 4 = 4+4+4. 5α= a + a + a + a + a.

For the purpose of arithmetic multiplication has been defined as the process of taking one quantity (the multiplicand) as many times as there are units in another quantity (the multiplier). This definition will not hold true when the multiplier is negative or fractional. Hence, the need for the following definition:

52. Multiplication is the process of performing on one factor (the multiplicand) the same operation that was performed upon unity to produce the other factor (the multiplier). The result of a multiplication is a product.

Illustration:

The Integral Multiplier.

3 x 5 means that 5 is taken three times in a sum. Or, 3x5=5+5+5. By the same process the multiplier was obtained from unity, for 3=1+1+1.

The Fractional Multiplier.

Obtained from unity a multiplier, 2 = 1 +1 +1 +1 +1. For unity was taken twice as an addend and of unity taken three times as an addend.

In like manner, 22 × 5 = 5 + 5 + 1⁄2 + 1⁄2 + 1⁄2 = 10 + 4 = 134.

THE NUMBER PRINCIPLES OF MULTIPLICATION

53. The Law of Order. Algebraic numbers may be multiplied

in any order.

In general:

abba.

Numerical Illustration: 3 x 5 = 5 x 3.

54. The Law of Grouping. The product of three or more algebraic numbers is the same in whatever manner the numbers are grouped.

In general:

abc= a(bc)=(ab)c = (ac)b.

Numerical Illustration: 2.3.5=2(3-5)=(23)5 (2.5)3.

=

55. The Law of Distribution. The product of a polynomial by a monomial equals the sum of the products obtained by multiplying each term of the polynomial by the monomial.

In general:

a(x + y + z) = ax + ay + az. Numerical Illustration: 2(3 + 4 + 5) = 6 + 8 + 10.

As in the case of addition, no rigid proof of these laws is ordinarily required until the later practice of elementary algebra.

Here, as in addition, the law of order is frequently called the commutative law, and the law of grouping is called the associative law.

SIGNS IN MULTIPLICATION

Upon the definition of multiplication we may establish the results for all possible cases in which the multiplicand or multiplier, or both, are negative numbers.

(1) A positive multiplier indicates a product to be added.

(2) A negative multiplier indicates a product to be subtracted.

(1) The Positive Multiplier. Expressed with all signs (multiplier =+3):

(+3) × (+ 5) = + 5 + 5 + 5 = + 15.
(+3) × (− 5) = − 5 − 5 − 5 — — 15. ̧

(2) The Negative Multiplier. Expressed with all signs (multiplier = -3):

=

- ( + 5 + 5 + 5) =

=

(+15)= 15.

(−3) × (+ 5) =
(-3) × (-5)=(-5-5-5) = − (− 15) = + 15.

Comparing the four cases in the ordinary form of multiplica

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56. Like signs in multiplication give a positive result.

57. Unlike signs in multiplication give a negative result.

Oral Drill (See also page 388)

Give the products in the following, each with its proper

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58. The coefficient of a term in a product of two algebraic expressions is the product of the coefficients in the given multiplier and multiplicand.

The principle is established by means of the Law of Grouping.

For the coefficient of mn in the product of am times en:

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