Elementary Algebra and Geometry for Schools

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Positive and Negative Numbers

ILLUSTRATIVE EXAMPLES: Subtraction of signed numbers

1. Subtract + 5 from + 12.

Solution. We write:

We must find the number which, when added to + 5, will produce + 12. In football language this could mean, " If Jim gained 5 yd. on the first play, what must he do on the second play in order to gain a total of 12 yd. in two tries?" We know that he must make a gain of 7 yd. This is shown on the number scale as follows:

The length of the black arrow is 7 units. Since it is directed to the right, the answer is + 7.

2. Subtract -3 from -8.

Solution. We write:

We must find the number which, when added to -3, will produce -8. In football language this could mean, "If Jack lost  3 yd. in running with the ball and if, in two attempts to advance the ball, he lost a total of 8 yd., what was his gain or loss when he carried the ball the second time?" We know that he must have lost 5 yd. This is shown on the number scale as follows:

The length of the black arrow is 5 units. Since it is directed to the left, the answer is -5.

3. Subtract + 7 from + 5.

Solution. We write:

We must find the number which, when added to + 7 will produce + 5. The diagram of the number scale shows that the answer is -2:

4. Subtract 4 from + 6.

We must find the number which, when added to -4 will produce + 6. The diagram of the number scale shows that the answer is + 10.

5. Subtract + 3 from -2.

We must find the number which, when added to +3 will produce -2. The diagram of the number scale shows that the answer is -5.

EXERCISES: Subtracting signed numbers

Referring to the number scale given below, perform each of the following subtractions (-):

Answers : 1. 7. 2. 8. 3. -13. 4. -17. 5. -7. 6. 13. 7. 10. 8. 1. 9. 14. 10. 0. 11. 6. 12. -11. 13. -4. 14. 0. 15. 11. 16. -6. 17. -6. 18. 0. 19.-12. 20. 0. 21. 5. 22. -6. 23. 8. 24. -2. 25. -12. 26. -6. 27. 7. 28. 3. 29. 9. 30. 3. 31. 18. 32. -16. 33. -1. 34. 11. 35.0.

Rule for the subtraction of signed numbers.

 We will now reexamine the illustrations of algebraic subtraction starting in the 1st example of this page. These examples are rewritten here for our convenience. Compare each subtraction example with the addition example written to its right. In example 1, notice that subtracting + 5 from +12 produces the same result as adding its opposite (-5) to + 12. In example 2, subtracting -3 from -8 gives the same result as adding its opposite (+ 3) to -8. We observe that subtracting a signed number from a quantity is the same as adding its opposite to that quantity.

• To subtract one signed number from another, change the sign of the subtrahend and then proceed as in addition.

ILLUSTRATIVE EXAMPLES: Applying the rule for subtraction of signed numbers

1. Subtract 18 from 27.

Solution. We can write this statement algebraically as

(27) -(18)

Changing the sign of the subtrahend, and changing to addition

(27) + (-18) = 9

Or we may write:

where we mentally change the sign of the 18 to minus and then add.

2. Subtract -21 from 17.

Solution. We can write this statement algebraically as

(17) -(-21)

Changing the sign of the subtrahend, and changing to addition,

(17) + (21) = 38

Or we may write:

where we mentally change the sign of the -21 to plus and then add.

3. Subtract -7xy from -3xy.

Solution. In pages of Chapter 2 we learned to subtract like terms by subtracting their numerical coefficients and multiplying this number by their common literal part. To subtract the numerical coefficients, we apply the rule for the subtraction of signed numbers.

Hence:

EXERCISES: Subtracting signed numbers

Applying the rule for the subtraction of signed numbers, perform each of the following subtractions:

Answers : 1. 11. 2. 52. 3. -38. 4. -32. 5. 3. 6. 46. 7. 29. 8. -47. 9. 56. 10. 0. 11. -16. 12. -35. 13. -29. 14. 0. 15. 88. 16. -148. 17. 6x. 18. -6y. 19. -29p. 20. 37 m. 21. -6a. 22. + 6b. 23. -74c. 24. 75x2. 25. 0. 26. 5ax. 27. 66y2. 28. 5xy. 29. 0. 30. -151a2 .31. -76pq. 32. 115b3

Answers : 33. -38. 34. -19. 35. 33. 36. -29. 37. -54. 38. -30. 39. 118. 40. -24. 41. 100. 42. -41. 43. 44. 44. 90. 45. -8. 46. 84. 47. 5. 48. 4x. 49. -8x. 50. -4y. 51. 72z. 52. -90x. 53. -2ax. 54. -2y. 55. 0. 56. -4a. 57. 17b. 58. -6x2. 59. -58x. 60. -9y. 61.0. 62. -38x.

Algebraic addition. We interpret (5a) + (-3a) to mean that the terms 5a and -3a are to be added. The plus sign is the sign of operation. We may omit the positive sign of operation in this case and write the above expression as 5a -3a. This expression 5a -3a, therefore, not only means that 3a is to be subtracted from 5a, but also indicates that -3a is to be added to 5a. As we have seen, the subtraction of a number from a quantity is equivalent to the addition of the "opposite" of that number to the given quantity.

Thus in performing additions and subtractions of like terms, we frequently consider the subtraction of a positive number to be simply the addition of a negative number. The expression 7x +2x -8x -5x +3x -12x may be considered to be the sum of six terms, some of which are negative; that is,

7x +2x + (-8x) + (-5x) +3x + (-12x)

We are now adding positive and negative numbers. If we proceed to add the terms, in order, we have: 7x plus 2x is 9x; 9x plus -8x is x; x plus -5x is -4x; -4x -plus 3x is -x; -x plus -12x is -13x, which is the final sum.

This method of collecting like terms is called algebraic addition.

EXERCISES: Collecting like terms

Collect like term in each of the following:

 1. 3a -5a +6a   2. 9x +2x -15x 3. 4b -3b -6b  4. -2y +y -3y 5. 8x -9x +5x 6. x -7x +4x 7. 5x + 4x -9x 8. 6b -10b +3b 9. 3y -8y -5y 10. 4a + 5a -12a

Answers : 1. 4a. 2. -4x. 3. -5b. 4. -4y. 5. 4x. 6. -2x. 7. 0. 8. -b. 9. -10y. 10. -3a.

Answers : 11. 2x. 12. -9x. 13. -9a. 14. -2ab. 15. -9x2 .16. -3a2. 17. -2a -6b. 18. -2x2. 19. -4b2 -5b. 20. -xy -12x + 2y.

Multiplication of signed numbers. You will remember that 3 X 5 means the sum of three fives. Similarly. (3) (-5) means the sum of three minus 5's, or:

(3)(-5) = (-5) +(-5) +(-5) =-15

The expression (-3)(5) can be written as (5) (-3), which means the sum of five minus 3's, or:

(-3) (5) =(-3)+(-3)+(-3)+(-3)+(-3) = -15

Let us now summarize our results thus far. We know that:

• I (3) (5) = 15
• II (3)(-5) = -15
• III (-3)(5) = -15

You will notice that we can derive equation II from equation I by changing the sign of one factor on the left side of the equal sign (changing the factor 5 to -5) and changing the sign of the right side of the equation (changing 15 to -15). Similarly, we can derive equaition III from equation I by changing the sign of one factor on the left side of the equal sign (changing the fa tor 3 to -3) and changing the sign of the right side of the equation (changing 15 to -15).

• Changing the sign of one factor of a product has the effect of changing the sign of the product.

Let us now apply this principle to equation II and change the sign of the factor 3:

• II (3)(-5) =-15
• IV (-3)(-5)=15

If we apply the same principle to equation III and change the sign of the factor 5, we derive the same equation IV:

• III (-3)(5) =-15
• IV (-3)(-5) = 15

Observe that in equation I the product of two positive numbers is positive; in equation IV the product of two negative numbers is positive; and in equations II and III the product of a positive number and a negative number is negative. Therefore we may conclude that:

• The product of two numbers with like signs (both positive or both negative) is positive. The product of two numbers with unlike signs (one positive and the other negative) is negative.

ILLUSTRATIVE EXAMPLES: Applying the rules for the multiplication of signed numbers

1. Multiply -6 by 4.

Solution. The numbers -6 and 4 have unlike signs, hence their product will be negative; we may write:

2. Find the product (-8) (-7).

Solution. The numbers -8 and -7 have like signs (both are negative), hence their product will be positive; we may write:

3. Multiply 6ax by -7a2

• Solution. Following the methods explained to multiply monomials and applying the rules for the multiplication of signed numbers, we may write:

EXERCISES: Multiplying signed numbers

Find the product of each of the following;

Rules for the division of signed numbers.

Division and multiplication are inverse operations. We know that in arithmetic if we multiply two numbers, then their product divided by either of the two numbers will give as a quotient the other number. For example, since 5· 2 = 10, then 10 ÷ 5 = 2 and 10 ÷ 2 = 5. This same relationship holds true for signed numbers.

We can learn the rules of signs in division of signed numbers by studying the following examples:

1. (+ 5)(+ 2) = + 10; therefore, (+ 10) ÷ (+ 2) = + 5.

2. (+ 5)(-2) =-10; therefore, (-10) ÷ (-2) = + 5.

3. (-5)(+ 2) =-10; therefore, (-10) ÷ (+ 2) =-5.

4. (-5)(-2) = + 10; therefore, (+ 10) ÷ (-2) = -5.

• The quotient of two numbers with like signs (both positive or both negative) is positive. The quotient of two numbers with unlike signs (one positive and the other negative) is negative.

ILLUSTRATIVE EXAMPLES: Division of signed numbers

1. Find the quotient : -63/9

Solution. The numbers -63 and 9 have unlike signs, hence their quotient is negative. We may write:

2. Divide-24 by .

Solution. The numbers -24 and have like signs; hence their quotient is positive. To divide a number by a fraction, we invert the fraction and proceed as in multiplication; hence:

3. Divide 84x3 by -4x2 Solution. We learned before to divide monomials.

Following the methods explained there and applying the rules for the division of signed numbers, we may write:

EXERCISES: Dividing signed numbers

Find each of the following indicated quotients:

Answers : 1. -2. 2. -4. 3. 8. 4. -7. 5. 11. 6. 9. 7. -7. 8. -3. 9. 13. 10. 3. 11. -2x. 12. -3a. 13. 4b2. 14. -4x. 15. -7y. 16. -3b2. 17. -9a2. 18. 2y2. 19. -x. 20. -7. 21. -15. 22. -17. 23. -18. 24. 24. 25. -75. 26. -4xy2. 27. 9ab2. 28. -4n4. 29. 10r3s4. 30. -9y2.

Parentheses preceded by a plus sign or a minus sign. In pages in Chapter 2 we learned that parentheses are used to indicate that the expression within the parentheses is to be considered as a single quantity. Thus the expression 20 + (7 + 4) means that the sum of 7 and 4 is to be added to 20; that is

20 + (7 + 4) = 20 + (11) = 31

We notice that we can obtain the same result by adding 7 to 20 and then adding to this sum the number 4; that is

20 + (7 + 4) = 20 + 7 + 4 = 31

• If a pair of parentheses is preceded by a plus sign, we may remove the parentheses without changing the sign of any term within the parentheses.

Let us consider the expression 20 -(7 + 4). This means that the sum of 7 and 4 is to be subtracted from 20; that is

20 -(7 + 4) = 20 -(11) = 9

We notice that we can obtain the same result by first subtracting 7 from 20 to get 13, and then subtracting 4 from 13; that is

20 -(7 + 4) = 20 -7 -4 = 9

Let us consider this same type of problem from another point of view. You learned to remove parentheses in an expression such as 2(a +b) by multiplying each term of the binomial a +b by 2; that is, 2(a +b) = 2a + 2b. Let us now consider the example -(2a -3b). You will remember that -x means -1x; similarly, -(2a -3b) means -1(2a -3b). Multiplying 2a by -1, we obtain the first product -2a; multiplying -3b by -1, we obtain the second product 3b; that is

-(2a -3b) = -1(2a -3b) = -2a+3b

• If a pair of parentheses is preceded by a minus sign, we may remove the parentheses providing we change the sign of every term that was within the parentheses.

ILLUSTRATIVE EXAMPLES: Removing parentheses

1. Simplify (a -b) -(2b +3a).

Solution. Given example (a -b) -(2b + 3a)

Removing parentheses a -b -2b -3a

Collecting like terms -2a -3b Answer

2. Simplify 2x(a -b) -5(3ax -2bx).

Solution. Given example 2x(a -b) -5(3ax -2bx)

Multiplying each term within the first parentheses by 2x, and multiplying each term within the second parentheses by-5

2ax -2bx -15ax + 10bx

Collecting like terms -13ax +8bx Answer

EXERCISES: Removing parentheses

Simplify each of the following by removing parentheses and then collecting like terms.

Answers : 1. -2a -b. 2. -3x + y. 3. -a -b + c. 4. -4x -3y. 5. -x2 + 5x -2. 6.-y2 -3y + 4.

Answers : 7. 3ax -4bx. 8. 8-4x + 3x2. 9. -5a2 + 3ab -b2. 10. x2 + x + 1. 11. 3x + 7. 12. 1. 13. 2ab -b2. 14. -y2 -1. 15. 2a -2b. 16. 4x2 -9x. 17. 11x -5y. 18. 0. 19. 2x2. 20. 3a + b. 21. x -12y. 22. a -18b. 23. 4x2 -24x. 24. -3a2 + 8. 25. -y. 26. 3x + 2. 27. 5x2 + 13x + 2. 28. 18ax -14ay -x -y. 29. -ax -bx. 30.0.

Enclosing terms within parentheses. From your work in the preceding exercises you have seen that the expression 3x + (2a -b) can be written as 3x + 2a -b. Similarly, if we reverse the order, we can write 3x + 2a -b as 3x + (2a -b). In like manner, 3x -(2a -b) can be written as 3x -2a + b; then, reversing the order, we can write 3x -2a + b as 3x -(2a -b). From these examples you will see that we can state the following rules:

• Terms of an algebraic expression may be placed within parentheses preceded by a plus sign if the sign before each term is left unchanged. Terms of an algebraic expression may be placed within parentheses preceded by a minus sign if the sign before each term is changed.

ILLUSTRATIVE EXAMPLES: Enclosing terms within parentheses

1. Enclose the expression 3a -4b + c within parentheses preceded by a minus sign.

Solution. Given example 3a -4b+c

We first write parentheses preceded by a minus sign  -(        )

We now place each term within parentheses, changing its original sign. Thus

2. Enclose the first two terms of the following expression within parentheses preceded by a plus sign, and the last two terms within parentheses preceded by a minus sign: 3x -4b -5a +y.

Solution. Given expression

EXERCISES: Enclosing terms within parentheses

In each of the ex. 1-3, enclose the terms within parentheses preceded by a plus sign.

1. x +2y -z            2. 3a -7b +c          3. 2r -s -t

In each of the ex. 4-6, enclose the terms within parentheses preceded by a minus sign.

4. 5x -7y +z           5. -x +y -z             6. -x +7y +8z

In each of the ex. 7-12, enclose the last two terms in parentheses preceded by a plus sign.

7. x +y +z               9. 3x -y +2             11. -2 a +3 b -c

8. a +b -c                10. 4a -b -6           12. 4x +y -5z

In each of the ex. 13-18, enclose the last two terms in parentheses preceded by a minus sign.

13. a +b +c             15. 2x -y +z            17. 12 +5x -y

14. r +s -t                16. 5a -2b -3c         18. 2 -4x +3y

In each of ex. 19-24, enclose the first two terms within parentheses preceded by a plus sign, and the last two terms within parentheses preceded by a minus sign.

19. x -y +a -b           22. y -x -2a -4b

20. 2x +y -3a +b      23. 2a -b -5x +y

21. x+2y+2a+3b      24. a -2b -6x -4y

Answers : 1. (x + 2y -z). 2. (3a -7b + e). 3. (2r -s -t). 4. -(-5x + 7y -z). 5. -(x -y + z). 6. -(x -7y -8z). 7. x + (y + z). 8. a + ( b - c). 9. 3x + (-y +2). 10. 4a +( -b -6). 11. -2a + (3b -c). 12. 4x + (y -5z). 13. a -(-b -c). 14. r -(-s + t). 15. 2x -(y -z). 16. 5a -(2b + 3c). 17. 12 -(-5x + y). 18. 2 -(4x -3y). 19. (x -y) -(-a + b). 20. (2x + y) -(3a -b). 21. (x + 2y) -(-2a -3b) . 22. (y -x) -(2a + 4b). 23. (2a -b) -(5x -y). 24. (a -2b) -(6x + 4y).

ORAL EXERCISES

1. Is a +b -c equal to a -(b +c)?

2. Is a -b +c -d equal to a -(b -c +d)?

3. Is a -b +c equal to a -(b -c)?

4. Is a -b -c equal to a -(b -c)?

5. Is x -y equal to -(y -x)?

6. Is x -2y +z equal to (x +z) -2y?

7. Is 2x + y -z equal to y -(z + 2x)?

8. Is 3 + (x -y) equal to 3 -(y -x)?

9. Is x -(a+ b) equal to x + (a -b)?

10. Is (2a + b) -(7-3x) equal to (b + 2a) + (3x -7)?

Answers : 1. no. 2. yes. 3. yes. 4. no. 5. yes. 6. yes. 7. no. 8. yes. 9. no. 10. yes.