Properties of Algebra

Source: Beginning and Intermediate Algebra, an open source (CC-BY) textbook by Tyler Wallace

In algebra we will often need to simplify an expression to make it easier to use. There are three basic forms of simplifying which we will review here.

World View Note: The term "Algebra" comes from the Arabic word al-jabr which means "reunion". It was first used in Iraq in 830 AD by Mohammad ibnMusa al-Khwarizmi.

The first form of simplifying expressions is used when we know what number each variable in the expression represents. If we know what they represent we can replace each variable with the equivalent number and simplify what remains using order of operations.

Example 30

p(q + 6) when p=3 and q =5 Replace p with 3 and q with 5
(3)((5) + 6) Evaluate parenthesis
(3)(11) Multiply
Solution 33

Whenever a variable is replaced with something, we will put the new number inside a set of parenthesis. Notice the 3 and 5 in the previous example are in parenthesis. This is to preserve operations that are sometimes lost in a simple replacement. Sometimes the parenthesis won't make a difference, but it is a good habbit to always use them to prevent problems later.

Example 31

x + zx(3 - z)(x / 3) when x=-6 andz =-2 Replace all x's with 6 and z's with 2

(-6) + (-2)(-6)(3-(-2))((-6)/3) Evaluate parenthesis

-6 + (-2)(-6)(5)(-2) Multiply left to right

-6 + 12(5)(-2) Multiply left to right

-6 + 60(-2) Multiply

-6 - 120 Subtract

Solution: -126

It will be more common in our study of algebra that we do not know the value of the variables. In this case, we will have to simplify what we can and leave the variables in our final solution. One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be 3xy and -7xy or 3a2b and 8a2b or - 3 and 5. If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. This is shown in the following examples.

Example 32

5x - 2y - 8x + 7y Combine like terms 5x - 8x and -2y +7y
Solution -3x + 5y

Example 33

8 x 2-3x + 7 - 2x2 + 4x - 3 Combine like terms 8x2 -2x2 and -3x + 4x and 7 - 3
Solution: 6x2 + x + 4

As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term, the sign always stays with the term. A final method to simplify is known as distributing. Often as we work with problems there will be a set of parenthesis that make solving a problem difficult, if not impossible. To get rid of these unwanted parenthesis we have the distributive property. Using this property we multiply the number in front of the parenthesis by each term inside of the parenthesis.

DistributiveProperty: a(b + c) = ab + ac

Several examples of using the distributive property are given below.

Example 34

4(2x - 7) Multiply each term by 4
Solution: 8x - 28

Example 35

-7(5x - 6) Multiply each term by -7
Solution: -35 + 42

In the previous example we again use the fact that the sign goes with the number, this means we treat the -6 as a negative number, this gives (-7)(-6) = 42, a positive number. The most common error in distributing is a sign error, be very careful with your signs!

It is possible to distribute just a negative through parenthesis. If we have a negative in front of parenthesis we can think of it like a -1 in front and distribute the -1 through. This is shown in the following example.

Example 36

-(4x - 5y + 6) Negative can be thought of as -1
-1(4x - 5y + 6) Multiply each term by -1
Solution: -4x + 5y - 6

Distributing through parenthesis and combining like terms can be combined into one problem. Order of operations tells us to multiply (distribute) first then add or subtract last (combine like terms). Thus we do each problem in two steps, distribute then combine.

Example 37

5 + 3(2x - 4) Distribute 3, multipling each term
5 + 6x - 12 Combine like terms 5 - 12
Solution: -7 + 6x

Example 38

3x - 2(4x - 5) Distribute -2, multilpying each term
3x - 8x + 10 Combine like terms 3x - 8x
Solution: -5x + 10

In the previous example we distributed - 2, not just 2. This is because we will always treat subtraction like a negative sign that goes with the number after it. This makes a big difference when we multiply by the - 5 inside the parenthesis, we now have a positive answer. Following are more involved examples of distributing and combining like terms.

Example 39

2(5x-8) - 6(4x+3) Distribute 2 into first parenthesis and - 6 into second
10x - 16 - 24x - 18 Combine like terms 10x - 24x and -16 - 18
Solution: -14x - 34

Example 40

4(3x-8) - (2x-7) Negative (subtract) in middle can be thought of as -1
4(3x - 8) - 1(2x - 7) Distribute 4 into first parenthesis, -1 into second
12x - 32 - 2x + 7 Combine like terms 12x - 2x and -32 + 7
Solution 10x - 25

Practice - Properties of Algebra

Evaluate each using the values given.

Expression Use
1. p + 1 + q - m m = 1, p= 3, q = 4
2. y2 + y - z y =5, z =1
3. p - pq/6 p = 6 and q = 5
4.
6 + z - y
---------
3
y = 1, z = 4
5. c2 - (a - 1) a = 3 and c = 5
6. x + 6z-4y x = 6, y = 4,z = 4
7. 5j + kh/2 h = 5, j = 4,k = 2
8. 5(b+a) + 1 + c a = 2, b = 6, c = 5
9.
4 - (p - m)
---------- + q
2
m = 4, p = 6, q = 6
10. z + x - (12)3 x = 5, z = 4
11. m + n + m + n/2 m = 1 and n = 2
12. 3 + z - 1 + y - 1 y = 5, z = 4
13. q - p - (q - 1 - 3) p = 3, q = 6
14. p + (q - r)(6 - p) p = 6, q = 5, r = 5
15. y - [4 - y - (z - x)] x = 3, y = 1, z = 6
16. 4z - (x + x - (z - z)) x = 3, z = 2
17. k × 32 - (j + k) - 5 j = 4, k = 5
18. a3(c2 - c) a = 3, c = 2
19.
zx - z - 4 + x 6
x = 2, z = 6
20. 5 + qp + pq - q p = 6, q = 3

Combine Like Terms:

21. r - 9 + 10
22. -4x + 2 - 4
23. n + n
24. 4b + 6 + 1 + 7b
25. 8v + 7v
26. -x + 8x
27. -7x - 2x
28. -7a - 6 + 5
29. k - 2 + 7
30. -8p + 5p
31. x - 10 - 6x + 1
32. 1 - 10n -10
33. m - 2m
34. 1 - r - 6
35. 9n - 1 + n + 4
36. -4b + 9b

Distribute:

37. -8(x - 4)
38. 3(8v + 9)
39. 8n(n + 9)
40. -(-5 + 9a)
41. 7k(-k + 6)
42. 10x(1 + 2x)
43. -6(1 + 6x)
44. -2(n + 1)
45. 8m(5 - m)
46. -2p(9p - 1)
47. -9x(4 - x)
48. 4(8n - 2)
49. -9b(b - 10)
50. -4(1 + 7r)
51. -8n(5 + 10n)
52. 2x(8x - 10)

Simplify:

53. 9(b + 10) + 5b
54. 4v - 7(1 - 8v)
55. -3x(1 - 4x) - 4x2
56. -8x + 9(-9x + 9)
57. -4k2 - 8k(8k + 1)
58. -9 - 10(1 + 9a)
59. 1 - 7(5 + 7p)
60. -10(x - 2) - 3
61. -10 - 4(n - 5)
62. -6(5 - m) + 3m
63. 4(x + 7) + 8(x + 4)
64. -2r(1 + 4r) + 8r(-r + 4)
65. -8(n + 6) - 8n(n + 8)
66. 9(6b + 5) - 4b(b + 3)
67. 7(7 + 3v) + 10(3 - 10v)
68. -7(4x - 6) + 2(10x - 10)
69. 2n(-10n + 5) - 7(6 - 10n)
70. -3(4 + a) + 6a(9a + 10)
71. 5(1 -6k) + 10(k - 8)
72. -7(4x + 3) - 10(10x + 10)
73. (8n2 - 3n) - (5 + 4n2)
74. (7x2 - 3) - (5x2 + 6x)
75. (5p - 6) + (1 - p)
76. (3x2 - x) - (7 - 8x)
77. (2 - 4v2) + (3v2 + 2v)
78. (2b - 8) + (b - 7b2)
79. (4 - 2k2) + (8-2k2)
80. (7a2 + 7a) - (6a2 + 4a)
81. (x2 - 8) + (2x2 - 7)
82. (3 - 7n2) + (6n2 + 3)

Answers

1. 7 2. 29 3. 1
4. 3 5. 23 6. 14
7. 25 8. 46 9. 7
10. 8 11. 5 12. 10
13. 1 14. 6 15. 1
16. 2 17. 36 18. 54
19. 7 20. 38 21. r + 1
22. -4x - 2 23. 2n 24. 11b + 7
25. 15v 26. 7x 27. -9x
28. - 7a - 1 29. k + 5 30. -3p
31. -5x - 9 32. -9 - 10n 33. - m
34. -5 - r 35. 10n + 3 36. 5b
37. -8x + 32 38. 24v + 27 39. 8n2 + 72n
40. 5 - 9a 41. -7k2 + 42k 42. 10x + 20x2
43. -6 - 36x 44. -2n - 2 45. 40m - 8m2
46. -18p2 + 2p 47. -36x + 9x2 48. 32n - 8
49. -9b2 + 90b 50. -4 - 28r 51. -40n - 80n2
52. 16x2 - 20x 53. 14b + 90 54. 60v - 7
55. -3x + 8x2 56. -89x + 81 57. -68k2 - 8k
58. -19 - 90a 59. -34 - 49p 60. -10x + 17
61. 10 - 4n 62. -30 + 9m 63. 12x + 60
64. 30r - 16r2 65. -72n - 48 - 8n2 66. -42b + 45 + 4b2
67. 79 - 79v 68. -8x + 22 69. -20n2 + 80n - 42
70. -12 + 57a + 54a2 71. -75 - 20k 72. -128x - 121
73. 4n2 - 3n - 5 74. 2x2 - 6x - 3 75. 4p - 5
76. 3x2 + 7x - 7 77. -v2 + 2v + 2 78. -7b2 + 3b - 8
79. -4k2 + 12 80. a2 + 3a 81. 3x2 - 15
82. -n2 + 6