Solving Equations containing Multiplication and Division

Contributed by:

NEO Wed, Feb 02, 2022 01:13 PM UTC

This pdf includes the following topics:-
One step algebra problems with addition and subtraction
Arithmetic Algebra
Examples and many more.

1. SOLVING LINEAR EQUATIONS
Goal: The goal of solving a linear equation is to find the value of the variable that will make the
statement (equation) true.
Method: Perform operations to both sides of the equation in order to isolate the variable.
Addition and Subtraction Properties of Multiplication and Division Properties of
Equality: Equality:
Let , , and represent algebraic expressions. Let , , and represent algebraic expressions.
1. Addition property of equality: 1. Multiplication property of equality:
If , If ,
then then
2. Subtraction property of equality: 2. Division property of equality:
If , If ,
then then

(provided 0)

Clearing Fractions or Decimals in an Equation:
When solving an equation with fractions or decimals, there is an option of clearing the fractions or
decimals in order to create a simpler equation involving whole numbers.
1. To clear fractions, multiply both sides of the equation (distributing to all terms) by the LCD of all
the fractions.
2. To clear decimals, multiply both sides of the equation (distributing to all terms) by the lowest
power of 10 that will make all decimals whole numbers.
Steps for Solving a Linear Equation in One Variable:
1. Simplify both sides of the equation.
2. Use the addition or subtraction properties of equality to collect the variable terms on one side of
the equation and the constant terms on the other.
3. Use the multiplication or division properties of equality to make the coefficient of the variable
term equal to 1.
4. Check your answer by substituting your solution into the original equation.
Note: If when solving an equation, the variables are eliminated to reveal a true statement such as,
13 13, then the solution is all real numbers. This type of equation is called an identity. On the
other hand, if the variables are eliminated to reveal a false statement such as, 7 3, then there is no
solution. This type of equation is called a contradiction. All other linear equations which have only one
solution are called conditional.
A. 5 2 Check:
5 5 7 5 2
7
2 2 (Solution Checks)
B. 3 7 Check:
3 3 10 3 7
10 7 7 (Solution Checks)

2. C. 3 24 Check:
3 24 38 24

3 3 24 24 (Solution Checks)
8
D. 7 28 Check:
7 28 74 28

7 7 28 28 (Solution Checks)
4

E. 4 Check:

2 6
· · (Multiply by the reciprocal) 4
3 1
6 4 4 (Solution Checks)
F. 2 Check:
!
5 " 10
· 25 2
1 5 5
" 10 2 2 (Solution Checks)
Identity Example: Contradiction Example:
G. 2 6 3 2 H. 5 3 4 2
2 6 3 6 5 3 4 8
2 6 2 6 5 3 5 8
2 2 5 5
6 6 True Statement 3 8 False Statement
Solution: all real numbers No Solution
I. 10 5 3 4 2 7
10 5 3 12 2 14 Check:
10 5 26 10 56 36 4 26 7
10 30 32 213
10 6 26 20 6 26
10 10 20 20 (Solution Checks)
6 36
#$ #
#
#
6
$ $
J. 2 (LCD is 10) K. 0.05 0.25 0.2
!
2 4 1000.05 0.25 0.2100
10 210 5 25 20
5 2
10 2 10 4 25 25
· · 20 !$ !
1 5 1 2
! !
2 2 5 4 20 1
2 4 5 20 20 Check:
3 16 20 0.051 0.25 0.2
16 16
$
0.05 0.25 0.2

0.2 0.2 (Solution Checks)

3. LINEAR EQUATIONS PRACTICE
1. 4 4 37.

.
68.
)
45
- ! !
2. 6 7
38. 2 69. 36
)
3. 47
$ 39.
$

$
70. 4 4 40
4. 9 #

71. 9 7 34
26
2 4 8
40.
5. ! '
* 72. "68
-
6. 14 3 2 41.
( !
73. 10 6
8 3 19

42.
# ) 74. 2 4 1
8. 6 9 $+
43. 75. 9 9 9
9. 12 ' '
$ $
$+!

' 2
3 2 6 44. 76.
10. # !
$ $
11. 32 8 12 45. 6
$ $ 77. 2
- ' )
12. 46 2 0 $ $
$
6
$
46. 78.
)

13. 3 2 6 15 $ $
! 1
3
79.
- )
4 2 3
47.
14.
!
!

"46
27 46 2
80.
15. 48. )

4 6 7 9 18 $ $ ! 81. 0.4 3.5

49.
#
17. 4 3 2 10 82. 5 3 45
$
50. 12
18. 3 3 2 1 83. 3 7 9
12 4 6 12
9 6 3 30
51. 84.
2 16 8 2 5 6
2 23 6
52. 85.
2 14 2 7 4 9
2 6 3 9 3
53. 86.

5 3 2 10 54. 8 87. 1 3 2 3 2
22. '
23. 3 12 24 9 55.

2 88. 3 4 2 5 3
'
24. 2 4 3 5 $ 89. 3.65 7.4 1.12 21.76
56. 4

25. 42 3 4 8 8 90. 8 3 2 6 3 4
57. 26
26. 6 11 6 5 91. 10 3 10 13 3 7
58. 3 15
27. 2 7 6 9 4 92. 6 3 5 1 3 2
59. 4 32
28. 53 4 6 20 9 93. 10 5 3 4 2 7

60. 5 9.2" 4.3 50.9
29. 4 3 5 0 94.

30. 24 6 2 3 61. 5 95. 0.05/ 0.2 0.15/ 10.5
)

$ ! 0.2560 0.10
15
31. 96.
'
62.

$ (
0.1560
#
32. 63. 30
# ! 97. 0.530 87 1.50 43.5
) ) #
33. 64. 90 98. 0.4" 10 0.6" 2
( ! !

*+
65.

4 99. 21.11 4.6 10.91 35.2

' 100. 0.125 0.0255 1
35. 66. 168
! #
, ,
36. 9 67.

2
( ! #

4. LINEAR EQUATIONS PRACTICE ANSWERS
1. 1 26.

51. 12 77. 63

2. 13 52. 8 78. 27
27. No Solution
3. 11 53. 7 79. 72
28. All real numbers
4. 27 ' 54. 56 80. " 18
29.
5. 2 55. 14 81. 2.5
(
6. 30. '
56. 8 82. 12

12 26 10
7. 2 31. 57. 83.
5 5 3
8. 3 32. 58. 84.
)
2 8 85.
9. 12 33. 59.

/0 15
10. 4
60.
34. 86.
5 45 -
11. 6 35. 61.
9 87. 4
12. 3 36. 2 30 62.
25 88. 6
13.
37. 15 63.
! 75 89. 3 2.3
38.
64.
14. 5 65. 12 90. All real numbers
19 3
144
15. 39. 91. No Solution
66.
1 30 92. 5
12
16. 40.
67.
0 / 30 93. 6
25
17. 41.
68.
18. !
1 94. " 6
81
42.
69.
1 95. / 107
9
43.
19. 3 70.
4 96. 120
3
(
44.
20. 16
71.
'
" 16
45. 97. All real numbers
72.
21. 0 46. !

98. " 2
73. 4
22. 1 ! 99. 1 3
47. - 74. #
23. No Solution 100. No Solution
24.
48. " 1 75.

!
49. 14
76. 15
50. 48
25. All real numbers

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