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A detailed explanation on how to solve multi-step equations, which involve more than one operation. It includes examples and steps to isolate variables, perform order of operations in reverse, and eliminate parentheses. Students will gain a solid understanding of the process and be able to apply it to various equations.
What you will learn
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This unit is about solving equations that involve more than one operation. First, a review will be provided for one-step equations and two-step equations, and then multi-step equations will be examined. To extend the process of solving equations, there may be times when the distributive property is used to eliminate any parentheses. Then, like terms may be combined and the equation can be solved.
Review of Basic Equations
Solving Multi-Step Equations
This section of the unit is a review one-step and two-step equations.
Remember, to solve an equation involving more than one operation, perform the order of operations IN REVERSE to solve for the unknown variable.
Example #1 : Solve y − 6 = − 21 for y.
y − 6 1 6
Therefore, y = –15.
Check the answer by replacing y with –15 in the original equation. 6 21 6 21 21
y
True
Example #3 : Solve 5 x + 6 = 31 for x.
5 x + 6 6
− from both sides of the equation. 5 25 5 Divide bo
6 Subt
th sides by
ract 6
x x
Therefore, x = 5.
Check the answer by replacing x with 5 in the original equation. 5 6 31 5 6 31 25 6 31 31
x
True
Example #4 : Solve for − 6 z − 18 = − 132 for z.
− 6 z − 18 18
19 both sides by 6. ( 114 6 19
18 Add
z z
Therefore, z = 19.
Check the answer by replacing z with 19 in the original equation. 6 18 132 6 18 132 114 18 132 132 132
z
true
Now, let’s take a look at solving equations with variables on both sides of the equals sign_._
Example 1 : Solve 8 x + 5 = 2 x – 16 for x.
Step #1 : Move the variables (with coefficients) to one side and the numbers (with no variable “attached”) to the other side. Use algebra to justify the adjustments.
8 x + 5 = 2 x 2
− x 2 x
− from both sides of the equation. (8 2
ubtract 2
5
x x x x x x x
− from both sides. ( 16 5 1
5 Subt 6 5 21) 6 21
ract 5 x
Step #2 : Divide both sides by 6 to solve for the unknown.
6 6
x 6
(^21) both sides by 6.
21
Divid
3.5 7 3 1 3
e
. 6 2 2
x
Therefore, x = –3.5.
Check the answer by replacing x with –3.5 in the original equation.
( 3.5) ( 3.
x x
true
Check the answer by replacing d with 17 in the original equation. 5( 4) 7( 2) 5( 4) 7( 2) 5(21) 7(
d d
true
Example #3 : Solve 3( x − 6) + 2 = 4( x + 2) − 21 for x.
Step #1 : Eliminate the parentheses by using the distributive property on each of the quantities.
2 4 x 8 21
x x
x − + = + − − + = + −
Step #2 : Combine any like terms on either side of the equals sign. In this case, combine (–18 + 2) on the left and (+8 – 21) on the right.
3 18 2 4 8 21 18 2 16 8 21 13 3 16 4 13
x x x x
Step #3 : Move the variables (with coefficients) to one side and the numbers (with no variables “attached”) to the other side.
3 x 16 4 13 3
x x
x − = x −
x x
Therefore, x = –3.
x x
x
x
x x
Check the answer by replacing x with –3 in the original equation.
3( 6) 2 4( 2) 21 3( 6) 2 4( 2) 21 3( 9) 2 4( 1) 21 27 2 4 21 25
x x
true
