The goal in solving an equation is to get the variable by itself on one side of the equation and a number on the other side of the equation.

To isolate the variable, we must reverse the operations acting on the variable. We do this by performing the inverse of each operation on both sides of the equation. Performing the same operation on both sides of an equation does not change the validity of the equation, or the value of the variable that satisfies it.

When more than one operation acts on a variable in an algebraic equation, apply the reverse of the order of operations to reverse the operations. Here is the order in which you should reverse operations:

- Reverse addition and subtraction (by subtracting and adding) outside parentheses.
- Reverse multiplication and division (by dividing and multiplying) outside parentheses.
- Remove (outermost) parentheses, and reverse the operations in order according to these three steps.

*Example 1*: Solve for
*x*
:
5*x* + 9 = 44

- Reverse addition:
5
*x*+ 9 - 9 = 44 - 9

5*x*= 35 - Reverse multiplication:
=

*x*= 7 - No parentheses.

Check: 5(7) + 9 = 44

Thus,

*Example 2*: Solve for
*y*
:
3( - 1) = 15

- No addition or subtraction
**outside the parentheses** - Reverse multiplication:
3( -1)÷3 = 15÷3

( - 1) = 5 - Within parentheses:
- Reverse subtraction:
- 1 + 1 = 5 + 1

= 6 - Reverse division:
×4 = 6×4

*y*= 24 - No parentheses.

- Reverse subtraction:
- 1 + 1 = 5 + 1

Check: 3( - 1) = 15

Thus,

*Example 3*: Solve for
*z*
:
4(3(*z* - 11) + 6) = 48

- No addition or subtraction.
- Reverse multiplication:
=

(3(*z*- 11) + 6) = 12 - Within parentheses:
- Reverse addition:
3(
*z*- 11) + 6 - 6 = 12 - 6

3(*z*- 11) = 6 - Reverse multiplication:
=

(*z*- 11) = 2 - Within parentheses
- Reverse subtraction:
*z*- 11 + 11 = 2 + 11

*z*= 13 - No multiplication or division.
- No parentheses.

- Reverse subtraction:

- Reverse addition:
3(

Check: 4(3(13 - 11) + 6) = 48

Thus,

Sometimes, the equation will not start out simplified. If this is the case, simplify the equation *before* reversing the operations.

*Example 4*: Solve for
*x*
:
6*x* - 5 - 2*x* + 3 - 2 = 4

First, simplify the equation by combining like terms:
4*x* - 4 = 4

- Reverse subtraction:
4
*x*- 4 + 4 = 4 + 4

4*x*= 8 - Reverse multiplication:
=

*x*= 2 - No parentheses.

Check: 6(2) + 5 - 2(2) - 3 + 2 = 12 - 5 - 4 + 3 - 2 = 4

Thus,