Combining the Rules to Solve
Single Variable Equations

In this unit we presented the procedure for solving single variable equations.

Steps for Solving Single Variable Equations

  1. Combine like terms.
    Using the properties of real numbers and order of operations (these are reviewed in Appendix Two.), you should combine any like terms.
  2. Isolate the terms that contain the variable you wish to solve for.
    Use the Properties of Addition, and/or Multiplication and their inverse operations to isolate the terms containing the variable you wish to solve for.
  3. Isolate the variable you wish to solve for.
    Use the Properties of Addition and Subtraction, and/or Multiplication and Division to isolate the variable you wish to solve for on one side of the equation.
  4. Substitute your answer into the original equation and check that it works.
    Every answer should be checked to be sure it is correct. After substituting the answer into the original equation, be sure the equality holds true.

Now let's look at working through some examples from beginning to end. Suppose, for example, that you wanted to solve the following equation:

7x – 2 = 8 + 2x

1. Combine like terms.

In the equation above there are two terms containing x. We need to first combine these terms. We do this by subtracting 2x from both sides.

7x - 2 - 2x = 8 + 2x - 2x
5x - 2 = 8

2. Isolate the terms that contain the variable you wish to solve for.

We isolate 5x by adding 2 to each side of the equation.

5x - 2 = 8
5x - 2 + 2 = 8 + 2
5x = 10

3. Isolate the variable you wish to solve for.

Since x is multiplied by 5 we use the inverse operation, division, to isolate x.

4. Substitute your answer into the original equation and check that it works.

We should always substitute the solution into the original equation to be sure the answer is correct. When we substitute x= 2 into the original equation we get 12= 12, which is true, so x = 2 is correct.

7x - 2 = + 2x
7(2) - 2 = 8 + 2(2)
14 - 2 = 8 + 4
12 = 12 

Now look through the example below. Be sure you understand how each solution is found. Then you should try the practice at the end of this unit.

Example

Solve the following equations for the variable, x.

Answers

Solve the following equations for the variable, x.

Detailed Answers

1.   5x–2 = 3x + 10                 x = 6

1. Combine like terms.

In the equation above there are two terms containing x. We need to first combine these terms. We do this by subtracting 3x from both sides.

5x - 2 = 3x + 10
5x - 3x - 2 = 3x - 3x + 10
(5 - 3)x - 2 = (3 - 3)x + 10
2x - 2 = 10 

2. Isolate the terms that contain the variable you wish to solve for.

We can isolate 2x by adding 2 to each side of the equation.

2x - 2 = 10
2x - 2 + 2 = 10 + 2
2x = 12

3. Isolate the variable you wish to solve for.

We now isolate the variable by dividing both sides by 2.

4. Substitute your answer into the original equation and check that it works.

We now substitute x = 6 into the original equation and get the equality 28 = 28. This means our answer is correct.

5x - 2 = 3x + 10
5(6) - 2 = 3(6) + 10
30 - 2 = 18 + 10
28 = 28 

2.   4(3x + 1) = 3x + 22        x = 2

1. Combine like terms.

In the equation, before we can combine terms, we must perform the operation indicated by the parentheses. Then we combine terms by subtracting 3x from both sides.

4(3x + 1) = 3x + 22
12x + 4 = 3x + 22
12x - 3x + 4 = 3x - 3x + 22
(12 - 3)x + 4 = (3 - 3)x + 22
9x + 4 = 22

2. Isolate the terms that contain the variable you wish to solve for.

We isolate 9x by subtracting 4 from each side of the equation.

9x + 4 = 22
9x + 4 - 4 = 22 - 4
9x = 18

3. Isolate the variable you wish to solve for.

We now isolate the variable by dividing both sides by 9.

4. Substitute your answer into the original equation and check that it works.

If we plug this result into our original equation, we show this result is correct.

[introduction] [steps for solving equations] [examples of solving equations]