Variables on Both Sides

Equations and Inequalities

instruction

Variables on Both Sides

You are familiar with the first type of equation in the table below. The variable is on one side of the equal sign, and there is a constant (number) on the other side. To solve, you simply divide both sides by two.

In the second equation, each side has both a variable and a constant. The variable on each side of the equation has the same value. In this example, the variable on the left side of the equation represents the same number of cookies as the variable on the right side of the equation.

The image depicts a scale with 2 rectangles on one end with question marks in them and 4 stacked circles on the other end.

Let x = one box of cookies.

Two boxes of cookies are equal to four cookies.

2x = 4
The image depicts a scale with 2 rectangles with question marks and a circle on one side and three circles and a rectangle with a question mark on the other side.

Let x = one box of cookies.

Two boxes of cookies plus one extra cookie are equal to one box of cookies plus three extra cookies.

2x + 1 = x + 3

To solve equations with variables on both sides, you will follow the same steps that you have applied to previous equations. You can use the following equation to review each of these steps.

Solve the Equation:

12x – 5 – 2x24x33 

1

Combine like terms on each side of the equation.

10x 5 24x33

2

Move the variables to one side of the equation by using inverse operations and combining like terms.

10x – 10x – 524x – 10x 33

514x33

3

Move the constants to the other side of the equation by using inverse operations and combining like terms.

– 5 + 33 14x33 + 33

28 = 14x

4

Isolate the variable by using inverse operations.

The equation states the fraction 28 over 14 =the  fraction 14 times x over 14.

2 = x


CA LogoTip: You can move the variables to either side of the equation. It may be easier to move the variable so that it ends up with a positive coefficient.

2 x – 5 = 7 x – 10
A speech bubble states, "You could subtract 2x from both sides and end up with negative 5 = 5x = 10. A speech bubble states, “or you could subtract 7 times x from both sides and end up with negative 5 times x minus 5 = negative 10.”

Now try solving the following equation on your own: 8x – 4 = 2x – 16.

Click on the Show Answer button below to check your answer.

Show AnswerAnswer:
8x – 4 = 2x – 16
8x – 2x – 4 = 2x – 2x –16
8x – 2x – 4 = –16
6x – 4 = –16
6x – 4 + 4 =  –16 + 4
6x = –12
x = –2

Click on the link below to watch the "Solving Equations with Variables on Both Sides" Teachlet® tutorial. Watch the tutorial up to the time 3:20.

Mouse icon Solving Equations with Variables on Both Sides