Solving Equations with Variables on Each Side

After studying this lesson, you will be able to:

• Solve equations with variables on each side of the equal sign.
• Solve equations with parentheses and other grouping symbols.

Steps for Solving Equations with Variables on Each Side and with Parentheses

1. Remove parentheses by multiplying

2. Collect like terms on each side of the equal sign

3. Get the variables together on one side of the equation and get the numbers together on the other side of the equation.

4. Isolate the variable by "undoing" the operation (do this until the variable is by itself)

1. "undo" addition and subtraction first
2. next, "undo" multiplication and division

5. Check by substituting the solution into the original equation

Example 1

2x + 4 = 6x + 8	This equation doesn't have parentheses, but it does have a variable each side. It doesn't have like terms to collect on either side.
2x + 4 - 2x = 6x + 8 - 2x	First, we need to get the variables together. It doesn't matter if we put them on the left side or the right side. Let's put them together on the right side this time. To do that, we move 2x to the other side by subtracting 2x from each side. Notice that we line up the like terms. (-2x is lined up with 6x so that it is easier to deal with.)
4 = 4x + 8	After collecting like terms (6x - 2x) we now have an equation where the variables are now together. Now, we work this as a 2-step equation.
4 - 8 = 4x + 8 - 8	We need to "undo" +8, so we subtract 8 from each side. This gives us -4 = 4x
	Now, we need to "undo" 4 times x, so we divide each side by 4
-1 = x	This is the solution

Check:

substitute -1 for each x in the original equation

2 (-1) + 4 = 6 (-1) +8 work out each side separately

-2 + 4 = -6 + 8 do the multiplying first

2 = 2

Example 2

6x - 3 = 2x + 13	This equation doesn't have parentheses, but it does have a variable each side. It doesn't have like terms to collect on either side.
6x - 3 - 2x = 2x + 13 - 2x	First, we need to get the variables together. It doesn't matter if we put them on the left side or the right side. Let's put them together on the left side this time. To do that, we move 2x to the other side by subtracting 2x from each side. Notice that we line up the like terms. (-2x is lined up with 6x so that it is easier to deal with.)
4x -3 = 13	After collecting like terms (6x - 2x) we now have an equation where the variables are now together. Now, we work this as a 2-step equation.
4x -3 + 3 = 13 + 3	We need to "undo" -3, so we add 3 to each side. This gives us 4x = 16
	Now, we need to "undo" 4 times x, so we divide each side by 4
x = 4	This is the solution

Check:

substitute 4 for each x in the original equation

6 (4) - 3 = 2 (4) + 13 work out each side separately

24 - 3 = 8 + 13 do the multiplying first

21 = 21