Variables on Both Sides

Some equations have the variable on both sides of the equals sign, like $6x + 4 = 2x + 24$ . Nothing new is required — you just add one move to the routine: gather the variable terms onto one side first.

Once the variables sit on a single side, the equation collapses into a two-step equation you already know. This lesson also covers the two strange cases: equations with no solution at all, and equations that are true for every number.

Gather the variables on one side

Pick a side and move the other side's variable term with the opposite operation. In $6x + 4 = 2x + 24$ , subtract $2x$ from both sides: $4x + 4 = 24$ . From there it's a two-step equation — subtract $4$ , divide by $4$ , and $x = 5$ .

A good habit: move the smaller variable term. Subtracting $2x$ , the smaller term, keeps the remaining coefficient positive, which means fewer sign mistakes.

No solution

Sometimes the variable terms cancel completely. In $5x + 3 = 5x - 8$ , subtracting $5x$ from both sides leaves $3 = -8$ — a false statement. No value of $x$ can fix it, so the equation has no solution.

Infinitely many solutions

The other special case: the two sides turn out to be the same expression. In $2(4x + 6) = 8x + 12$ , distributing on the left gives $8x + 12 = 8x + 12$ . That's true no matter what $x$ is, so there are infinitely many solutions.

The summary: variables cancel and leave a false statement — no solution. Variables cancel and leave a true statement — infinitely many solutions.

Worked examples

Example 1: one solution

Solve $6x + 4 = 2x + 24$ .

Subtract

2x

from both sides

4x + 4 = 24

Subtract

4

from both sides

4x = 20

Divide both sides by

4

x = 5

Check:

6(5) + 4 = 34

and

2(5) + 24 = 34

✓

Answer: $x = 5$

Example 2: no solution

Solve $5x + 3 = 5x - 8$ .

Subtract

5x

from both sides

3 = -8

The variables are gone and the statement is false

No value of

x

can make it true

Answer: No solution

Example 3: infinitely many solutions

Solve $2(4x + 6) = 8x + 12$ .

Distribute on the left

8x + 12 = 8x + 12

Subtract

8x

from both sides

12 = 12

The statement is true for every value of

x

Answer: Infinitely many solutions

Try one yourself

Common questions

Which side should the variables go to?

Either side works — the answer comes out the same. Moving the smaller variable term keeps the coefficient positive, so most people find it less error-prone.

What if the variable terms disappear?

Read what's left. A false statement like $3 = -8$ means no solution. A true statement like $12 = 12$ means infinitely many solutions — every number works.

Do I distribute before moving variable terms?

Yes. Clear any parentheses and combine like terms on each side first, so each side is as simple as possible. Then gather the variables and solve.

How do I check my answer?

Substitute it into both sides of the original equation separately. If both sides come out to the same number, the answer is right.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Variables on Both Sides

Gather the variables on one side

No solution

Infinitely many solutions

Worked examples

Example 1: one solution

Example 2: no solution

Example 3: infinitely many solutions

Try one yourself

Common questions

Keep learning

Two-Step Equations

Equations with the Distributive Property

Writing Equations from Word Problems

Equations with Variables on Both Sides