Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.

How to Solve Equations with Variables on Both Sides in 3 Simple Steps

By Grade 7 at Think Academy (Grade 8 in school), students are expected to solve equations with variables on both sides. The hardest part isn’t the math itself but avoiding common errors—like forgetting to move terms correctly or making sign mistakes—that can derail the solution. This guide breaks the process into simple steps so students can solve with confidence.

What Is an Equation?

An equation is a mathematical sentence that says two expressions are equal. It has an equal sign in the middle.

Example:

𝑥 + 3𝑥 + 8 − 1 = 5𝑥 − 3𝑥 + 9 + 6

What Is a Solution?

A solution is the number that makes the equation true.

For example, in:

𝑥 + 3 = 5,

the solution is 𝑥=2, because 2+3=5.

What Are Like Terms?

Like terms are terms that have the same variables raised to the same powers. Constants are also like terms. Combining like terms makes calculations easier.

For example, in

𝑥 + 3𝑥 + 8 − 1 = 5𝑥 − 3𝑥 + 9 + 6,

• On the left-hand side, 𝑥 and 3𝑥 are like terms, and 8 and -1 are like terms.
• On the right-hand side, 5𝑥 and -3𝑥 are like terms, and 9 and 6 are like terms.

Three Steps to Solve Equations with Variables on Both Sides

Let’s continue with this example:

𝑥 + 3𝑥 + 8 − 1 = 5𝑥 − 3𝑥 + 9 + 6,

Step 1: Combine Like Terms

Combine the like terms on each side to make the future calculation easier.

Therefore:

4𝑥 + 7 = 2𝑥 + 15

Step 2: Move Variables to the Left Side, Constants to the Right

Move the variable terms to the left

Subtract 2𝑥 from both sides:

Therefore:

2𝑥 + 7 = 15

Move the constants to the right

Subtract 7 from both sides:

Therefore:

2𝑥 = 8

Step 3: Solve for the Variable

Divide both sides of the equation by the coefficient 2 to get the final answer:

Therefore:

𝑥 = 4

Example Problems: Solving Equations with Variables on Both Sides

Example 1

Find the solution to the equation.

3𝑥 + 3𝑥 + 10 − 1 = 7𝑥 − 4𝑥 + 17 + 4

Solution:

Step 1: Combine like terms

6𝑥 + 9 = 3𝑥 + 21

Step 2: Move variables left, constants right:

6𝑥 + 9 − 3𝑥 = 3𝑥 + 21 − 3𝑥
3𝑥 + 9 = 21
3𝑥 + 9 − 9 = 21 − 9
3𝑥 = 12

Step 3: Solve for the variable

3𝑥 ÷ 3 = 12 ÷ 3
𝑥 = 4

The final result is 𝑥 = 4.

Example 2

Find the solution to the equation.

𝑥 + 𝑥 + 3𝑥 + 3 + 9 = 9𝑥 − 5𝑥 + 23 − 4

Solution:

Step 1: Combine like terms

5𝑥 + 12 = 4𝑥 + 19

Step 2: Move variables left, constants right:

5𝑥 + 12 − 4𝑥 = 4𝑥 + 19 − 4𝑥
𝑥 + 12 = 19
𝑥 + 12 − 12 = 19 − 12
𝑥 = 7

Step 3: Solve for the variable

The final result is 𝑥 = 7.

Summary: How to Solve Equations with Variables on Both Sides

When solving equations with variables on both sides:

1. Combine like terms;
2. Move variables to the left, constants to the right;
3. Solve for the variable.

For extra safety, always substitute our solution back into the original equation to make sure it works perfectly.

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Additional Math Topics for Grade 8 – with Free Worksheets

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