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.

Want more printable practice?

Additional Math Topics for Grade 8 – with Free Worksheets

• How to Solve Equations with Fractions and Decimals in 3 Simple Steps
• How to Solve Equations with Parentheses in 4 Simple Steps
• Area of a Circle: Definition, Formula and Example

About Think Academy

Think Academy, a leading K–12 math education provider wholly owned by TAL Education Group, is dedicated to helping students build strong mathematical foundations and critical thinking. Our structured curriculum provides multiple course levels designed to accommodate students with diverse academic goals and proficiency levels, ensuring targeted and effective learning experiences. Supported by advanced teaching methods, expert instructors, and innovative AI technology, Think Academy consistently demonstrates excellence, trustworthiness, and proven expertise in mathematics education.

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