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

To solve an equation with variables on both sides, first simplify each side, then collect all variable terms on one side and constants on the other, and finally divide by the variable’s coefficient.

For example: 4x + 7 = 2x + 15

Subtract 2x from both sides: 2x + 7 = 15

Subtract 7 from both sides: 2x = 8

Divide both sides by 2: x = 4

The main idea is simple: perform the same operation on both sides so that the equation stays balanced.

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 + 3 = 5

The expression x + 3 is on the left, and 5 is on the right.

An equation works like a balanced scale. If you add, subtract, multiply, or divide one side, you must perform the same operation on the other side to keep the equation balanced.

What Is a Solution to an Equation?

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.

Examples of terms that are not like terms include:

• x and x²
• x and y
• 3x and 3

How to Solve Equations With Variables on Both Sides in 3 Steps?

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

How to Check the Solution?

Substitute x = 4 into the original equation: x + 3x + 8 – 1 = 5x – 3x + 9 + 6

Both sides equal 23, so the solution is correct: x = 4

Which Side Should You Move the Variable To?

You can move all variable terms to either side and still get the same answer.

However, choosing the side with the larger variable coefficient often helps you avoid negative coefficients.

Consider: 3x + 5 = 7x – 11

Because 7x has the larger coefficient, subtract 3x from both sides: 5 = 4x – 11

Add 11 to both sides: 16 = 4x

Divide by 4: x = 4

You could instead subtract 7x from both sides, but that would create -4x. The solution would still be correct, although the arithmetic might be slightly less convenient.

Example Problems: Solving Different Types of Equations With Variables on Both Sides

The same three-step strategy works for simple equations and equations with negative coefficients. If an equation contains parentheses, simplify it first using the distributive property—see how to solve equations with parentheses. For equations containing fractions or decimals, clear or simplify them first by following our guide to solving equations with fractions and decimals.

The following examples cover the most common types of equations with variables on both sides.

Example 1: How Do You Solve a Simple Equation With Variables on Both Sides?

Solve: 3x + 3x + 10 – 1 = 7x – 4x + 17 + 4

First, combine like terms on each side.

The equation becomes: 6x + 9 = 3x + 21

Subtract 3x from both sides: 3x + 9 = 21

Subtract 9 from both sides: 3x = 12

Divide both sides by 3: x = 4

Check the answer:

6(4) + 9 = 3(4) + 21

24 + 9 = 12 + 21

33 = 33

Therefore: x = 4

Example 2: Which Side Should You Move the Variable Terms To?

Solve: x + x + 3x + 3 + 9 = 9x – 5x + 23 – 4

First, combine like terms on each side:

5x + 12 = 4x + 19

You could move either variable term. However, subtracting 4x keeps the remaining variable coefficient positive.

Subtract 4x from both sides: x + 12 = 19

Subtract 12 from both sides: x = 7

Check the answer:

5(7) + 12 = 4(7) + 19

35 + 12 = 28 + 19

47 = 47

Therefore: x = 7

If you subtract 5x instead, you will still get the correct answer, but you will need to work with a negative coefficient. When possible, moving the smaller variable term toward the larger one can make the equation easier to solve.

Example 3: How Do You Solve an Equation With a Negative Solution?

Solve: 5y – 9 = 8y

Subtract 5y from both sides: -9 = 3y

Divide both sides by 3: y = -3

Check the answer in the original equation:

5(-3) – 9 = 8(-3)

-15 – 9 = -24

-24 = -24

Therefore: y = -3

A negative solution does not mean that something went wrong. Substitute the value into the original equation to check whether both sides are equal.

Example 4: How Do You Solve an Equation With a Negative Coefficient?

Solve: -2x + 9 = 5x – 12

Subtract 5x from both sides: -7x + 9 = -12

Subtract 9 from both sides: -7x = -21

Divide both sides by -7: x = 3

Check the answer:

-2(3) + 9 = 5(3) – 12

-6 + 9 = 15 – 12

3 = 3

Therefore: x = 3

Remember that dividing a negative number by another negative number produces a positive result.

Example 5: How Do You Know When an Equation Has No Solution?

Solve: 7x + 4 = 7x – 9

Subtract 7x from both sides:

4 = -9

The variable terms have disappeared, and the remaining statement is false. There is no value of x that can make 4 equal to -9.

Therefore: No solution.

An equation has no solution when the variable terms cancel and leave a false statement.

Example 6: How Do You Know When an Equation Has Infinitely Many Solutions?

Solve: 6x + 2 = 6x + 2

Subtract 6x from both sides: 2 = 2

The variable terms have disappeared, but the remaining statement is always true. The original equation is true for every possible value of x.

Therefore: Infinitely many solutions

An equation has infinitely many solutions when both sides simplify to the same expression.

What Common Mistakes Should You Avoid?

Why Should You Simplify Both Sides Separately?

Do not combine terms across the equal sign.

In: 3x + 5 = 2x + 9

You can combine terms only after using the same operation on both sides. You cannot combine 3x and 2x immediately because they are on different sides of the equation.

Why Must You Apply an Operation to Both Sides?

An equation must stay balanced.

If you subtract 2x from the left side, you must also subtract 2x from the right side.

Incorrect: 4x + 7 – 2x = 2x + 15

Correct: 4x + 7 – 2x = 2x + 15 – 2x

Why Should You Combine Like Terms Correctly?

Only terms with the same variable and exponent can be combined.

Correct: 3x + 2x = 5x

Incorrect: 3x + 2 = 5x

The terms 3x and 2 are not like terms.

Why Should You Check the Solution?

Checking catches sign errors and arithmetic mistakes.

Substitute the answer into the original equation—not only the simplified equation—and verify that both sides have the same value.

Some Practice Problems

Try solving each equation before checking the answers.

1. 6x + 5 = 2x + 21
2. 4(x – 2) = 2x + 6
3. -3x + 11 = x – 5
4. x/3 + 2 = x/6 + 5
5. 7x + 4 = 7x – 9
6. 2(3x + 1) = 6x + 2
7. 5x – 8 = 2x + 13
8. 3(x + 4) – 2 = 2(x + 7)

Answers

1. x = 4
2. x = 7
3. x = 4
4. x = 18
5. No solution
6. Infinitely many solutions
7. x = 7
8. x = 4

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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More related articles

• 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

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