Equations with Variables on Both Sides

To solve equations with variables on both sides, get all the variable terms on one side and all the numbers on the other. Add or subtract a variable term from both sides first. Then solve the two-step equation that is left. For example, 5x + 2 = 3x + 10 becomes 2x + 2 = 10, so x = 4.

Before this lesson: You should know how to solve multi-step equations, combine like terms and use the distributive property.

Extra practice: The Algebra 1 Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Does “Variables on Both Sides” Mean?

In the equations you have solved so far, the variable was on one side only. Now it shows up on both sides, like 5x + 2 = 3x + 10. You can’t divide yet, because there are two x-terms in different places.

Think of an equation as a balance scale. You can take the same thing off both sides and it stays balanced. That includes variable terms. Subtract 3x from both sides, and only one x-term is left.

How to Solve Equations with Variables on Both Sides

1. Simplify each side. Distribute and combine like terms.
2. Move the variable terms to one side. Add or subtract a variable term on both sides.
3. Move the numbers to the other side.
4. Divide by the coefficient (the number in front of the variable).
5. Check your answer in the original equation.

Tip: Move the smaller variable term. In 5x + 2 = 3x + 10, 3x is smaller than 5x. Subtracting 3x keeps the coefficient positive, so there are fewer negative signs to track.

\begin{aligned}
5x + 2 &= 3x + 10 \\
2x + 2 &= 10 & &\text{subtract } 3x \\
2x &= 8 & &\text{subtract 2} \\
x &= 4 & &\text{divide by 2}
\end{aligned}

Check: 5(4) + 2 = 22 and 3(4) + 10 = 22. Both sides match.

Worked Examples

Example 1: Basic equation

Solve 6x - 5 = 2x + 11.

6x - 5 - 2x = 2x + 11 - 2x

4x - 5 = 11

4x - 5 + 5 = 11 + 5

4x = 16

\frac{4x}{4} = \frac{16}{4}

x = 4

Answer: x = 4

Check: 6(4) - 5 = 19 and 2(4) + 11 = 19. It works.

Example 2: A negative variable term

Solve 4n - 9 = -2n + 21.

4n - 9 + 2n = -2n + 21 + 2n

6n - 9 = 21

6n - 9 + 9 = 21 + 9

6n = 30

\frac{6n}{6} = \frac{30}{6}

n = 5

Answer: n = 5

Check: 4(5) - 9 = 11 and -2(5) + 21 = 11. It works.

Example 3: Parentheses on both sides

Solve 3(a + 4) = 5(a - 2).

3a + 12 = 5a - 10

3a + 12 - 3a = 5a - 10 - 3a

12 = 2a - 10

12 + 10 = 2a - 10 + 10

22 = 2a

\frac{22}{2} = \frac{2a}{2}

a = 11

Answer: a = 11

Check: 3(11 + 4) = 45 and 5(11 - 2) = 45. It works.

Example 4: Fractions

Solve \frac{1}{2}x + 3 = \frac{1}{4}x + 5.

4 \cdot \frac{1}{2}x + 4 \cdot 3 = 4 \cdot \frac{1}{4}x + 4 \cdot 5

2x + 12 = x + 20

2x + 12 - x = x + 20 - x

x + 12 = 20

x + 12 - 12 = 20 - 12

x = 8

Answer: x = 8

Check: \frac{1}{2}(8) + 3 = 7 and \frac{1}{4}(8) + 5 = 7. It works.

Example 5: Word problem

Bike Shop A charges $12 plus $3 per hour. Bike Shop B charges $4 plus $5 per hour. After how many hours do the two shops cost the same?

12 + 3h = 4 + 5h

12 + 3h - 3h = 4 + 5h - 3h

12 = 4 + 2h

12 - 4 = 4 + 2h - 4

8 = 2h

\frac{8}{2} = \frac{2h}{2}

h = 4

Step 8. Find the cost. Shop A: 12 + 3(4) = 24. Shop B: 4 + 5(4) = 24. Both cost $24.

Answer: The shops cost the same, $24, after 4 hours.

Common Mistakes

When the Variables Cancel Out

Sometimes the variable terms cancel on both sides. Then you are left with only numbers. Look at that statement to decide the answer.

An equation that is true for every number is called an identity. Both sides are the same expression, written two ways.

You will see the same idea later with lines. Two parallel lines never meet, which matches “no solution.” See number of solutions for systems of equations.

Practice Problems

Solve each equation. If there is no solution or every number works, say so.

1. 6x + 1 = 4x + 9
2. 9k - 5 = 4k + 20
3. 2m + 7 = 5m - 14
4. 8 - 2y = 3y + 23
5. 4(b - 1) = 2(b + 5)
6. 0.5x + 6 = 0.2x + 9
7. \frac{x}{3} + 2 = \frac{x}{6} + 5
8. Maya has $50 and saves $8 a week. Leo has $90 and saves $3 a week. After how many weeks will they have the same amount? How much will each have?
9. 5(t + 2) - t = 4t + 10
10. 3(2w - 1) = 6w + 4

Answers

2x + 1 = 9

2x = 8

x = 4

5k - 5 = 20

5k = 25

k = 5

7 = 3m - 14

21 = 3m

7 = m

m = 7

8 = 5y + 23

-15 = 5y

-3 = y

y = -3

4b - 4 = 2b + 10

2b - 4 = 10

2b = 14

b = 7

0.3x + 6 = 9

0.3x = 3

x = 10

6 \cdot \frac{x}{3} + 6 \cdot 2 = 6 \cdot \frac{x}{6} + 6 \cdot 5

2x + 12 = x + 30

x + 12 = 30

x = 18

50 + 8w = 90 + 3w

50 + 5w = 90

5w = 40

w = 8

50 + 8(8) = 50 + 64 = 114

5t + 10 - t = 4t + 10

4t + 10 = 4t + 10

10 = 10

6w - 3 = 6w + 4

-3 = 4

Want more practice? The Algebra 1 Unit 2 Practice Pack has 51 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Equations with Variables on Both Sides FAQ

How do you solve equations with variables on both sides?

Add or subtract a variable term on both sides so the variable is on one side only. Then move the numbers to the other side and divide by the coefficient.

Which side should the variable go on?

Either side works and gives the same answer. Moving the smaller variable term usually keeps the coefficient positive, which means fewer sign mistakes.

What does it mean when the variables cancel out?

Look at what is left. A false statement, like 5 = 1, means there is no solution. A true statement, like 6 = 6, means every number is a solution.

Can the answer be on the left side, like 11 = a?

Yes. 11 = a means the same thing as a = 11. You can flip the sides of an equation at any time.

Summary

• Get the variable terms on one side by adding or subtracting a variable term on both sides.
• Simplify first: distribute and combine like terms on each side.
• Move the smaller variable term to keep the coefficient positive.
• If the variables cancel and the result is false, there is no solution.
• If the variables cancel and the result is true, every number is a solution.

Previous lesson: Multi-Step Equations

Next lesson: Literal Equations

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