Equations with Variables on Both Sides

Learn how to solve equations where the variable appears on both sides of the equals sign.

Intermediate25 minLesson
Printable worksheet

Definition

When an equation has variables on both sides of the equals sign, we need to collect all variable terms on one side before solving.
The Strategy:
  1. 1.Collect variable terms on one side (usually the left)
  2. 2.Collect constant terms on the other side
  3. 3.Solve the simplified equation
Example: 5x−3=2x+95x - 3 = 2x + 9
5x−2x=9+35x - 2x = 9 + 3
3x=123x = 12
x=4x = 4
Key Principle: Whatever you do to one side, you must do to the other side to keep the equation balanced.

Try it now

Solve: 4x+5=2x+134x + 5 = 2x + 13

Worked Examples

Solve: 5x−3=2x+95x - 3 = 2x + 9
1

Identify variable terms on both sides

Left: 5x5x, Right: 2x2x → Variables on both sides
2

Subtract 2x2x from both sides

5x−2x−3=2x−2x+95x - 2x - 3 = 2x - 2x + 9 → 3x−3=93x - 3 = 9
3

Add 33 to both sides

3x−3+3=9+33x - 3 + 3 = 9 + 3 → 3x=123x = 12
4

Divide both sides by 33

3x3=123\frac{3x}{3} = \frac{12}{3} → x=4x = 4
5

Check the solution

5(4)−3=2(4)+95(4) - 3 = 2(4) + 9 gives 17=1717 = 17 \checkmark → Solution verified!

Common Mistakes

Subtracting variables incorrectly: 5x−2x=35x - 2x = 3 instead of 3x3x

Why it's wrong: When combining like terms, only the coefficients are subtracted: 5x−2x=(5−2)x=3x5x - 2x = (5-2)x = 3x.

Correct: 5x−2x=3x5x - 2x = 3x, not 33. The variable stays!

Forgetting to perform operations on BOTH sides

Why it's wrong: An equation is like a balance scale. If you only change one side, it becomes unbalanced (and wrong).

Correct: Always write out both sides: 5x−2x−3=2x−2x+95x - 2x - 3 = 2x - 2x + 9

Sign errors when moving terms: 5x=2x+125x = 2x + 12 becoming 5x−2x=125x - 2x = 12

Why it's wrong: When a term crosses the equals sign, its sign changes. But +12+12 stays positive if it's not moving.

Correct: 5x−2x=125x - 2x = 12 only if you're subtracting 2x2x from both sides.

Not distributing before collecting like terms

Why it's wrong: You cannot combine terms inside and outside parentheses until you distribute.

Correct: Always distribute first: 3(x+2)=3x+63(x + 2) = 3x + 6, then collect like terms.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Solve Linear Equations | Step-by-Step Examples

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Interactive Visual

Balance Scale

5x − 3=2x + 9
5x
−3
2x
9
Apply to both sides:

Solution: x = 4

Equation Solver

Solve for xx:

5x−3=2x+95x - 3 = 2x + 9
1
5x−3=2x+95x - 3 = 2x + 9
Subtract 2x2x from both sidesMove variables to one side

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

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Practice Problems

17 problems
Problem 1 of 17
Easy
Solve: 4x+5=2x+134x + 5 = 2x + 13

Why It Matters

Equations with variables on both sides appear constantly in real-world problem solving:
  • Comparing costs: When does renting become cheaper than buying?
  • Break-even analysis: At what point do two options cost the same?
  • Distance problems: When do two objects meet if traveling toward each other?
  • Science: Balancing chemical equations, physics formulas
This skill is essential for more advanced algebra, including systems of equations and inequalities.

Real World Applications

Comparing Phone Plans

Phone companies offer different pricing structures. Equations help find the break-even point.

Example:

Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 10 euros plus 0.15 euros per minute. When do they cost the same?

1Try It Yourself

Plan A: 20 + 0.10m euros. Plan B: 10 + 0.15m euros, where m = minutes used.

At how many minutes do both plans cost the same?

Step 1: Write the mathematical expression

Set the costs equal: 20+0.10m=10+0.15m20 + 0.10m = 10 + 0.15m

Catching Up in a Race

When two runners start at different positions or speeds, we can calculate when they'll meet.

Example:

Runner A starts 50 meters ahead but runs at 6 m/s. Runner B runs at 8 m/s. When does B catch A?

2Try It Yourself

Position A: 50 + 6t meters. Position B: 8t meters (t = seconds).

After how many seconds does Runner B catch Runner A?

Step 1: Write the mathematical expression

Set positions equal: 50+6t=8t50 + 6t = 8t

Temperature Conversion

Celsius and Fahrenheit scales meet at a specific temperature.

Example:

At what temperature does Celsius equal Fahrenheit? Use F=95C+32F = \frac{9}{5}C + 32.

3Try It Yourself

If C=FC = F, then: C=95C+32C = \frac{9}{5}C + 32

At what temperature are Celsius and Fahrenheit equal?

Step 1: Write the mathematical expression

Solve: C=95C+32C = \frac{9}{5}C + 32

Key Takeaways

  • 1Equations with variables on both sides require collecting variable terms on one side first
  • 2Use inverse operations: subtract/add variable terms, then subtract/add constants
  • 3If there are parentheses, distribute before collecting like terms
  • 4Always check your solution by substituting back into the original equation
  • 5Whatever you do to one side, you must do to the other side

Frequently Asked Questions

No! You can collect variables on either side. However, collecting on the side with the larger coefficient often avoids negative coefficients, making arithmetic easier.
No! You can collect variables on either side. However, collecting on the side with the larger coefficient often avoids negative coefficients, making arithmetic easier.
This means the equation is an identity - it's true for ALL values of x. For example, 2x+4=2(x+2)2x + 4 = 2(x + 2) simplifies to 0=00 = 0, meaning any number works.
This means there is no solution. The equation is a contradiction. For example, x+3=x+5x + 3 = x + 5 leads to 3=53 = 5, which is never true.

Glossary

Like terms
Terms with the same variable raised to the same power (e.g., 5x5x and 2x2x are like terms)
Coefficient
The number multiplied by a variable (e.g., in 5x5x, the coefficient is 55)
Inverse operation
An operation that undoes another (addition/subtraction, multiplication/division)
Identity equation
An equation that is true for all values of the variable (e.g., x+1=x+1x + 1 = x + 1)
Contradiction
An equation with no solution because it's never true (e.g., x=x+1x = x + 1)

Formula Card

Solving Strategy

1. Distribute parentheses 2. Collect variable terms on one side 3. Collect constant terms on the other side 4. Solve and check

Steps to solve equations with variables on both sides

Today's challengeOne problem, three tries

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