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Equations with Variables on Both Sides

A free Pre-Algebra lesson from the “Algebra Readiness” unit, with a worked example and practice problems including step-by-step solutions.

When a variable appears on both sides of an equation, the first step is to use addition or subtraction to collect all variable terms on one side. Then solve like a two-step equation. If the variable cancels and a true statement remains (like 5 = 5), the equation has infinitely many solutions. If a false statement remains (like 5 = 7), there is no solution.

What you'll learn

Why it matters: Comparing two pricing plans (gym A versus gym B), two filling rates (when do tanks have equal water), or break-even analysis all create equations with the unknown on both sides.

Worked example

Problem. Solve 5x + 2 = 3x + 10.

  1. Subtract 3x from both sides: 2x + 2 = 10.
  2. Subtract 2 from both sides: 2x = 8.
  3. Divide by 2: x = 4.

Answer: 4

Practice problems

1. Solve 4x = 2x + 10.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 4x = 2x + 10.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 2x from both sides: 2x = 10.
  4. Divide by 2: x = 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. Solve 6x - 3 = 2x + 9.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 6x - 3 = 2x + 9.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 2x: 4x - 3 = 9.
  4. Add 3: 4x = 12, so x = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Solve 5x + 1 = 3x + 9.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 5x + 1 = 3x + 9.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 3x: 2x + 1 = 9.
  4. Subtract 1 and divide by 2: x = 4.
  5. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. Solve 7x - 8 = 3x + 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 7x - 8 = 3x + 4.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 3x: 4x - 8 = 4.
  4. Add 8 and divide by 4: x = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. Solve 2(x + 1) = x + 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2(x + 1) = x + 5.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Distribute: 2x + 2 = x + 5.
  4. Subtract x: x + 2 = 5.
  5. Subtract 2: x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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