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
- Move variable terms to one side using addition or subtraction
- Solve multi-step equations with variables on both sides
- Recognize equations with no solution and equations with infinitely many solutions
Worked example
Problem. Solve 5x + 2 = 3x + 10.
- Subtract 3x from both sides: 2x + 2 = 10.
- Subtract 2 from both sides: 2x = 8.
- Divide by 2: x = 4.
- Connect the calculation back to Equations with Variables on Both Sides so the method, not just the arithmetic, is clear.
Answer: 4
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Solve 4x = 2x + 10.
Show solution
- Subtract 2x from both sides: 2x = 10.
- Divide by 2: x = 5.
Answer: 5
2. Solve 6x - 3 = 2x + 9.
Show solution
- Subtract 2x: 4x - 3 = 9.
- Add 3: 4x = 12, so x = 3.
Answer: 3
3. Solve 5x + 1 = 3x + 9.
Show solution
- Subtract 3x: 2x + 1 = 9.
- Subtract 1 and divide by 2: x = 4.
Answer: 4
4. Solve 7x - 8 = 3x + 4.
Show solution
- Subtract 3x: 4x - 8 = 4.
- Add 8 and divide by 4: x = 3.
Answer: 3
5. Solve 2(x + 1) = x + 5.
Show solution
- Distribute: 2x + 2 = x + 5.
- Subtract x: x + 2 = 5.
- Subtract 2: x = 3.
Answer: 3
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