Solving Equations with Variables on Both Sides
When the variable appears on both sides, first collect all variable terms onto one side, then solve as normal.
This is a free explanation of Solving Equations with Variables on Both Sides — a preview, not the full interactive lesson. The lesson, practice, quizzes, and AI Tutor are inside the app.
When the variable appears on both sides of the equation, your first move is to collect all variable terms on one side. Do this by adding or subtracting one of the variable terms from both sides. Once the variable is only on one side, solve it as a normal two-step equation.
Carry the sign with any term you move. Subtracting 3x from both sides of 5x − 3 = 3x + 9 leaves 2x − 3 = 9 — note that 5x − 3x = 2x, not 8x. Then isolate the variable term and divide.
Sometimes the variable disappears entirely. If you are left with a false statement (like 5 = 9), the equation has no solution; if you are left with a true statement (like 6 = 6), every value of x is a solution (all real numbers).
Worked example
Solve: 5x − 3 = 3x + 9
- Get all variable terms on one side: subtract 3x from both sides. 5x − 3x − 3 = 9, so 2x − 3 = 9.
- Add 3 to both sides to isolate the variable term: 2x = 12.
- Divide both sides by 2: x = 6.
- Check: left side 5(6) − 3 = 27; right side 3(6) + 9 = 27. Both match.
x = 6
On the test: Watch for special cases — no solution (a false statement) or all real numbers (a true statement).
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