Equations with Unknowns on Both Sides –

Equations with unknowns on both sides appear on virtually every GCSE Maths paper. The key idea is to collect all the variable terms on one side and all the number terms on the other, then solve as a standard two-step equation.

What Are Equations with Unknowns on Both Sides?

An equation with unknowns on both sides has the variable (usually x) appearing in expressions on both the left-hand side and the right-hand side. For example, 5x + 3 = 2x + 18. You cannot solve it directly because x terms are split across the equals sign. The strategy is to rearrange so that all x terms are on one side and all constants are on the other.

The golden rule is: whatever you do to one side, you must do to the other. This keeps the equation balanced. You can add, subtract, multiply, or divide both sides by the same value. The order of operations matters — deal with the x terms first, then isolate x.

When brackets are involved, expand them before collecting terms. Always check your solution by substituting it back into the original equation to verify that both sides are equal.

Key Formulas

Step-by-Step Method

1. If there are brackets, expand them first.
2. Subtract the smaller x term from both sides to collect all x terms on one side.
3. Add or subtract constants to move all number terms to the other side.
4. Divide both sides by the coefficient of x to find the value of x.
5. Check by substituting your answer into the original equation.

Worked Example 1 — Foundation Level

Question: Solve 7x + 2 = 3x + 18.

Working:

Step 1 — Subtract 3x from both sides: 7x - 3x + 2 = 18, so 4x + 2 = 18.

Step 2 — Subtract 2 from both sides: 4x = 16.

Step 3 — Divide both sides by 4: x = 4.

Check: LHS = 7(4) + 2 = 30. RHS = 3(4) + 18 = 30. Both sides equal 30.

Answer: x = 4

Worked Example 2 — Higher Level

Question: Solve 3(2x - 1) = 5x + 9.

Working:

Step 1 — Expand the bracket: 6x - 3 = 5x + 9.

Step 2 — Subtract 5x from both sides: x - 3 = 9.

Step 3 — Add 3 to both sides: x = 12.

Check: LHS = 3(2(12) - 1) = 3(23) = 69. RHS = 5(12) + 9 = 69. Both sides equal 69.

Answer: x = 12

Worked Example 3 — Exam Style

Question: Solve 4(x + 3) = 2(3x - 1). (3 marks)

Working:

Step 1 — Expand both brackets: 4x + 12 = 6x - 2.

Step 2 — Subtract 4x from both sides: 12 = 2x - 2.

Step 3 — Add 2 to both sides: 14 = 2x.

Step 4 — Divide by 2: x = 7.

Check: LHS = 4(7 + 3) = 4(10) = 40. RHS = 2(3(7) - 1) = 2(20) = 40. Both equal 40.

Answer: x = 7

Common Mistakes

• Subtracting from only one side. If you subtract 3x from the left, you must also subtract 3x from the right. Forgetting this breaks the balance of the equation.
• Sign errors when expanding brackets with a negative. In -2(x - 5), students often write -2x - 10 instead of the correct -2x + 10. A negative times a negative gives a positive.
• Collecting x terms on the wrong side and getting a negative coefficient. It is easier to subtract the smaller x term so that the coefficient stays positive, though both approaches give the same answer.

Exam Tips

• Always collect x terms to the side that keeps the coefficient positive — this reduces the chance of sign errors.
• Show your check step explicitly. Even though it is not always required, it reassures the examiner and catches mistakes.
• If the answer is a fraction, leave it as a simplified fraction unless the question says otherwise.

Practice Questions

Q1 (Foundation): Solve 5x + 1 = 2x + 13.

Q2 (Foundation): Solve 8x - 5 = 3x + 20.

Q3 (Higher): Solve 2(5x + 3) = 3(2x + 8).

Practise equations with unknowns on both sides questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Solving Linear Equations
• Forming and Solving Equations
• Expanding Brackets

Summary

• Equations with unknowns on both sides require collecting all x terms on one side and all number terms on the other.
• Expand any brackets before you start collecting terms.
• Subtract the smaller x term from both sides to keep the coefficient positive.
• Solve the resulting two-step equation by isolating x.
• Always check your answer by substituting back into the original equation.