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.

§Key definitions

Question:

Solve 7x + 2 = 3x + 18.

Q1 (Foundation):

Solve 5x + 1 = 2x + 13.

Q2 (Foundation):

Solve 8x - 5 = 3x + 20.

Q3 (Higher):

Solve 2(5x + 3) = 3(2x + 8).

§Formulas to memorise

If ax + b = cx + d, then (a - c)x = d - b, so x = (d - b) / (a - c)

Check: substitute your answer back into both sides of the original equation

Worked example

Solve 7x + 2 = 3x + 18.

Working:

⚠ 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.

MMXXVI specification · AQA · Edexcel · OCRgcsemathsai.co.uk/formulas/equations-with-unknowns-on-both-sides

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