Simultaneous Equations — GCSE Mathematics Revision

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What is Simultaneous Equations?

Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time. There are two main methods: elimination (adding or subtracting equations to remove one variable) and substitution (rearranging one equation and substituting into the other). At Higher tier, you also need to solve one linear and one quadratic simultaneously.

Board notes: Linear-quadratic simultaneous equations are Higher tier only on all boards. AQA and Edexcel often set these as 4-5 mark questions.

Worked example

Common mistakes

• 1Forgetting to multiply the ENTIRE equation (both sides) when scaling up for elimination.
• 2Sign errors when subtracting equations — especially when subtracting a negative term (e.g. 3y - (-2y) = 5y, not y).
• 3Only finding x and forgetting to substitute back to find y.
• 4In linear-quadratic pairs, not expanding brackets correctly after substitution.

Simultaneous Equations exam questions

Exam-style questions for Simultaneous Equations with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel and OCR specifications.

Frequently asked questions

• When should I use elimination vs substitution?

  Use elimination when the coefficients of one variable are the same (or easy to make the same). Use substitution when one equation is already in the form y = ... or x = ..., or when one equation is quadratic.

• How do I solve simultaneous equations with a quadratic?

  Rearrange the linear equation to make one variable the subject, substitute into the quadratic, expand and simplify to get a quadratic in one variable, then solve using factorising or the quadratic formula.