GCSE Maths: Simultaneous Equations

Simultaneous equations are one of the clearest step-ups in GCSE Mathematics: they demand method choice, disciplined working and a final check against both original equations. This quiz covers the topic end to end, from Foundation-accessible linear pairs to the Higher tier linear-with-quadratic systems.

The linear questions practise both standard techniques. Elimination appears in its cleanest form — adding equations like 5x − 2y = 11 and 3x + 2y = 13 so a variable cancels immediately — and in the scaled form where one equation must first be multiplied through before subtracting. Substitution gets its own workout with systems arranged for it, and a comparison question checks that you can tell which method a given system invites. A worded problem about pens and notebooks tests the modelling step: translating "three pens and two notebooks cost £8" into the correct pair of equations before any algebra begins — the exact format exam papers use for these questions.

The Higher tier material closes the set: solving y = x² against y = x + 2 and y = x² − 4 against y = 2x − 1 by setting the expressions equal, rearranging to a quadratic, and factorising. These questions make the crucial point that a linear-quadratic system generally has two solution pairs, and that each x must be matched with its own y.

All questions are self-contained with small integer solutions, so the focus stays on method rather than arithmetic grind. Explanations show the full elimination or substitution working, and the distractors encode the standard failure modes: sign errors when subtracting equations, and pairing an x-value with the wrong y-value.

• Solve two linear simultaneous equations by elimination, scaling one equation when needed
• Solve linear systems by substitution and recognise when each method is most efficient
• Translate a worded costs problem into a correct pair of simultaneous equations
• Solve one linear and one quadratic equation simultaneously (Higher tier)
• Match each x-solution with its corresponding y-value and check both equations

Topic scope follows the Algebra strand of the Department for Education's GCSE mathematics subject content: solving two simultaneous equations in two variables — linear/linear at both tiers, linear/quadratic at Higher tier — algebraically, and deriving them from real contexts.