Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Simultaneous Equations

In one sentence. Two unknowns, two equations — elimination, substitution, and the Higher-tier pairing of a line with a curve.

Algebra — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Carl Friedrich Gauss. Learn Simultaneous Equations, teach it, then defend it against Carl Friedrich Gauss.

Where this sits in the specification

Specification 4MA1, 2.6 Simultaneous equations (linear/quadratic pairs: Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 2.6 — Simultaneous equations (linear/quadratic pairs: Higher)

1 · Understand it

What it is

Two equations that share the same x and y pin down a single pair of values. Elimination scales the equations until one letter's coefficients match, then adds or subtracts to remove it. Substitution rearranges one equation and feeds it into the other — the standard method when one equation is linear and the other is a curve.

Why it works

Adding equals to equals keeps truth: if both equations hold, their sum holds too — and if the y-coefficients are opposite, the sum contains no y. Graphically, each equation is a line, and solving finds their crossing point; a line with a circle explains why linear/quadratic pairs can have TWO solutions.

When a question is asking for it

Worded price problems ('3 coffees and 2 teas cost…') are elimination in disguise. Same-sign matching coefficients → subtract; opposite signs → add. If one equation contains x² or y², substitute the linear one into it.

How to hold it in your head

Elimination is a controlled collision: line the equations up, make one letter's armies equal, and let them cancel. Whatever survives is a one-letter equation you already know how to solve. Then one substitution back recovers the fallen letter.

2 · The mistakes that cost marks

  • Subtracting when the matched coefficients have opposite signs. +3y and −3y cancel by ADDING; same signs cancel by subtracting.
  • Finding one unknown and stopping. The question isn't finished until you substitute back for the other letter — that step carries marks too.
  • Scaling only part of an equation. Multiplying by 2 must hit every term, right-hand side included.
  • In linear/quadratic pairs, forgetting that each x usually brings its own y — solutions come in PAIRS.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Solve: 3x + 2y = 19 and x + 2y = 9.

  1. Same +2y in both: subtract the second from the first.
  2. (3x − x) + (2y − 2y) = 19 − 9 → 2x = 10 → x = 5.
  3. Substitute back: 5 + 2y = 9 → y = 2.
  4. Check in the OTHER equation: 3(5) + 2(2) = 19 ✓

Answer: x = 5, y = 2

Solve: y = x + 1 and x² + y² = 25.

  1. Substitute the line into the curve: x² + (x + 1)² = 25.
  2. x² + x² + 2x + 1 = 25 → 2x² + 2x − 24 = 0 → x² + x − 12 = 0.
  3. (x + 4)(x − 3) = 0 → x = 3 or x = −4.
  4. Each x brings its y: (3, 4) and (−4, −3).

Answer: (3, 4) and (−4, −3)

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Solve the simultaneous equations 3x + 2y = 16 and x − y = 2. Work out the value of x.
  2. Grade 7 — Solve the simultaneous equations 3x + 2y = 19 and x + 2y = 9. Work out the value of y.
  3. Grade 8 — Solve the simultaneous equations 5x + 2y = 20 and 3x + 4y = 26. Work out the value of y.
  4. Grade 9 — 3 coffees and 2 teas cost £8.05. 5 coffees and 2 teas cost £11.75. Work out the cost of one tea. (answer in £)
  5. 9+ — The line y = 2x − 5 meets the circle x² + y² = 50 at the points A and B. Work out the length of AB, giving your answer as a surd in its simplest form. (Type it like 3sqrt7.)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Simultaneous Equations, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Simultaneous Equations works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Simultaneous Equations against Carl Friedrich Gauss — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 11 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

Completely free. No subscriptions, no paywalls, no account needed to start.

Carry on through the course

  • Previous module: Quadratic Equations
  • Next module: Inequalities
  • Expressions & Index Laws — Collecting like terms and running the index laws on letters — the fluency layer every other algebra module assumes.
  • Expanding & Factorising — Brackets out, brackets back in — including the double brackets and quadratic factorising that unlock every quadratic equation.
  • Linear Equations — Solving any equation with one unknown — both sides, brackets, fractions — by doing the same thing to both sides, every time.
  • Formulae & Rearranging — Substituting into formulae and changing the subject — including the Higher-tier case where the new subject appears twice.
  • Quadratic Equations — Four ways to solve a quadratic — factorising, the formula, completing the square, and reading a graph — why each one works, how to choose between them, and how to reason about how many solutions there are.
  • All of Algebra · Incandio Mathematics · The 4MA1 specification