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Algebra 8 min read

How to Solve Simultaneous Equations — GCSE Maths Guide

Master simultaneous equations for GCSE Maths. Learn elimination, substitution, and graphical methods with worked examples for AQA, Edexcel and OCR.

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What Are Simultaneous Equations?

Simultaneous equations are two (or more) equations that share the same unknown variables. You need to find values that satisfy both equations at the same time — that's why they're called "simultaneous". In GCSE Maths, you'll usually deal with two equations in two unknowns (x and y).

  • Two equations, two unknowns (usually x and y)
  • Worth 4–6 marks in GCSE papers
  • Appears on both Foundation (simple) and Higher (with quadratics) tiers

Method 1: Elimination

Elimination is the most common method at GCSE. You add or subtract the two equations to cancel out one variable, then solve for the other. This works best when the coefficients of one variable are the same (or can be made the same by multiplying).

Worked Example

Solve: 3x + 2y = 16 and 5x + 2y = 24

  1. 1.Label: (1) 3x + 2y = 16 and (2) 5x + 2y = 24
  2. 2.Subtract (1) from (2): (5x − 3x) + (2y − 2y) = 24 − 16 → 2x = 8
  3. 3.Divide both sides by 2: x = 4
  4. 4.Substitute x = 4 into (1): 3(4) + 2y = 16 → 12 + 2y = 16 → 2y = 4 → y = 2
Answer: x = 4, y = 2

Method 2: Substitution

Substitution works well when one equation is already solved for a variable (e.g., y = 2x + 3). Rearrange one equation, substitute it into the other, and solve the resulting single-variable equation.

Worked Example

Solve: y = 2x + 1 and 3x + y = 16

  1. 1.Substitute y = 2x + 1 into 3x + y = 16
  2. 2.3x + (2x + 1) = 16 → 5x + 1 = 16 → 5x = 15 → x = 3
  3. 3.Substitute x = 3 back: y = 2(3) + 1 = 7
Answer: x = 3, y = 7

Higher Tier: Linear and Quadratic

On Higher papers, you may need to solve a linear and a quadratic equation simultaneously. Use substitution: rearrange the linear equation for one variable, substitute into the quadratic, and solve the resulting quadratic (usually giving two solution pairs).

Worked Example

Solve: y = x + 2 and x² + y² = 20

  1. 1.Substitute y = x + 2 into x² + y² = 20
  2. 2.x² + (x + 2)² = 20 → x² + x² + 4x + 4 = 20
  3. 3.2x² + 4x − 16 = 0 → x² + 2x − 8 = 0
  4. 4.Factorise: (x + 4)(x − 2) = 0 → x = −4 or x = 2
  5. 5.When x = 2: y = 4. When x = −4: y = −2
Answer: (2, 4) and (−4, −2)

Common Mistakes to Avoid

Many students lose marks by forgetting to find both variables, or by not checking their answer. After solving, always substitute both values back into both original equations to verify.

  • Always find BOTH x and y — half the answer earns half the marks
  • Check your solution in both equations, not just one
  • If coefficients don't match, multiply one or both equations first
  • Write "x = ..." and "y = ..." clearly as your final answer

Frequently Asked Questions

How do I know which method to use?

Use elimination when the coefficients of one variable are equal or can easily be made equal. Use substitution when one equation is already in the form y = ... or x = ...

Can simultaneous equations have no solution?

Yes — if the equations represent parallel lines (same gradient, different intercept), there's no solution. If they're the same line, there are infinitely many solutions.

Are simultaneous equations on Foundation GCSE?

Yes, simple linear simultaneous equations appear on Foundation papers. Higher tier extends this to include one linear and one quadratic equation.

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