What is important to know about Simultaneous equations?

Two methods: elimination and substitution.

What is important to know about Simultaneous equations?

Elimination: make the coefficient of one variable the same in both equations, then add or subtract.

What is important to know about Simultaneous equations?

Substitution: rearrange one equation and substitute into the other.

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Algebra · Foundation & Higher

Simultaneous equations

Simultaneous equations are two equations with two unknowns. You solve them to find values that satisfy both equations at the same time.

Revising simultaneous equations the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Simultaneous equations GCSE” on the site for the complete guide.

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Key facts to remember

1Two methods: elimination and substitution.
2Elimination: make the coefficient of one variable the same in both equations, then add or subtract.
3Substitution: rearrange one equation and substitute into the other.
4The solution is the point where the two lines (graphs) intersect.

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Worked examples

Example 1

Solve: 3x + 2y = 16 and x + 2y = 8

Working

Subtract equation 2 from equation 1: (3x − x) + (2y − 2y) = 16 − 8
2x = 8, so x = 4
Substitute into equation 2: 4 + 2y = 8 → 2y = 4 → y = 2
Check in eq 1: 3(4) + 2(2) = 12 + 4 = 16 ✓

Answerx = 4, y = 2

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Common mistakes

✗Subtracting when you should add (when signs of the matching terms are opposite).

✗Substituting back into the equation you derived from, rather than an original equation.

✗Making arithmetic errors — write every step clearly.

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Exam tips

✓Always check your solution in both original equations.

✓If coefficients don't match, multiply one or both equations to make them match before eliminating.

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