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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.

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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

  1. Subtract equation 2 from equation 1: (3x − x) + (2y − 2y) = 16 − 8
  2. 2x = 8, so x = 4
  3. Substitute into equation 2: 4 + 2y = 8 → 2y = 4 → y = 2
  4. 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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