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.

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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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