Topic summary · Pure foundations (P1)

Simultaneous Equations summary

Simultaneous equations model two constraints. Elimination and substitution are the algebraic workhorses; graphs show the intersection.

Open Learning Sheet — Simultaneous Equations Download Simultaneous Equations answers PDF

Files: P1-Simultaneous Equations.pdf · P1-Simultaneous Equations answers.pdf · free public downloads · diagrams in the PDFs

Key learning points

Elimination: equalise one coefficient, then add or subtract.
Substitution: rearrange one equation and replace in the other.
Linear–quadratic: substitute the linear expression into the quadratic.
Check both original equations with your solution.

Worked example 1 — Elimination

Solve $x+y=7$ , $x-y=1$ .

Working

x=4,\ y=3

Add to eliminate $y$ .

Worked example 2 — Substitution

Solve $y=2x+1$ , $x+y=7$ .

Working

x=2,\ y=5

Substitute for $y$ .

Worked example 3 — Scaled elimination

Solve $2x+3y=12$ , $x+y=5$ .

Working

x=3,\ y=2

Match coefficients, then subtract.

Worked example 4 — Linear pair

Solve $x+2y=8$ and $3x-y=3$ .

Working

x=2,\ y=3

From the second, $y=3x-3$ ; substitute into the first.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.

Open Learning Sheet Answers PDF

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