Simultaneous equations

Overview

Solving two equations together — at this level usually one linear and one quadratic.

The method (substitution)

1. Rearrange the linear equation to make one variable the subject.
2. Substitute into the quadratic equation.
3. Solve the resulting quadratic.
4. Substitute back to find the other variable.

The solutions are the points of intersection of the two graphs.

Worked example

y = x + 1 and x² + y² = 25.

Substitute: x² + (x + 1)² = 25 → 2x² + 2x − 24 = 0 → x² + x − 12 = 0 → (x + 4)(x − 3) = 0, so x = −4 or 3.

Then y = −3 or 4 → points (−4, −3) and (3, 4).

Common mistakes

• Forgetting to find the paired value of the second variable for each solution.

Exam tips

• One solution (repeated) means the line is a tangent to the curve.