Simultaneous Equations: Advanced

This lesson covers advanced simultaneous equations as required by the AQA A-Level Mathematics specification (7357). At A-Level, you must be able to solve systems involving one linear and one quadratic equation (or other non-linear equation). You also need to understand the geometric interpretation of solutions and use the discriminant to determine the number of intersection points.

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Linear-Quadratic Systems

A linear-quadratic system consists of one linear equation and one quadratic equation (which may represent a parabola, circle, or other conic). The solution involves substituting the linear equation into the quadratic.

Method

1. Rearrange the linear equation to make one variable the subject (e.g. y = ...).
2. Substitute into the quadratic equation.
3. Solve the resulting quadratic equation.
4. Substitute back to find the other variable.

Example: Solve simultaneously: y = 2x + 1 and x² + y² = 10.

Substitute y = 2x + 1 into x² + y² = 10:
x² + (2x + 1)² = 10
x² + 4x² + 4x + 1 = 10
5x² + 4x − 9 = 0
(5x + 9)(x − 1) = 0
x = −9/5 or x = 1