Solving a quadratic and linear pair of simultaneous equations graphically

I can relate the solutions (0, 1 or 2) of a linear and quadratic pair of simultaneous equations to the graphs of these equations.

Download all resources

Share activities with pupils

New

Year 10

Higher

Solving a quadratic and linear pair of simultaneous equations graphically

I can relate the solutions (0, 1 or 2) of a linear and quadratic pair of simultaneous equations to the graphs of these equations.

Download all resources

Share activities with pupils

These resources will be removed by end of Summer Term 2025.

Switch to our new teaching resources now - designed by teachers and leading subject experts, and tested in classrooms.

View new resources

Slide deck

Download slide deck

Skip slide deck

Lesson details

Key learning points

You can use your knowledge of graphical representations to draw the graphs on the same axes.
If there exists two solutions, then the points of intersection are those solutions.
If there exists one solution, then the point where the linear graph touches the quadratic graph is that solution.
If there are no solutions then the graphs do not touch at all.
When substituting or eliminating, you are left with a combined quadratic which has roots where the graphs intersect.

Keywords

Linear - The relationship between two variables is linear if, when plotted on a pair of axes, a straight line is formed.
Quadratic - A quadratic is an equation, graph, or sequence whereby the highest exponent of the variable is 2; the graph of which forms a parabola.

Common misconception

For a pair of simultaneous equations (one linear and one quadratic), there will always be two intersections and therefore two solutions.

The linear graph might be tangent to the curve therefore only one intersection (solution). Alternatively, the straight line may never intersect the parabola and in this case, there is no solution.

If you have access to technology this a good opportunity to improve fluency. Get pupils to solve pairs of simultaneous equations (where one is linear and one is quadratic) using a written algebraic method, then use graphing technology to demonstrate or check that their working is correct.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).