14.4 Linear and quadratic simultaneous equations — Learn

Prerequisites: Expand and factorise quadratics, solve quadratic equations, and substitute into expressions.

Objectives: Combine a linear equation and a quadratic equation by substitution, then find every solution pair. This page covers A19.

Substitute, then solve the quadratic

A quadratic equation has a squared variable, such as $x^2$. When one equation is linear, substitute its expression into the quadratic. The resulting quadratic can have two solutions.

Example. Solve $y=x+1$ and $y=x^2-3x+1$.

The equations both equal $y$, so set their right sides equal:

Collect terms on one side:

Factorise:

So $x=0$ or $x=4$. Use $y=x+1$: when $x=0$, $y=1$; when $x=4$, $y=5$. The solutions are $(0,1)$ and $(4,5)$. Check each pair in both equations.

Watch out. A quadratic can give two values of $x$. Do not stop after the first root. Substitute every root to find its matching $y$ value, then check each pair.

Progress check

In the example, the two intersections are separate points. Which equation is quickest for finding $y$ once you know $x$? Explain why.

You should now be able to

• set two expressions equal when both equal the same variable;
• solve the resulting quadratic equation;
• find and check every coordinate pair.