Simultaneous equations: one linear, one quadratic

By the end you can solve a pair of simultaneous equations where one is linear and one is quadratic, by substitution, and give the solutions as matching pairs.

Simultaneous equations are two equations that must both be true at the same time, for the same $x$ and $y$ .

A linear equation has no powers above 1, like $y = 2x - 5$ . Its graph is a straight line.
A quadratic equation has terms of degree 2: squared terms like $x^2$ or $y^2$ , or a product like $xy$ . Examples: $x^2 + y^2 = 10$ , $x^2 + xy = 6$ .

You can't use elimination (adding or subtracting the equations) here, because no variable can be made to cancel. Use substitution:

Make $x$ or $y$ the subject of the linear equation. Pick whichever avoids fractions.
Substitute it into the quadratic equation. Now there's only one variable.
Rearrange to $= 0$ and solve the quadratic.
Put each value back into the linear equation to find its partner.
Write the answers as pairs: coordinates, or " $x = \ldots, y = \ldots$ ".

Each equation is a graph. A solution pair is a point on both graphs: an intersection point.

Substitution hands you a quadratic, which has at most two roots.

So a straight line can meet a parabola or a circle at most twice. The algebra and the picture agree.

Solve the simultaneous equations $y = 2x - 5$ and $x^2 + y^2 = 10$ .

Solve the simultaneous equations $y = x - 1$ and $x^2 + y^2 = 13$ . Enter the solution with the smaller $x$ first.

For the pair $3x + y = 7$ and $x^2 + xy = 6$ , which is the neatest first step?

The method