How do I find the intersection of a line and a curve?

Say you are given the equations of both a line and a curve, for example y=x²+8x-1 and y=3x-7, and asked to find where these two intersect. This just means where the two lines would cross or touch if drawn on the same graph.

To find these points you simply have to equate the equations of the two lines, where they equal eachother must be the points of intersection.

For this example this would mean x²+8x-1=3x-7

Collecting like terms leads to x²+5x+6=0

And from then this is a simple case of solving the quadratic. This expression factorises to (x+2)(x+3)=0 which implies either x=-2 or x=-3. To find the corresponding y coordinates for each point simply input these x values one at a time into either one of the original equations.

For x=-2, we get y=3(-2)-7=-13 so the point is (-2,-13)

For x=-3, we get y=3(-3)-7=-16 so the point is (-3,-16)

Both of these are valid intersection points for the line and curve given.