Finding an intersection between something and a line

I have a set of points which are interpolated with an unknown method, or to be more precise, the method is known but it can be one of the several - it can be polynomial interpolation, spline, simple linear ... - and a line, which, let's for now imagine it is given in the simple form of y = ax + b.

For interpolation, I don't know what method is used (i.e. the function is hidden), so I can only determine y for some x, and equally, x for a given y value.

What is the usual way to go about finding an intersection between the two?