you can just try to solve those 2 equations by elimination (taking one to minus the other).

it is also helpful to remember the equation

when , this is a line passing through the intersection of the circles.

when , this is a circle passing through the intersection of the 2 circles.

that's if i remember correctly, and this equation is just used generally. for example if the 2 circles do not intersect no such line will even exist. or if the circles intersect at one point only there is an infinite number of lines that pass through that intersection.

others please correct me if i'm wrong