How to find points of intersection between ellipse and line?

I'm completely stuck. I have an ellipse, and a line. Line is set by two points, ellipse - by bottom-left and top-right corners. I have to find their points of intersection, using java. I tried to solve an equation system: (1) y = kx + m; x^2/a^2 + y^2/b^2 = 1; but I could't make things work properly. I assume it's because of java's coordinate system, but it also may be my own mistake