Attached, find a graphic of a Plane and a Line.
I'm hoping the solution can be conveyed without Matrix math, and am thinking it ought to be solvable using 'conventional' simutaneous equations (which I almost 'get').
Too, I'm hoping the solution will be fairly 'general', in the sense that, after understanding this first effort, that I will be able to apply it to a more complex surface.
I appreciate your work in visualizing the plane and line in space and ...
A solution to the intersect of a line and a surface is found using a substitution method.