Researched Gaussian elim more, does what I need. Thanks. Too bad you didn't make it a full answer, or I would have accepted it.

Right. I was aware in the back of my mind that vertical distance is what's used in regression and that the reason for this is that you the interpretation of a regression is that it gives you the best possible predictor of Y given X (minimum vertical distance) assuming the relevant assumptions are met. However, for some reason I kept visualizing the problem as using perpendicular distance.

Thanks. I was pretty sure I had just overlooked something relatively obvious. This appears to be the obvious thing I had overlooked.