My equation produces the undefined fraction 0/0 in that case. However, my equation isn't necessary because both lines become y = -x. They're the same line, so there's not one point of intersection. If you find a time my equation doesn't work for lines that aren't the same, I'm interested.
You originally asked about a way to prove a general method (one without exceptions mentioned), but your method isn't general, as it fails when there are no common points (distinct parallel lines) and when there are common points that you don't call intersection points (because the lines are identical).