x=y
x=x
1=1
Dividing by x wasn't what really broke the problem though. Dividing by x is normally perfectly legal unless it has been forced to equal 0.
2x=x
2=1
It only broke by dividing by x there because I already divided by x-y a few steps prior which can only equal 0. It changed the graph of the line from y=x [an infinite number of solutions] to a single point [Only one solution, (0,0)] and since x can now only equal 0 at that step, dividing by 0 a second time only breaks it even more and changes it again from one solution to no solution.