Second, are you sure you have copied the problem correctly? You say "if x+ y= x" but if that were true, then obviously, subtracting x from both sides, y= 0 which is impossible "where x, y and z are positive".
And, in any case, there is NO "y" in 2|x- z|- |z-x|. The crucial point is that x- z= -(z- x) so that |x- z|= |z- x|.
2a- a= a.