Hi,

your principle is wrong because by = you're saying that a point (x,2x) must map to itself. This condition is too strong and as you showed it has no solution fork. What you have to consider is (because (x,2x) can be mapped to arbitrary point (w,2w))

and after eliminatingwfrom two simultaneous equations you get the right result (or you get necessary condition fork, and by linearity you see that it is also sufficient)