Prove that, where $a,b, \ldots, e$ are real numbers and $a \neq 0$, if $ax + by = c$ has the same solution set as $ax+dy=e$ then they are the same equation. What if $a=0$?
Note: If $a \ne 0$ then the solution set of the first equation is $\{(x,y) \mid x=c-by/a\}$.