Consider the homogeneous system of linear equations

$\displaystyle ax + by = 0 $

$\displaystyle cx + dy = 0 $

Prove that if $\displaystyle ad - bc \not= 0 $, then $\displaystyle x = 0,y=0 $ is the only solution to the system.

First, I tried rewriting the system of equations to get $\displaystyle y = -\frac{a}{b} x $ and $\displaystyle y = -\frac{c}{d} x $. This would have probably helped me in the proof, but I realized that I may not be able to divide by $\displaystyle b $ and $\displaystyle d $ because they may be $\displaystyle 0$.

Maybe I could use the contrapositive to prove this. Proving the statement "If $\displaystyle ad - bc = 0$, then $\displaystyle x = 0, y = 0 $ is not the only solution to the system. I'd have to show that there are more solutions. I am not sure how to do this though. Although, this seems like the easiest way to prove it.

Can anyone give me some help? Thanks in advance.