Moise cites a the following theorem which holds in the algebra of the real numbers (please direct me to a proof of this as well):
If $ab-cd\ne0$ then the following system of equations:
${\begin{cases}ax+dy=e\\cx+by=f\end{cases}}$
is satisfied by exactly one pair of numbers $(x,y)$.
Does this theorem hold true in any commutative ring with unity? Does it hold true in any field?
Your help is greatly appreciated.