If $x+y=12$ and $x-y=8$, what is the value of $2x-xy$?
First, notice $2x - xy = x(2 - y)$. So this problem reduces to finding the values of $x$ and $y$.

Add the two equations together to find $x$: \begin{align*}
2x &= 20, \\
x &= 10.
\end{align*}
Subtract the two equations to find $y$: \begin{align*}
2y &= 4, \\
y &= 2.
\end{align*}
Since $y = 2$ and $x = 10$, $x(2 - y) = 10(2 - 2) = \boxed{0}$.