Evaluate

$\Delta=\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$

Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$, we have:

$\Delta=\left|\begin{array}{lll}1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x\end{array}\right|$

Expanding along $\mathrm{C}_{1}$, we have:

$\Delta=1(x y-0)=x y$