12. $[8]$ Solve the following system of equations for $w$.
$$
\begin{array}{l}
2 w+x+y+z=1 \\
w+2 x+y+z=2 \\
w+x+2 y+z=2 \\
w+x+y+2 z=1 .
\end{array}
$$
Answer: $\frac{-1}{5}$ Add all the equations together to find that $5 x+5 y+5 z+5 w=6$, or $x+y+z+w=\frac{6}{5}$. We can now subtract this equation from the first equation to see that $w=\frac{-1}{5}$.