7. The real numbers $x, y, z, w$ satisfy
$$
\begin{array}{l}
2 x+y+z+w=1 \\
x+3 y+z+w=2 \\
x+y+4 z+w=3 \\
x+y+z+5 w=25 .
\end{array}
$$

Find the value of $w$.
Solution: $11 / 2$. Multiplying the four equations by $12,6,4,3$ respectively, we get
$$
\begin{aligned}
24 x+12 y+12 z+12 w & =12 \\
6 x+18 y+6 z+6 w & =12 \\
4 x+4 y+16 z+4 w & =12 \\
3 x+3 y+3 z+15 w & =75 .
\end{aligned}
$$

Adding these yields $37 x+37 y+37 z+37 w=111$, or $x+y+z+w=3$. Subtract this from the fourth given equation to obtain $4 w=22$, or $w=11 / 2$.