$4 \cdot 143$ Solve the system of equations
$$
\left\{\begin{array}{l}
x-y+z=1, \\
y-z+u=2, \\
z-u+v=3, \\
u-v+x=4, \\
v-x+y=5 .
\end{array}\right.
$$
[Sol] (2) + (3) - (5) gives
$x=0$
(3) + (4) - (1) gives
$y=6$,
(4) + (5) - (2) gives
$z=7$
(5) + (1) - (3) gives
$u=3$,
(1) + (2) - (4) gives $\quad v=-1$.

Upon verification, the solution to the original system of equations is
$$
\left\{\begin{array}{l}
x=0 \\
y=6 \\
z=7 \\
u=3 \\
v=-1
\end{array}\right.
$$