5. Solve the system of equations: $\left\{\begin{array}{l}x-y+z=1, \\ y-z+u=2, \\ z-u+v=3, \\ u-v+x=4, \\ v-x+y=5 .\end{array}\right.$
Solving the system of equations by adding them in sequence, we get $\left\{\begin{array}{l}x+u=3, \\ y+v=5, \\ z+x=7, \\ u+y=9, \\ v+z=6 .\end{array}\right.$
Therefore, the solution to the original system of equations is $(x, y, z, u, v)=(0,6,7,3,-1)$.