To solve this system of equations, we can use the method of substitution. Let's start by expressing each variable in terms of the others.

From the first equation:

x = 5 - y - z - u

Substitute x into the second equation:

y + z + u + v = 1

y + z + u + v = 1
y + z + u + 5 - y - z - u = 1
5 + v = 1
v = -4

Substitute v into the third equation:

z + u + x = 2

z + u + 5 - y - z - u = 2
5 - y = 2
y = 3

Substitute y into the fourth equation:

u + v + x + y = 0

u +(-4) + 5 - 3 = 0
u = 2

Finally, substitute u into the fifth equation:

v + x + y + z = 4

-4 + 5 + 3 + z = 4
4 + z = 4
z = 0

Therefore, the solution to the system of equations is:

x = 5 - 3 - 0 - 2 = 0
y = 3
z = 0
u = 2
v = -4