Stubborn Equations

$y+u+x+v=0$ ................... $(1)$
$z+y+v+u=1$ ................... $(2)$
$x+y+z+u=5$ ................... $(3)$
$z+u+v+x=2$ ................... $(4)$
$v+x+y+z=4$ .................... $(5)$

Adding these five equations we get,
$4(x+y+z+u+v)=12$
so, $(x+y+z+u+v=3$ ................ $(6)$

If , $(6) - (1)$ then, $z=3$
If, $(6) - (2)$ then, $x=2$
If, $(6) - (3)$ then, $v=-2$
If, $(6) - (4)$ then, $y=1$
If, $(6) - (5)$ then, $u=-1$

So, $xyzuv=2*1*3*-1*-2=\boxed{12}$

y + u + x + v = 0

z + y + v + u = 1

x + y + z + u = 5

z + u + v + x = 2

v + x + y + z = 4

add all the equations and you will get

4( u + v + x + y + z ) = 12

u + v + x + y + z = 3 call this equation A

subtract each of the 5 given equations from equation A and you will get

z = 3

x = 2

v = -2

y = 1

u = -1

multiply all the equations, the answer is 12.