<s>[INST] A Five Equations problem? If,
$$\begin{align*}
    y+u+x+v&=0\\
    z+y+v+u&=1\\
    x+y+z+u&=5\\
    z+u+v+x&=2\\
    v+x+y+z&=4\,,
\end{align*}$$
What is the value of $xyzuv$?
 [/INST] $$\begin{align*}
    y+u+x+v&=0\tag1\\
    z+y+v+u&=1\tag2\\
    x+y+z+u&=5\tag3\\
    z+u+v+x&=2\tag4\\
    v+x+y+z&=4\tag5
\end{align*}$$                                                            
Adding these five equations, we get:
$$\begin{align*}
    4(x+y+z+u+v)&=12\\
    x+y+z+u+v&=3\tag{6}
\end{align*}$$
$$
\begin{align}
(6) - (1) &\implies z=3\\
(6) - (2) &\implies x=2\\
(6) - (3) &\implies v=-2\\
(6) - (4) &\implies y=1\\
(6) - (5) &\implies u=-1
\end{align}
$$
So, from these results,
$$
x\cdot y\cdot z \cdot u\cdot v = 2\times 1 \times 3 \times (-1) \times (-2)$$
$$
\therefore  \boxed{\ x\,y\,z\,u\,v = 12\ }
$$
</s>