Large numbers, easy trick

$\begin{cases}5732x + 2134y+2134z=7866&\implies \boxed{1}\\ 2134x + 5732y+2134z=670&\implies \boxed{2}\\ 2134x + 2134y+5732z=11464&\implies \boxed{3}\end{cases}$

The trick: $\boxed{1} + \boxed{2} + \boxed{3}$

$(2134 + 2134 + 5732)(x+y+z) = 7866+670+11464\\ 10000(x+y+z) = 20000\\ x+y+z = 2\\ x+y=2-z \implies \boxed{4}$

Substitute $\boxed{4}$ into $\boxed{3}$ :

$2134(x+y) + 5732z = 11464\\ 2134(2-z) +5732z = 11464\\ 4268 - 2134z + 5732z = 11464\\ 3598z = 7196\\ z=2$

Substitute $z=2$ into $\boxed{4}$ :

$x+y = 2-2\\ x=-y\implies \boxed{5}$

Substitute $z=2$ and $\boxed{5}$ into $\boxed{2}$ :

$2134(-y) + 5732y +2134(2) = 670\\ 3598y = -3598\\ y=-1$

Therefore, $x = -(-1) = 1$

$xyz = 1(-1)(2) = \boxed{-2}$

Wow! I knew I didn't have to write a solution, because I have you to fill in for me, :)