Is $x=y=z=0$ ?

Algebra Level 5

$\large{\begin{cases} (x-2)(y-2)=4 \\ (y-3)(z-3)=9 \\ (z-4)(x-4)=16 \end{cases}}$

Let $x,y$ and $z$ be positive rational numbers satisfying the system of equations above, find the value of $5x+7y-z$ .