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 \(5x+7y-z\).