smc

If the sum of three squared expressions is zero, then each expression itself must be zero, since $a^2\ge 0$ with the equality iff $a=0$. In this case, $x-3=0$, $y-4=0$, and $z-5=0$. Adding the three equations and moving the constant to the right gives $x+y+z=12$, and the answer is $\boxed{12}$.