Example 5. Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=\sqrt{x+y+z+1}+5, \\
\frac{x}{2}=\frac{y}{3}=\frac{z}{4} .
\end{array}\right.
$$
Let $\sqrt{x+y+z+1}=t$, then (1) can be transformed into
$$
i^{3}-t-6=0 \text {. }
$$

We get $t=3$ or $t=-2$ (discard $t=-2$)
F: So, $\sqrt{x}-y+z+1=3, x+y+z=8$.
From (2), we get $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}=\frac{x+y+z}{2+3+4}=\frac{8}{9}$.
Therefore, the solution to the original system of equations is
$$
x=\frac{16}{9}, \quad y=\frac{8}{3}, \quad z=\frac{32}{9} .
$$