Example 2. (1978 Canadian Mathematical Competition) Determine the maximum real number $z$, where $(x, y$ are also real numbers )
$$
\left\{\begin{array}{l}
x+y+z=5, \\
x y+y z+z x=3 .
\end{array}\right.
$$
Solving, we get $x+y=5-z, xy=3-(5-z)z$. By the inverse of Vieta's formulas, we obtain the equation for $t$
$$
t^{2}-(5-z) t+3-(5-z) z=0
$$

For real roots, we have $\Delta \geqslant 0$, solving which gives $-1 \leqslant z \leqslant \frac{13}{3}$,
$$
z_{\max }=\frac{13}{3}
$$