Example 8 (Question from the 7th Canadian Junior High School Mathematics Competition) Determine the largest real number $z$ such that the real numbers $x, y$ satisfy:
$$\left\{\begin{array}{l}
x+y+z=5 \\
x y+y z+x z=3 .
\end{array}\right.$$
Solve: From $x+y+z=5$, we get
$$x^{2}+y^{2}+z^{2}+2(xy+yz+xz)=25$$

Also, $xy+yz+xz=3$, substituting into (1) we get
$$x^{2}+y^{2}+z^{2}=19 \text{.}$$

Thus, $x^{2}+y^{2}=19-z^{2}$ can be seen as a circle in terms of $x, y$. By the corollary, we get
$$2(19-z^{2}) \geqslant (5-z)^{2}.$$

Solving this, we get $3z^{2}-10z-13 \leqslant 0$,
which gives $-1 \leqslant z \leqslant \frac{13}{3}$,
so the maximum real value of $z$ is $\frac{13}{3}$.