6・94 Determine the largest real number $z$, such that
$$x+y+z=5, \quad xy+yz+zx=3,$$

and $x, y$ are also real numbers.
[Solution] From $x+y+z=5$, we get
$$(x+y)^{2}=(5-z)^{2},$$

From $x y+y z+z x=3$, we get
$$x y=3-z(5-z) .$$

Thus, we have
$$\begin{aligned}
(x-y)^{2} & =(x+y)^{2}-4 x y=(5-z)^{2}-4[3-z(5-z)] \\
& =-3 z^{2}+10 z+13=(13-3 z)(1+z),
\end{aligned}$$

Therefore,
$$\begin{array}{l}
(13-3 z)(1+z) \geqslant 0 \\
-1 \leqslant z \leqslant \frac{13}{3}
\end{array}$$

When $x=y=\frac{1}{3}$, $z=\frac{13}{3}$. Therefore, the maximum value of $z$ is $\frac{13}{3}$.