Example 8 Determine the largest real number $z$, such that $x+y+z=5, xy+yz+zx=3$, and $x, y$ are also real numbers.
(7th Canadian Competition Question)
From the given, we have $x^{2}+y^{2}+z^{2}=(x+y+z)^{2}-2(xy+yz+zx)=5^{2}-2 \cdot 3=19$. Let the line $l: x+y+(z-5)=0, \odot O: x^{2}+y^{2}=19-z^{2}$.
From the given, we know that $l$ and $\odot O$ have a common point, thus $\frac{|z-5|}{\sqrt{2}} \leqslant \sqrt{19-z^{2}}$, which means $3z^{2}-10z-13 \leqslant 0$.
Solving this, we get $-1 \leqslant z \leqslant \frac{13}{3}$, hence $z_{\max}=\frac{13}{3}$.