Example 8 Find the real solutions of the system of equations $\left\{\begin{array}{l}x^{3}+x^{3} y^{3}+y^{3}=17, \\ x+x y+y=5\end{array}\right.$.
Let $x+y=u, xy=v$, then the original system of equations can be transformed into
$$
\left\{\begin{array}{l}
u^{3}+v^{3}-3uv=17, \\
u+v=5 .
\end{array}\right.
$$
(2) $)^{3}$ - (1) gives
$$
uv=6 \text{.}
$$

From equations (2) and (3), we get
$$
\left\{\begin{array}{l}
u=2, \\
v=3
\end{array} \text{ or } \left\{\begin{array}{l}
u=3, \\
v=2,
\end{array}\right.\right.
$$

That is, $\left\{\begin{array}{l}x+y=2, \\ xy=3\end{array}\right.$, or $\left\{\begin{array}{l}x+y=3, \\ xy=2 .\end{array}\right.$
By Vieta's formulas, $x$ and $y$ in these two systems of equations are the roots of the equations $t^{2}-2t+3=0$ or $t^{2}-3t+2=0$. The first equation has no real roots, solving the second equation gives $t=1$ or $t=2$.
Therefore, the real solutions to the original system of equations are
$$
x_{1}=1, y_{1}=2 ; x_{2}=2, y_{2}=1 \text{. }
$$