Example 1 Find the real solutions of the system of equations
$$
\left\{\begin{array}{l}
x^{3}+x^{3} y^{3}+y^{3}=17, \\
x+x y+y=5
\end{array}\right.
$$

(1990, Shaoxing City Junior High School Mathematics Competition)
Solution: Let $x+y=u, xy=v$, the original system of equations is transformed into $\left\{\begin{array}{l}u^{3}+v^{3}-3uv=17, \\ u+v=5 .\end{array}\right.$
$(2)^{3}-(1)$ gives $uv=6$.
From (2) and (3), we get
$\left\{\begin{array}{l}u=2, \\ v=3\end{array}\right.$ or $\left\{\begin{array}{l}u=3, \\ v=2 .\end{array}\right.$
That is (I) $\left\{\begin{array}{l}x+y=2, \\ xy=3\end{array}\right.$ or (II) $\left\{\begin{array}{l}x+y=3, \\ xy=2 .\end{array}\right.$
From (I), $x, y$ are the roots of the equation $t^{2}-2t+3=0$. Since $\Delta=4-4 \times 3<0$, (I) has no real solutions.

From (II), $x, y$ are the roots of the equation $t^{2}-3t+2=0$. Solving this equation gives $t_{1}=1, t_{2}=2$. Therefore, the real solutions of the original system of equations are