10. (2003 German Mathematical Olympiad) Find all pairs of real numbers $(x, y)$ such that $x, y$ satisfy the system of equations $\left\{\begin{array}{l}x^{3}+y^{3}=7, \\ x y(x+y)=-2 .\end{array}\right.$
10. Let $x^{3}+y^{3}=7$ be equation (1), and $x y(x+y)=-2$ be equation (2). Adding (1) and 3 times (2) gives $x^{3}+y^{3}+3 x y(x+y)=1$, so $(x+y)^{3}=1$, which means $x+y=1$. Substituting $x+y=1$ into (2), we get $x y=-2$, hence $x$ and $y$ are the two real roots of the quadratic equation $t^{2}-t-2=0$. Solving this, we find $t_{1}=2, t_{2}=-1$, so the solutions to the original system of equations are $(x, y)=(2,-1)$ or $(-1,2)$.