Example 8 (1997 National High School League Original Problem) Let $x, y$ be real numbers, and satisfy $\left\{\begin{array}{l}x^{3}-3 x^{2}+2000 x=1997, \\ y^{3}-3 y^{2}+2000 y=1999,\end{array}\right.$ then $x+y=$ $\qquad$
Solve: Fill in 2. Reason: The conditional expression can be transformed into $\left\{\begin{array}{l}(x-1)^{3}+1997(x-1)=-1, \\ (y-1)^{3}+1997(y-1)=1 .\end{array}\right.$
Consider the function $f(t)=t^{3}+1997 t$, then $f(t)$ is monotonically increasing on $\mathbf{R}$, thus the conditional expression becomes $f(x-1)=f(1-y)$, hence we have $x-1=1-y$, which means $x+y=2$.