9. Solve the system of equations in the set of real numbers $\left\{\begin{array}{l}\sqrt{x+1}+\sqrt{y-1}=5, \\ x+y=13 .\end{array}\right.$
9. Let $\sqrt{x+1}=\frac{5}{2}+t$,
$$
\sqrt{y-1}=\frac{5}{2}-t \text {. }
$$

By squaring (1) and (2) and adding them, we get $x+y=2 t^{2}+\frac{25}{2}$.
Substituting (3) into the equation $x+y=13$ yields $2 t^{2}+\frac{25}{2}=13$, solving for $t$ gives $t= \pm \frac{1}{2}$.
When $t=\frac{1}{2}$, from $\left\{\begin{array}{l}\sqrt{x+1}=\frac{5}{2}+\frac{1}{2} \\ \sqrt{y-1}=\frac{5}{2}-\frac{1}{2}\end{array}\right.$ we solve to get $\left\{\begin{array}{l}x=8, \\ y=5 .\end{array}\right.$

Upon verification, the solutions $(x, y)$ to the original system of equations are $(8,5),(3,10)$.