[Example 4.4.5] Solve the equation: $\sqrt{x-2}+\sqrt{x-7}=5$.
Solving the equation directly is not difficult; just move $\sqrt{x-7}$ to the right side of the equation, square both sides, and after rearrangement, square again to obtain the solution of the equation.

In addition to the direct solution method mentioned above, considering the overall handling of the equation is also a simple and effective way to solve this problem.
For this purpose, consider a paired equation with the given equation (1):
$$
\sqrt{x-2}-\sqrt{x-7}=m \text {. }
$$

Viewing equation (1) and equation (2) as a whole forms a system of equations with $x$ and $m$ as unknowns.
Multiplying equation (1) and equation (2) can eliminate the radicals.
$$
\begin{array}{c}
(x-2)-(x-7)=5 m, \\
m=1 .
\end{array}
$$

Substituting $m=1$ into equation (2) yields
$$
\sqrt{x-2}-\sqrt{x-7}=1,
$$

Then, combining equation (1) and equation (3) into a system of equations, adding the two equations gives
$$
\begin{array}{c}
2 \sqrt{x-2}=6, \\
x=11 .
\end{array}
$$

Upon verification, $x=11$ is the only root of the original equation.