Example 9 Solve the equation $\sqrt{x-2}+\sqrt{x-7}=5$.

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Example 9 Solve the equation $\sqrt{x-2}+\sqrt{x-7}=5$.

This is the direct translation of the provided Chinese text into English, maintaining the original format and line breaks.
Analysis: Directly solving this equation requires squaring twice. However, if we consider the equation paired with it,
$$
\sqrt{x-2}-\sqrt{x-7}=m \text{, }
$$

treating the two as a system of equations with $x, m$ as unknowns.
Multiplying equation (1) and equation (2) eliminates the radicals, yielding $(x-2)-(x-7)=5m$.
Solving for $m$ gives $m=1$.
Substituting $m=1$ into equation (2), and adding it to equation (1) results in
$$
2 \sqrt{x-2}=6 .
$$

Solving this gives $x=11$.
Substituting $x=11$ into equation (1) verifies that $x=11$ is a root of the original equation.