For example, solving the equation $\frac{x^{2}-7}{6}=\sqrt{6 x+7}$. This is the result of transitioning from concrete to abstract thinking.

From a geometric perspective, this is equivalent to finding the x-coordinates of the intersection points of the line $y=\frac{x^{2}-7}{6}$ and the curve $y=\sqrt{6 x+7}$. These two happen to be inverse functions of each other. According to the geometric property that the graphs of a function $y=f(x)$ and its inverse $y=f^{-1}(x)$ are symmetric about the line $y=x$, their intersection points must lie on the line $y=x$.
Solve $\sqrt{6 x+7}=x$ to get
$$
\begin{array}{l}
x^{2}-6 x-7=0, \\
\therefore x_{1}=-1 \text { (discard), } x_{2}=7 .
\end{array}
$$

Upon verification, the root of the original equation is $x=7$.