Example. Regarding the equation about $x$: $x+\frac{1}{x-1}=a+\frac{1}{a-1}$.
Solve the original equation $\stackrel{\text { transformation }}{\Longleftrightarrow} x-1+\frac{1}{x-1}=a-1+\frac{1}{a-1}$ $\stackrel{\text { Proposition 2 }}{\Longleftrightarrow} x-1=a-1$ or $x-1=\frac{1}{a-1}$, solving these gives $x_{1}=a$, $\mathbf{x}_{2}=\frac{\mathrm{a}}{\mathrm{a}-1}$ as the roots of the original equation.