Example 11. Are the equations

$$
\frac{2(x-10)}{x^{2}-13 x+30}=1 \quad \text { and } \quad x^{2}-15 x+50=0
$$

equivalent?
Solution. We will solve the first equation. By getting rid of the denominator, i.e., multiplying both sides of the original equation by the expression $x^{2}-13 x+30$, we obtain the equation

$$
2 x-20=x^{2}-13 x+30
$$

The set of all roots of this equation consists of two numbers: $x_{1}=10$ and $x_{2}=5$. As a result of the transformation, extraneous roots could have appeared; therefore, it is necessary to check. The check shows that the number $x_{1}=10$ is not a root of the original equation, while the number $x_{2}=5$ is its root, i.e., the first equation has a single root $x=5$.

The equation $x^{2}-15 x+50=0$ has two solutions: $x_{1}=5$ and $\boldsymbol{x}_{2}=10$. Comparing the sets of roots of the given equations, we get: the second equation is a consequence of the first.