We actually do a checking for "extraneous solutions", which I will do my best to explain.
Let us start simple, with $x=5$.
Compare this to $x^2=5x$, which you correctly note to be not the same.
Consider solving for $x$ in the first example, then solving for $x$ after we multiply on both sides. This yields
$$x^2=5x\implies x^2-5x=x(x-5)=0\implies x=5,0\tag2$$
Note that $(1)$ was the original, and thus correct, but $(2)$ was affected, and so by nature, is incorrect. But upon further inspection, $(2)$ is partially incorrect, and partially correct.
We call the partially correct portion the 'answer' and the partially incorrect portion the 'extraneous solution'.
To tell the two apart, return to the original equation and plug in every 'solution' you found, just to see which ones are right and which ones are wrong. In this manner, you may work with multiplying equations by variables or anything, as a matter of fact, so long as you remember the rule to 'check' back to the original equation.
Here, we have $5=5$, which is correct, but $0\ne5$, so $0$ is an extraneous solution.
For a problem where this is more apparent than your example, we attempt to solve the following equation for $x$:
We can proceed as follows:
Then, to check for the extraneous solutions,
Only one of the solutions will make sense.
So the point is that you can proceed so long as you check your work.