This is giving me some headache.
Suppose I have an equation $$\frac{1}{x}=5$$
Then is this equation the same as $$1=5x\quad ?$$
Now the domain of $x$ in the first equation is $\mathbb{R}\setminus \{0\}$, however the domain of $x$ in the second equation is the whole $\mathbb{R}$.
Does rearranging terms change the meaning (I don't know the right word here) of equations?