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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?

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    The two equations are *equivalent* for $x\ne 0$. The solution of both equations is $x=\frac{1}{5}\ne 0$.2012-07-08
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    To be precise, an equation is meaningless unless you specify a domain; for most applications, though, the domain is so obvious that it is omitted. If you are working over $\mathbb{R}\setminus \{0\}$ then the equations you give are the same, but of course the first equation is not defined for all of $\mathbb{R}$.2012-07-08

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