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$$\frac1{x} = 4$$ If multiplied by $x$ both sides, $$1 = 4x$$ Then it looks like linear equation in one variable. But is such multiplication by variable on both sides allowed?

Air Mike
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    No, it is not, but it can be transformed into a linear equation except that $x$ cannot be $0$. – Peter Sep 28 '20 at 09:29
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    Whenever you multiply with (or divide by) an expression containing $x$, you must be careful that this expression is not $0$. Many fake proofs are based on "division by $0$" or "multiplication with $0$" – Peter Sep 28 '20 at 09:30
  • $x$ is an unknown, but is not a variable – 1123581321 Sep 28 '20 at 10:09

1 Answers1

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The two expressions are equivalent within the condition that $x\neq 0$ indeed for $x=0$ we would obtain

  • $\frac1{0} = 4$
  • $1=0$

which are undefined.

We can observe that as equations in the unknown $x$ they are equivalent in the sense that they lead to the same solution $x=\frac14$.

Otherwise for the expressions $\frac {x^3}{x}=1$ and $x^3=x$ are not equivalent as equations because $x=0$ is a solution for the second one but not for the first one.

user
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