Is $frac{1}{x} = 4$ strictly a linear equation (in one variable)?

The two expressions are equivalent within the condition that $xneq 0$ 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.