# Can a variable appear as a denominator in a linear equation or is the system non-linear?

I am trying to figure out whether the following equation is non-linear or if it's linear, how would I solve it?

$x+\frac{2}{y}=0$

It can be rewritten as $x+y^{-2}=0$ so I guess if this is non-linear.

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Indeed, the function is not linear. – Arturo Magidin Sep 25 '11 at 21:43
But it would be $xy+2=0$, not $x+y^{-2}=0$. – Ross Millikan Sep 25 '11 at 21:50
Or it would be $x + 2y^{-1}=0$. – Arturo Magidin Sep 25 '11 at 21:54
But if all your $y$'s appear as denominators, you can make a linear equation system by substituting $u=1/y$. – Henning Makholm Sep 25 '11 at 22:31
@DBLim: When you are dealing with a question from a beginner that is trying to puzzle out some of the basics, please don't "correct" what was likely an error rather than a typo as an edit; it's important to point out those errors to prevent them from being committed again. If you simply edit them out, the OP may not realize he had made an algebraic error along the line. – Arturo Magidin Sep 25 '11 at 22:34

It is non-linear, and the criteria of linearity I've put in answer to your previous question. Here $f(x,y) = x+\frac2y$ and for $\alpha = 1,\beta = 1$ we have $$f(x'+x'',y'+y'') = x'+x''+\frac2{y'}+\frac2{y''}\neq x'+x''+\frac2{y'+y''} = f(x',y')+f(x'',y'')$$ thus the equation is non-linear.
Saw the possible misprint in your question: $\frac2y\neq y^{-2} = \frac1{y^2}$.
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