# How to remove the numerator to find a variable?

If I have

$$60 = 40/X + X$$

How do I remove $40$ and find $X$?

As I understand it, I have to multiply both sides by $X$. But if I do that then I'm left with

$$\begin{eqnarray*} &60X = 40 + X &\Rightarrow \\ &60X - X = 40 &\Rightarrow \\ &X - X = 40 / 60 &\Rightarrow \\ & 1 = 2/3 & \end{eqnarray*}$$

And I think I ended at the wrong spot...

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Minor slip. When you multiply both sides by $X$, you should end up with $60X=40+X^2$. You may have forgotten to multiply the $X$ at the right end by $X$. – André Nicolas Oct 17 '11 at 22:26
@Andre Nicolas - Do I have to do that to all other variables in the equation? for instance if that equation was instead 60 = 40 / X + x^2 + X it would come out to be 60X = 40 + X^3 + x^2? – AedonEtLIRA Oct 17 '11 at 23:14
@Aedon, yes -- you do it for each term, whether or not they contain variables. It's not a special magical rule, it's just the ordinary $a(b+c)=ab+ac$ applied with $a=x$, $b=\frac{40}{x}$ and $c=x$. – Henning Makholm Oct 17 '11 at 23:18

You know $X\ne 0$ so you can multiply by it. That will leave you a quadratic equation for $X$.
Multiplying both sides by $X$ gives $60X=40+X^2\implies X^2-60X+40=0$. – yunone Oct 17 '11 at 22:27