# How to find value of $x$ in this formula

I have this formula:

$$1-\frac 1x=y$$

How do I invert this so that, if I have value of $y$, I want to find value of $x$.

I know, but I am pretty dense in math :(

I dont even know what category to put this under! Any help is appreciated!

• Isolate $1/x$ to one side, then reciprocate both sides. – Gregory Grant Apr 13 '15 at 16:00
• $x=1/(1-y)$ right? – Gregory Grant Apr 13 '15 at 16:00
• @GregoryGrant: Thank you so much! – open_sourse Apr 13 '15 at 16:09
• another approach may be to multiply both side by $x$, yielding $$x-1=xy$$. – John Joy Apr 13 '15 at 16:51

\begin{align} 1 - \frac{1}{x} &= y \\ 1 &= \frac{1}{x} + y \\ 1 - y &= \frac{1}{x} \\ x &= \frac{1}{1-y} \end{align}

If $y = 1$, the above solution is undefined because there is no $x$ such that $1 - \frac{1}{x} = y$ because we would have \begin{align} 1 - \frac{1}{x} &= 1 \\ \frac{1}{x} &= 0 \\ \end{align}

which is undefined (looking at a graph of $f(x) = \frac{1}{x}$ is another way to see this).

• Thank you Michael, that is so detailed and helped me understand this well – open_sourse Apr 13 '15 at 16:09
• Done, and thanks once more :) – open_sourse May 12 '15 at 18:46

First write as $1-y=\frac1x$. Then reciprocate $\frac{1}{1-y}=x$