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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!

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

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  • $\begingroup$ Thank you Michael, that is so detailed and helped me understand this well $\endgroup$ – open_sourse Apr 13 '15 at 16:09
  • $\begingroup$ Done, and thanks once more :) $\endgroup$ – open_sourse May 12 '15 at 18:46
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First write as $1-y=\frac1x$. Then reciprocate $\frac{1}{1-y}=x$

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