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I've been solving various quadratic equation for some time now, and I found 2 equations that I am unable to solve:

$x^2-y^2=5 $

$\frac{1}{x}+\frac{1}{y}=-\frac{1}{6}$

This is one of the equations. The problem I have is if I exchange $y$ with and expression with $x$ I will always get $xy$, and I just can't get rid of $xy$ and I can't express $xy$ by anything else.

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    If you want to solve these two equations *simultaneously,* for both $x$ and $y$, you should probably edit your question. Both @Didier and I misunderstood your goal as wanting to solve each equation for $y$, separately.2012-04-01

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I understand that you want to solve these equations simultaneously?

First take the equation $\frac{1}{x}+\frac{1}{y}=−\frac{1}{6}$

We'll rearrange this to find $y$: $\frac{1}{y}=−\frac{1}{6}-\frac{1}{x}$

Now put the RHS into one fraction: $\frac{1}{y}=-\frac{6+x}{6x}$ Then reciprocate both sides: $y=-\frac{6x}{6+x}$

Now, you can substitute this into the first equation (or, another method that you prefer). Be careful though - this may result in a quartic equation. Consider writing the first equation as $(x-y)(x+y) = 5$ before substituting.

Hopefully it should be clear what to do from here.

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    It's a shame that :) is too few characters for a valid comment. ;)2012-04-01
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$y=\pm\sqrt{x^2-5}\qquad\text{and}\qquad y=-6x/(6+x)$