# Problem with solving quadratic equation

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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What's your question? Do you just want the answers, or are you looking for help with the solving process? For $x^2-y^2=5$, did you get as far as $y^2 = x^2 - 5$? I don't see where $xy$ comes in. – LarsH Apr 1 '12 at 9:55
I don't want the answer. Of course I've expressed $y^2$ as $x^2-5$. If i use that in second equation i will always get $xy$ – Transcendental Apr 1 '12 at 9:58
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. – LarsH Apr 1 '12 at 10:12

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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Good insight into what the OP (may have) meant, and good pedagogy (anticipating the pitfalls). – LarsH Apr 1 '12 at 10:08
Thanks: your comment discussion helped to confirm this for me. – Ronald Apr 1 '12 at 10:10
Ooooh, so that's what he means by "If i use that in second equation". – LarsH Apr 1 '12 at 10:13
It's a shame that :) is too few characters for a valid comment. ;) – Ronald Apr 1 '12 at 10:18

$$y=\pm\sqrt{x^2-5}\qquad\text{and}\qquad y=-6x/(6+x)$$

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