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I'm solving the following equations, $$x+y=zw$$ $$z+w=xy$$ How many solutions $(x,y,z,w)$ exist, if the variables are reals?

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    Since your system is underdetermined, there will be infinitely-many solutions.2012-12-22
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    Of course, a priori, those infinitely many solutions could all be complex or at infinity (in projective space).2012-12-22
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    The same equations where one wants integer solutions appeared on this forum. http://math.stackexchange.com/questions/219762/ab-c-times-d-and-a-times-b-c-d/220512#2205122012-12-23

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Solving the 1st equation for $y$ and substituting in the second gives $$z+w=xzw-x^2\iff x^2-xzw+z+w=0$$ This equation has a solution for $x$ when $$\Delta\ge 0\iff z^2w^2-4z-4w\ge 0$$ It remains to check that it always has solutions for $z,w$. Therefore, your system has an infinite number of solutions all of which satisfy: $$z^2w^2-4z-4w\ge 0$$ $$x=\frac{zw-\sqrt{z^2w^2-4z-4w}}2$$ $$y=zw-x$$

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    oops... I think you meant $y = zw - x$ in your last line.2012-12-22
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    @ShaunAult Now fixed2012-12-22