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?
Simultaneous equations in four variables
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algebra-precalculus
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1The 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#220512 – 2012-12-23
1 Answers
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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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0@ShaunAult Now fixed – 2012-12-22