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The Question :

Solve for real $x, y$ :
$$xy^2 = 15x^2 + 17xy + 15y^2$$ $$x^2y= 20x^2 + 3y^2$$

My initial attempts involved adding and subtracting the two equations, eliminating one variable, completing some squares and so on. But there was no significant progress as I soon reached a dead end.

The way I was actually able to solve this problem was somewhat unusual : I divided the equations to get $\frac {y}{x} = \frac{15x^2 +17xy + 15y^2}{20x^2 + 3y^2}$. Seemed a very hopeless start, but soon I realised that I could divide the numerator and denominator by $x^2$ on Right Hand Side of the equation, which allowed me to substitute $\frac {y}x= m$. Clearing the denominator, I obtained a cubic equation in $m$ which factored as $(m^2+1)(m-5)=0$. As $x$ and $y$ are real numbers, I rejected the $m^2 +1 = 0$ possibility and got $m=5$ and hence $y = 5x$. After using this relation, I finally got the ordered pair $(x,y) \equiv (19,95)$ as the solution.

I feel that my method is very "robotic" and unnecessarily complicated. Is there any shorter, better, or more elegant way of solving this problem ? I am unable to find any other approach. Thanks in advance.

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    $\begingroup$ Robotic or not, congratulations on solving it! I'm trying to see if there's some obvious simplifying features of the original equations, but nothing's coming to me yet ... $\endgroup$ – bob.sacamento May 4 '18 at 16:09
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    $\begingroup$ I think your method is clever, not robotic. The equations are nearly homogeneous so that substitution stands a good chance of being useful. $\endgroup$ – Ethan Bolker May 4 '18 at 16:16
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    $\begingroup$ You can also solve the4 second equation for $x$ $$x=\pm\sqrt{\frac{3y^2}{y-20}}$$ $\endgroup$ – Dr. Sonnhard Graubner May 4 '18 at 16:18
  • $\begingroup$ @Dr.SonnhardGraubner How does that help? $\endgroup$ – user399078 May 4 '18 at 16:24
  • $\begingroup$ How does that not help? $\endgroup$ – Dr. Sonnhard Graubner May 4 '18 at 16:34
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I think your method is fine. Over the complex numbers we obtain the solutions $$ (x,y)=(0,0),(19,95),(-17i,17),(17i,17), $$ by taking the resultant with respect to $y$, which is $$ 15x^4(x^2+17^2)(x-19)=0. $$

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