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I should solve the following system: $$\begin{cases} x^3-y^3=19(x-y) \\ x^3+y^3=7(x+y) \end{cases}$$ by reducing the system to a system of second degree.

We can factor: $$\begin{cases} (x-y)(x^2+xy+y^2)=19(x-y) \\ (x+y)(x^2-xy+y^2)=7(x+y) \end{cases}$$ I really don't want to divide the equations by $x-y$ and $x+y$, respectively. I am taught to divide by expressions containing variables only in special cases. Is there any other way here?

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You can divide by variables if you ensure they are not zero. Here, you can consider the cases $x=y$ and $x=-y$ first. If $x=y$ the first equation is trivial and the second becomes $2x^3=14x$ or $x=0,\pm \sqrt 7$. You can do the same for $x=-y$ and find a pair of solutions.

Then decree that $x+y \neq 0, x-y \neq 0$ and divide away. Once you do that, you can subtract the two equations to get $2xy=12$ and use that to get expressions for $(x+y)^2, (x-y)^2$

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  • $\begingroup$ Thank you for the response! $2x^3=14x$ and I think you missed $x=0$. $\endgroup$ – Katherine Jan 7 '20 at 20:15
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    $\begingroup$ I left that for you in the second one:). Actually I missed it and have edited it in. $\endgroup$ – Ross Millikan Jan 7 '20 at 20:27
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After considering of cases $x=y$ or $x=-y$ use $$7(x^2+xy+y^2)=19(x^2-xy+y^2).$$

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  • $\begingroup$ Can down-voter explain us, why did you do it? $\endgroup$ – Michael Rozenberg Jan 20 '20 at 5:31
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The thing that catches my eye is that $7,19$ are primes congruent to $1 \pmod 3,$ therefore integrally represented by both $x^2 + xy+y^2, \; x^2 -xy + y^2,$ meaning there are integer points on the two ellipses. It is worth drawing both, by hand (a valuable skill), to see whether that gives a simplified answer when $y \neq \pm x.$

enter image description here

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To avoid all possible problems, add the two equations to get $$x^3-13 x+6 y=0 \implies y=\frac{1}{6} \left(13 x-x^3\right)$$ Plug in the first equation to end with $$x \left(x^8-39 x^6+507 x^4-2665 x^2+4788\right)=0$$ So $x=0$ if a root.

For the remaining, let $z=x^2$ to make $$z^4-39z^3+507z^2-2665z+4788=0$$ By inspection, $z=4$ and $z=7$ are solutions. Now, long division $$\frac{z^4-39z^3+507z^2-2665z+4788 } {(z-4)(z-7) }=z^2-28 z+171=(z-19) (z-9)$$

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