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$x^2+y^2-z^2=20$

$x^4+y^4-z^4=560$

$x^3+y^3+z^3=3xyz$

I know the fact that if $x^3+y^3+z^3=3xyz$ then $x+y+z=0$ (coming from Euler's identity) and first equation can be written as

$(x+y-z)^2-2(xy-xz-yz)=20$ and since $x+y=-z$ then $(xy-xz-yz)=2z^2-10$.

And the second equation becomes

$(x^2+y^2-z^2)^2-2(x^2y^2-2x^2z^2-2y^2z^2)=560$.

Since we know $x^2+y^2-z^2=20$, therefore $20^2-2(x^2y^2-2x^2z^2-2y^2z^2)=560$ or

$(x^2y^2-2x^2z^2-2y^2z^2)=80$.

Now I am stuck! Help please!

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  • $\begingroup$ if $x^3+y^3+z^3=3xyz$ then $x+y+z=0$ or $x=y=z$,but $x=y=z$ can't such other condition, and you can let $z=-x-y$ $\endgroup$ – math110 Jan 8 '15 at 14:49
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Hint

Using $z=-x-y$, the first equation write $2 x y+20=0$ and so $y=-\frac{10}{x}$. Now the second equation write, after minor simplifications, $$x^2+\frac{100}{x^2}=29$$ in which you recognize a quadratic in $x^2$.

I am sure that you can take from here.

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since$$x^3+y^3+z^3=3xyz\Longrightarrow (x+y+z)(x^2+y^2+z^2-xy-yz-xz)=0$$ $$\Longrightarrow x+y+z=0 ,\text {or} ,x=y=z(\text{impossible})$$ let $z=-x-y$,then you have $$x^2+y^2-z^2=20\Longrightarrow x^2+y^2-(x+y)^2=20\Longrightarrow xy=-10$$ since $$x^4+y^4=(x^2+y^2)^2-2x^2y^2=(20+z^2)^2-200=z^4+40z^2+200$$ so $$z^4+40z^2+200-z^4=560\Longrightarrow z^2=9\Longrightarrow z=\pm 3$$ then It is easy to find $x,y$

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This is a well-knonw example for solving a system of polynomial equations via Groebner bases. The real solutions are given by (check this yourself): $$ (x,y,z)=(5,-2,-3),\; (-2,5,-3),\;(2,-5,3),\;(-5,2,3). $$ Over the complex numbers, there are another $8$ solutions for $z$, i.e., the roots of the polynomial $$ 9z^8 - 240z^6 + 1600z^4 + 57600z^2 + 230400=0, $$ and $y=-\frac{1}{120(z^2 + 8)}(120xz^2 + 960x + 3z^5 - 280z^3 - 1440z)$, $x^2 =z^2- y^2 +20$.

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  • $\begingroup$ Since the only category proposed is algebra-pre-calculus, it is very probable that the person who asked this question does not know about Groebner bases. $\endgroup$ – Bernard Massé Jan 8 '15 at 15:15
  • $\begingroup$ This is of course very likely, I admit. On the other hand, the question does not really belong to algebra-pre-calculus alone. It is also about solving systems of polynomial equations. How to find solutions, real and complex ones, might be interesting for the OP. And algebra-pre-calculus answers have been given anyway (assuming real solutions). $\endgroup$ – Dietrich Burde Jan 8 '15 at 15:33

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