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Is there any elegant way of solving this system of equations?

\begin{gather*} a_1 x^2+a_2 x+ a_3 y^2+a_4 y+a_5 z^2+a_6 z+a_7=0 \\
b_1 x^2+b_2 x+b_3 y^2+b_4 y+b_5 z^2+b_6 z+b_7=0\\
c_1 x^2+c_2 x+c_3 y^2+c_4 y+c_5 z^2+c_6 z+c_7=0 \end{gather*}

Still, any solution will be appreciable.

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    @J.M. well it guess it is, myself have formulated few steps, but ... still this o$n$e is out of curiosity.2011-04-20

1 Answers 1

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Plug it in Wolfram Alpha.

More seriously, Wolfram Alpha will use Groebner bases to find all solutions to any polynomial system of equations.

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    sorry url can be found [here](http://www.wolframalpha.com/input/?i=%20%28x-5%29^2%2b%28y-8%29^2%2b%28z-1%29^2%20=%20r^2,%20%28x-2%29^2%2b%28y-0%29^2%2b%28z-4%29^2%20=%20r^2,%20%28x-3%29^2%2b%28y%2b3%29^2%2b%28z%2b1%29^2%20=%20r^2)2011-04-21