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.