I need to get the conditions on $a_i$ for the set: $$x_1+x_2+x_3=a_1a_2a_3$$ $$x_1+x_3+x_4=a_1a_3a_4$$ $$x_1+x_2+x_4=a_1a_2a_4$$ $$x_2+x_3+x_4=a_2a_3a_4$$
The best I get is $$x_1-x_4=a_2a_3(a_1-a_4)$$ and 5 other combinations where the LHS subscript is the same for the RHS subscripts in the bracket.
How do i continue?
\begin{align*} \left( \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right) &= \left( \begin{array}{cccc} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \\ \end{array} \right)^{-1} \left( \begin{array}{c} a_{2} a_{3} a_{4} \\ a_{1} a_{3} a_{4} \\ a_{2} a_{2} a_{4} \\ a_{1} a_{2} a_{3} \\ \end{array} \right) \\& \\ &= \left( \begin{array}{cccc} -\frac{2}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & -\frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & -\frac{2}{3} \\ \end{array} \right) \left( \begin{array}{c} a_{2} a_{3} a_{4} \\ a_{1} a_{3} a_{4} \\ a_{2} a_{2} a_{4} \\ a_{1} a_{2} a_{3} \\ \end{array} \right) \end{align*}