Question 118
Suppose the real numbers $x_1, x_2, x_3, x_4, x_5$ satisfy the following set of five equations:
$$
\begin{cases}
x_1x_2 + x_1x_3 + x_1x_4 + x_1x_5 = -1 \\
x_2x_1 + x_2x_3 + x_2x_4 + x_2x_5 = -1 \\
x_3x_1 + x_3x_2 + x_3x_4 + x_3x_5 = -1 \\
x_4x_1 + x_4x_2 + x_4x_3 + x_4x_5 = -1 \\
x_5x_1 + x_5x_2 + x_5x_3 + x_5x_4 = -1
\end{cases}
$$
Determine all the possible values that the number $x_1$ can take.
$$
\begin{cases}
x_1x_2 + x_1x_3 + x_1x_4 + x_1x_5 = -1 \\
x_2x_1 + x_2x_3 + x_2x_4 + x_2x_5 = -1 \\
x_3x_1 + x_3x_2 + x_3x_4 + x_3x_5 = -1 \\
x_4x_1 + x_4x_2 + x_4x_3 + x_4x_5 = -1 \\
x_5x_1 + x_5x_2 + x_5x_3 + x_5x_4 = -1
\end{cases}
$$
Determine all the possible values that the number $x_1$ can take.