\[ \begin{cases}x_{2}x_{3}x_{4}...x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}...x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}...x_{n}=a_{3}x_{3}\\...\\x_{1}x_{2}x_{3}...x_{n-1}=a_{n-1}x_{n-1} \end{cases} \] let $x_1x_2x_3...x_n=P$ multiplying all we get $P^{n-1}=a_1a_2a_3...a_{n-1}P$ , so $P=\sqrt[n-2]{a_1a_2a_3...a_{n-1}}$ from equations by order we get: $x_i^2=\frac{P}{a_i}$ , so $\boxed{x_i=\pm\sqrt{\frac{P}{a_i}}}$ , $1\le i\le n$ .

Clearly the last equation must be $x_1x_2x_3\cdots x_{n-1} = a_nx_n$. Moreover, nowhere is it said that the values are non-zero. So, roza, you need revise your attempt.