I have a system of $n-1$ equations which have $n$ unknown variables, as follows: \begin{align} &\text{Equation $1$} &&x_1A_{1,1}+...+x_nA_{1,n}=0\\ & \hspace{1cm}\vdots && \hspace{2cm}\vdots\\ &\text{Equation $n-1$} &&x_1A_{n-1,1}+...+x_nA_{n-1,n}=0 \end{align} As an example, if $n=2$, we have a single equation: $$ x_1A_{1,1}+x_2A_{1,2}=0 \rightarrow x_1=-\frac{x_2A_{1,2}}{A_{1,1}}$$ and we have infinitive possible solutions for $x_2$.
I am trying to find a generic expression for the $x$ values for any value of $n$, probably this will be some recursive formula. Any suggestions on how to solve this are welcome.