I have a system of equations in a program I'm writing. It's of the form: $$ \left\{ \begin{array}{c} \frac{a_1}{x}+b_1 y+c_1 z=d_1 \\ \frac{a_2}{x}+b_2 y+c_2 z=d_2 \\ \frac{a_3}{x}+b_3 y+c_3 z=d_3 \end{array} \right. $$ $a_n$, $b_n$, $c_n$ and $d_n$ are known coefficients.
I'm going to use Gauss-Jordan elimination to solve it. But I'm not sure how exactly I should form the matrix with $x$ being in the denominator. I guess I could just put in $a_n$ like the other coefficients and take the inverse of the solution for the first variable to get the value of $x$. But it feels like I'm missing something and that it might be an extra step I could avoid somehow. What would be the "correct" way to do this?