I've given some equations look like this.

$a_{1,1} x_1 + a_{1,2} x_2 + a_{1,3} x_3 + ... + a_{1,n} x_n\equiv 1 \mod p$

$a_{2,1} x_1 + a_{2,2} x_2 + a_{2,3} x_3 + ... + a_{2,n} x_n\equiv 1\mod p$

$...$

$a_{m,1} x_1 + a_{m,2} x_2 + a_{m,3} x_3 + ... + a_{m,n} x_n\equiv 1\mod p$

($p$ is prime, I know the values of $a_{1..m, 1..n}$, I have to get $x_{1..n}$) (all of the values of $a_{1..m, 1..n}, x_{1..n}$ should not be negative, and they must be integers)

I think I can solve this using Gaussian elimination, but I'm not sure how to use this.

I appreciate any help or tip. Thank you in advance. :)