Fastest way to solve a system of linear equations
A LU decomposition of a $n \times n$ matrix can be computed in $O(M(n))$ time, where $M(n)$ is the time to multiply two $n \times n$ matrices.  Therefore, you can find a solution to a system of $n$ linear equations in $n$ unknowns in $O(M(n))$ time.  For instance, Strassen's algorithm achieves $M(n) = O(n^{2.8})$, which is faster than Gaussian elimination.  See https://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion.

Rather than trying to implement this yourself, I would suggest using a library: e.g., a BLAS library.