function solveLinearSystem(mat) {
  const n = mat.length;

  for (let i = 0; i < n; i++) {
    // Find max in the i-th column (ignoring i - 1 first rows)
    let maxRow = i;
    let maxEl = Math.abs(mat[i][i]);
    for (let r = i + 1; r < n; r++) {
      const absValue = Math.abs(mat[r][i]);
      if (absValue > maxEl) {
        maxEl = absValue;
        maxRow = r;
      }
    }

    if (maxEl === 0) {
      return null; // matrix is singular
    }

    // Swap max row with i-th (current) row
    const tmp = mat[maxRow];
    mat[maxRow] = mat[i];
    mat[i] = tmp;

    // Subtract the i-th row to make all the remaining rows 0 in the i-th column
    for (let j = i + 1; j < n; j++) {
      const coef = -mat[j][i] / mat[i][i];
      for (let k = i; k < n + 1; k++) {
        if (i == k) {
          mat[j][k] = 0;
        } else {
          mat[j][k] += coef * mat[i][k];
        }
      }
    }
  }

  // Solve Ax=b for upper triangular matrix A (mat)
  const x = new Array(n);
  for (let l = n - 1; l >= 0; l--) {
    x[l] = mat[l][n] / mat[l][l];
    for (let m = l - 1; m >= 0; m--) {
      mat[m][n] -= mat[m][l] * x[l];
    }
  }
  return x;
}
Solves system of linear equations using Gaussian elimination method.

@param {Array<Array<number>>} mat Augmented matrix (n x n + 1 column)
                                    in row-major order.
@return {Array<number>|null} The resulting vector.