\[ \begin{array}{l} x+2 y-3 z=2 \\ x-3 y-z=3 \\ -x+3 y+(4-k) z=7 \end{array} \] If we don't apply the Cramer's method to the given linear system then find the value for $k$.

Step 1 ：Eliminate x from the first two equations and then from the second and third equations to get two new equations: \(5y - 2z = -1\) and \(2x - 6y - z(4 - k) - z = -4\)

Step 2 ：Eliminate y from the new equations to find the value of k: \(\boxed{k = \frac{-2x + 16y + z - 2}{z}}\)