Solve for the variable x in the system of equations: $\begin{array}{l}4 x+z=4 \\ x-y+4 z=-11 \\ -3 x+y-5 z=11\end{array}$

Step 1 ：We are given the system of equations: \(\begin{array}{l}4 x+z=4 \ x-y+4 z=-11 \ -3 x+y-5 z=11\end{array}\)

Step 2 ：We can represent this system in matrix form as follows: \(A = \begin{bmatrix} 4 & 0 & 1 \ 1 & -1 & 4 \ -3 & 1 & -5 \end{bmatrix}\) and \(b = \begin{bmatrix} 4 \ -11 \ 11 \end{bmatrix}\)

Step 3 ：We solve the system of equations by finding the inverse of matrix A and multiplying it with matrix b. This gives us the values of x, y, and z.

Step 4 ：The solution for the variable x in the system of equations is \(\boxed{2}\)