$\left\{\begin{array}{l}x+y+z=4 \\ 2 x-y+z=3 \\ -4 x+2 y-z=-1\end{array}\right.$

Step 1 ：Given the system of equations: \(\left\{\begin{array}{l}x+y+z=4 \ 2x-y+z=3 \ -4x+2y-z=-1\end{array}\right.\)

Step 2 ：We can represent this system in matrix form as follows: A = [[1, 1, 1], [2, -1, 1], [-4, 2, -1]], X = [x, y, z], B = [4, 3, -1]

Step 3 ：To solve for X, we find the inverse of A and multiply it with B.

Step 4 ：The inverse of A is calculated as: A_inv = [[ 0.33333333, -1, -0.66666667], [ 0.66666667, -1, -0.33333333], [ 0, 2, 1]]

Step 5 ：Multiplying A_inv with B, we get X = [-1.00000000e+00, -1.66533454e-16, 5.00000000e+00]

Step 6 ：The small non-zero value for y is due to the numerical precision of the computation and can be considered as zero.

Step 7 ：So, the solution to the system of equations is \(\boxed{x = -1, y = 0, z = 5}\)