$\left\{\begin{array}{l}x-2 y+z=1 \\ y+2 z=-9 \\ x+y+3 z=-14\end{array}\right.$

Step 1 ：Given the system of linear equations: \(\begin{cases} x-2y+z=1 \\ y+2z=-9 \\ x+y+3z=-14 \end{cases}\)

Step 2 ：We can represent this system in matrix form as: \(\begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 2 \\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -9 \\ -14 \end{bmatrix}\)

Step 3 ：Solving this system of equations, we find the solution to be: \(x = -2, y = -3, z = -3\)

Step 4 ：Final Answer: \(\boxed{x = -2, y = -3, z = -3}\)