Use row operations to solve the system. \[ \begin{aligned} x+y-z & =-1 \\ 4 x-y+z & =6 \\ x-3 y+2 z & =-10 \end{aligned} \]

Step 1 ：We are given the system of equations: \n\[\begin{aligned}x+y-z & =-1 \4 x-y+z & =6 \x-3 y+2 z & =-10\end{aligned}\]

Step 2 ：We can solve this system using Gaussian elimination, which is a method for solving linear systems by eliminating variables from the system. The goal is to transform the system of equations into a form where we can easily solve for the variables.

Step 3 ：Performing Gaussian elimination, we get the matrix A and vector b as follows: \n\[A = \begin{bmatrix}1 & 1 & -1 \4 & -1 & 1 \1 & -3 & 2\end{bmatrix}, b = \begin{bmatrix}-1 \6 \-10\end{bmatrix}\]

Step 4 ：Solving this system, we get the solution vector as follows: \n\[\text{solution} = \begin{bmatrix}1 \15 \17\end{bmatrix}\]

Step 5 ：Thus, the solution to the system of equations is \(\boxed{x = 1, y = 15, z = 17}\).