Solve. \[ \begin{aligned} x+y+z & =13 \\ 2 x+y-z & =9 \\ x-y+z & =3 \end{aligned} \]

Step 1 ：We are given a system of linear equations with three variables: \[\begin{aligned} x+y+z & =13 \\ 2x+y-z & =9 \\ x-y+z & =3 \end{aligned}\]

Step 2 ：We can solve this system using the matrix method. The matrix representation of the system is: A = \[\begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix}\] and b = \[\begin{bmatrix} 13 \\ 9 \\ 3 \end{bmatrix}\]

Step 3 ：Solving this system, we find the solution to be x = 4, y = 5, and z = 4.

Step 4 ：Final Answer: The solution to the system of equations is \(\boxed{x = 4, y = 5, z = 4}\)