To solve this system of linear equations:

1. $x + y + z = 1$
2. $x - 2y + z = 4$
3. $x - y + z = 3$

we can use either the substitution method, elimination method, or matrix methods such as Gaussian elimination. Here, I'll solve it using elimination.

Step 1: Set up the equations

1. $x + y + z = 1$
2. $x - 2y + z = 4$
3. $x - y + z = 3$

Step 2: Eliminate $x$

Subtract equation (1) from equations (2) and (3) to eliminate $x$.

Subtract equation (1) from equation (2): $(x - 2y + z) - (x + y + z) = 4 - 1$ $-3y = 3 \Rightarrow y = -1$

Subtract equation (1) from equation (3): $(x - y + z) - (x + y + z) = 3 - 1$ $-2y = 2 \Rightarrow y = -1$

Now we know $y = -1$.

Step 3: Substitute $y = -1$ back into the equations

Substitute $y = -1$ into equation (1) to solve for $x$ and $z$.

From equation (1): $x + (-1) + z = 1$ $x + z = 2 \quad \text{(Equation A)}$

From equation (3): $x - (-1) + z = 3$ $x + z = 2 \quad \text{(Equation B)}$