To solve the system of equations by elimination:

$\text{(1)} \ -x + y - 3z = 6$ $\text{(2)} \ -x - y + z = 4$ $\text{(3)} \ -x - y - 2z = 16$

Step 1: Eliminate one variable

We will eliminate $y$ by adding and subtracting the equations.
From equations (1) and (2):

$\text{(1)} + \text{(2)}: \ (-x + y - 3z) + (-x - y + z) = 6 + 4$ Simplify: $-2x - 2z = 10 \quad \Rightarrow \quad x + z = -5 \ \text{(4)}$

From equations (2) and (3):

$\text{(2)} + \text{(3)}: \ (-x - y + z) + (-x - y - 2z) = 4 + 16$ Simplify: $-2x - y - z = 20 \quad \Rightarrow \quad -x - z = 10 \ \text{(5)}$

Step 2: Solve for $x$ and $z$

From equations (4) and (5):

1. $x + z = -5 \quad \text{(4)}$
2. $-x - z = 10 \quad \text{(5)}$

Add the two equations:

$(x + z) + (-x - z) = -5 + 10$ Simplify: $0 = 5$