To solve this system of linear equations using the elimination method, let's start by defining the system clearly and then apply elimination steps:

Given equations:

$x + y + z = 66$ $2x - y - 6z = -34$ $-x - y + 8z = 39$

Step 1: Eliminate $y$ by combining equations

1. Add the first and second equations to eliminate $y$: $(x + y + z) + (2x - y - 6z) = 66 + (-34)$ Simplifying this: $3x - 5z = 32$

2. Add the first and third equations to eliminate $y$ as well: $(x + y + z) + (-x - y + 8z) = 66 + 39$ Simplifying this: $9z = 105$ Solving for $z$: $z = \frac{105}{9} = 11.67$