Choose a variable to eliminate and two equations to work with.
There are many possibilities.
We will use the first and third equations and eliminate $x$ in the example.
Multiply the third equation by $−1$ then add it to the first.

$x−2y+3z=8\\−x−y−2z=−12\\−3y+z=−4$

Choose a different pair of equations and eliminate the same variable.
Here, we use the second and third equations.
Multiply the third equation by $−2$ then add it to the second.

$2x+y+z=6\\−2x−2y−4z=−24\\−y−3z=−18$

Now combine the two result equations.

$−3y+z=−4\\3y+9z=54$

Solve for $z$ and substitute to find the other values.

Check your result in each equation.