The system of equations is:

1. $x + y + z = -6$
2. $x + y - 2z = 3$
3. $x - 2y + z = 9$

Let's solve this system step-by-step using the elimination method:

Step 1: Subtract Equation 1 from Equation 2

Subtract Equation 1 from Equation 2 to eliminate $x + y$:

$(x + y - 2z) - (x + y + z) = 3 - (-6)$

$-3z = 9 \Rightarrow z = -3$

Step 2: Substitute $z = -3$ into Equations 1 and 3

Now that we know $z = -3$, we can substitute it back into Equations 1 and 3 to find $x$ and $y$.

Substitute into Equation 1:

$x + y + (-3) = -6$ $x + y = -3$

Substitute into Equation 3:

$x - 2y + (-3) = 9$ $x - 2y = 12$

Now we have the system:

1. $x + y = -3$
2. $x - 2y = 12$

Step 3: Solve for $x$ and $y$ using the new system

Subtract Equation 1 from Equation 2:

$(x - 2y) - (x + y) = 12 - (-3)$ $-3y = 15 \Rightarrow y = -5$

Step 4: Substitute $y = -5$ into Equation 1 to find $x$

$x + (-5) = -3$ $x = 2$

Solution

The solution set for the system is: $(x, y, z) = (2, -5, -3)$

Would you like more details on each step or have any questions?

Here are five related questions that could expand on this topic:

1. How do you solve a system of linear equations with more than three variables?
2. Can this system of equations be solved using matrix methods, like Gaussian elimination?
3. What are the advantages of using substitution versus elimination methods?
4. How can you verify the solution to a system of equations?
5. How would you graphically represent a solution to a three-variable system?

Tip: When solving systems with multiple methods, always double-check the solution by substituting back into the original equations.