We are given the following system of equations:

1. $x + 2y + 3z = 0$
2. $2x + y + 3z = 0$
3. $3x + 2z = 1$

We will solve this system step-by-step to find the value of $z$.

Step 1: Solve for $y$ in terms of $x$ and $z$

From equation (1), we can express $y$ in terms of $x$ and $z$: $y = -\frac{1}{2}(x + 3z)$ Similarly, from equation (2), we can express $y$ as: $y = -2x - 3z$

Step 2: Substitution to eliminate $y$

Equate the two expressions for $y$ from equations (1) and (2): $-\frac{1}{2}(x + 3z) = -2x - 3z$

Multiply through by 2 to eliminate the fraction: $-(x + 3z) = -4x - 6z$ $-x - 3z = -4x - 6z$

Simplify: $3x = -3z$ $x = -z$

Step 3: Substitute $x = -z$ into equation (3)

Substitute $x = -z$ into equation (3): $3(-z) + 2z = 1$ $-3z + 2z = 1$ $-z = 1$ $z = -1$

Final Answer:

The value of $z$ is $\boxed{-1}$.

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:

1. What is the value of $x$ in this system of equations?
2. How can you find $y$ after knowing $z$ and $x$?
3. How can the substitution method be generalized for larger systems of equations?
4. What are the possible types of solutions for a system of linear equations?
5. How would the solution process change if there were no unique solutions?

Tip: Always double-check for consistency in substitution steps to avoid errors in solving systems of equations.