What is the value of *zz* in the following system of equations?

x+2y+3z=0x+2y+3z=0

2x+y+3z=02x+y+3z=0

3x+2z=1

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.

What is the value of *z* in the following system of equations?

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

Learn how to solve a system of linear equations using the substitution method with a step-by-step guide. The equations include x + 2y + 3z = 0, 2x + y + 3z = 0, and 3x + 2z = 1, focusing on algebraic concepts and methods. Suitable for students in Grades 9-12.

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