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Let's solve this system of equations step by step to determine whether the system is dependent or inconsistent. The given system of equations is:

\[
\begin{aligned}
1) & \quad 2x - 3y - 9z = -23 \\
2) & \quad x + 3z = 8 \\
3) & \quad -3x + y - 4z = -11
\end{aligned}
\]

### Step 1: Solve for \( x \) in terms of \( z \) from the second equation.
From equation (2):  
\[
x + 3z = 8 \implies x = 8 - 3z
\]

### Step 2: Substitute \( x = 8 - 3z \) into equations (1) and (3).

#### Substitution into equation (1):
\[
2(8 - 3z) - 3y - 9z = -23
\]
Simplifying:
\[
16 - 6z - 3y - 9z = -23
\]
\[
16 - 15z - 3y = -23
\]
\[
-15z - 3y = -23 - 16
\]
\[
-15z - 3y = -39
\]
Dividing by -3:
\[
5z + y = 13 \quad \text{(Equation 4)}
\]

#### Substitution into equation (3):
\[
-3(8 - 3z) + y - 4z = -11
\]
Simplifying:
\[
-24 + 9z + y - 4z = -11
\]
\[
-24 + 5z + y = -11
\]
\[
5z + y = 13 \quad \text{(Equation 5)}
\]

### Step 3: Compare equations (4) and (5).
Both equations are identical:
\[
5z + y = 13
\]
This indicates that the system has infinitely many solutions because equations (1), (2), and (3) reduce to the same condition. This suggests that the system is **dependent**.

### Step 4: Express the solution using a free variable \( z \).
Since the system is dependent, we can express the solution in terms of \( z \). Using \( x = 8 - 3z \) and \( y = 13 - 5z \), the general solution is:
\[
x = 8 - 3z, \quad y = 13 - 5z, \quad z = z
\]
Thus, the solution is:
\[
(x, y, z) = (8 - 3z, 13 - 5z, z)
\]

### Final Answer:
- **(a)** The system is **dependent**.
- **(b)** The solution in terms of \( z \) is \( (8 - 3z, 13 - 5z, z) \).

Would you like more details or have any questions? Here are five related questions to consider:

1. How can you tell if a system of equations is consistent or inconsistent without solving?
2. What is the difference between a dependent and independent system of equations?
3. Can all systems of linear equations be solved by substitution? If not, why?
4. What happens if a system of equations has no solutions?
5. How can you use matrix methods (like Gaussian elimination) to solve this system?

**Tip:** In a dependent system, one equation is often a linear combination of the others, leading to infinitely many solutions.

Given the system of equations: 2x - 3y - 9z = -23, x + 3z = 8, and -3x + y - 4z = -11, determine the type of system (dependent or inconsistent). If dependent, express the solution in terms of z.

Explore how to solve a system of linear equations with three variables. Learn to determine if the system is dependent or inconsistent and how to express the solution in terms of a free variable. Suitable for high school algebra students.

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