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Let's solve this system of linear equations for the values of \( x \), \( y \), and \( z \).

The given equations are:

\[
\begin{cases}
2x - 5y - z = -12 \quad \text{(Equation 1)} \\
3x - 9y + 2z = -15 \quad \text{(Equation 2)} \\
4x - 4y - z = -3 \quad \text{(Equation 3)}
\end{cases}
\]

### Step 1: Eliminate \( z \) from two of the equations
To eliminate \( z \), we can combine pairs of equations.

#### Multiply Equation 1 by 2:
\[
2(2x - 5y - z) = 2(-12)
\]
\[
4x - 10y - 2z = -24 \quad \text{(Equation 4)}
\]

Now, we subtract Equation 2 from Equation 4 to eliminate \( z \):
\[
(4x - 10y - 2z) - (3x - 9y + 2z) = -24 - (-15)
\]
\[
4x - 10y - 2z - 3x + 9y - 2z = -24 + 15
\]
\[
x - y = -9 \quad \text{(Equation 5)}
\]

### Step 2: Eliminate \( z \) using Equations 1 and 3
Let's use Equations 1 and 3 to eliminate \( z \) again.

Subtract Equation 1 from Equation 3:
\[
(4x - 4y - z) - (2x - 5y - z) = -3 - (-12)
\]
\[
4x - 4y - z - 2x + 5y + z = -3 + 12
\]
\[
2x + y = 9 \quad \text{(Equation 6)}
\]

### Step 3: Solve for \( x \) and \( y \) using Equations 5 and 6
Now we have:
\[
x - y = -9 \quad \text{(Equation 5)}
\]
\[
2x + y = 9 \quad \text{(Equation 6)}
\]

Add Equation 5 and Equation 6:
\[
(x - y) + (2x + y) = -9 + 9
\]
\[
3x = 0
\]
\[
x = 0
\]

### Step 4: Substitute \( x = 0 \) back into Equation 5 to find \( y \)
Substitute \( x = 0 \) into Equation 5:
\[
0 - y = -9
\]
\[
y = 9
\]

### Step 5: Substitute \( x = 0 \) and \( y = 9 \) into Equation 1 to find \( z \)
Substitute \( x = 0 \) and \( y = 9 \) into Equation 1:
\[
2(0) - 5(9) - z = -12
\]
\[
-45 - z = -12
\]
\[
z = -33
\]

### Solution
The values are:
\[
x = 0, \quad y = 9, \quad z = -33
\]

Would you like a detailed explanation of each step, or have any questions?

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Here are some related questions for practice:

1. What are the conditions for a unique solution in a system of linear equations with three variables?
2. How can matrix methods like Gaussian elimination be applied to solve this system?
3. Can the system be solved by substitution alone without elimination?
4. How does Cramer's rule apply to systems of three equations with three variables?
5. What happens if the equations in a system are dependent?

**Tip:** When solving systems of equations, always check if the equations are independent to ensure a unique solution.

Find the values of x, y, and z in the system of linear equations:

1. 2x - 5y - z = -12
2. 3x - 9y + 2z = -15
3. 4x - 4y - z = -3

Learn how to solve a system of linear equations with three variables. This example includes detailed steps for finding values of x, y, and z using elimination and substitution methods. Suitable for students in Grades 9-11.

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