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The problem shows a system of equations represented by an augmented matrix. The goal is to solve for the variables \( x \), \( y \), \( z \), and \( w \), and express the solution in terms of free parameters if necessary.

The augmented matrix is:

\[
\begin{bmatrix}
-8 & 8 & -1 & 5 & | & -17 \\
-7 & 6 & -1 & 3 & | & -14 \\
-3 & 4 & 0 & 3 & | & -8
\end{bmatrix}
\]

### Step 1: Convert the Augmented Matrix to Row Echelon Form (REF)

We will perform Gaussian elimination to simplify the matrix.

1. **Row 1:** Keep it as is: \([-8, 8, -1, 5 | -17]\).
   
2. **Row 2:** Multiply Row 1 by \(\frac{7}{8}\) and subtract it from Row 2.
   
3. **Row 3:** Multiply Row 1 by \(\frac{3}{8}\) and subtract it from Row 3.

### Step 2: Solve the System from REF

After converting to REF, identify leading variables and express the general solution in terms of free variables (parameters).

Let me calculate the exact solution now.The augmented matrix after Gaussian elimination (in row echelon form) is:

\[
\begin{bmatrix}
1 & -1 & 0.125 & -0.625 & 2.125 \\
0 & 1 & 0.125 & 1.375 & -0.875 \\
0 & 0 & 1 & -1 & -3
\end{bmatrix}
\]

### Step 3: Back-substitute to Find the General Solution

From the matrix:

1. The third row gives: \( z - w = -3 \) ⟹ \( z = w - 3 \).
2. The second row gives: \( y + 0.125z + 1.375w = -0.875 \). Substituting \( z = w - 3 \):
   \[
   y + 0.125(w - 3) + 1.375w = -0.875
   \]
   ⟹ \( y + 0.125w - 0.375 + 1.375w = -0.875 \)
   ⟹ \( y + 1.5w = -0.5 \) ⟹ \( y = -0.5 - 1.5w \).
3. The first row gives: \( x - y + 0.125z - 0.625w = 2.125 \). Substituting \( y = -0.5 - 1.5w \) and \( z = w - 3 \):
   \[
   x - (-0.5 - 1.5w) + 0.125(w - 3) - 0.625w = 2.125
   \]
   Simplifying:
   \[
   x + 0.5 + 1.5w + 0.125w - 0.375 - 0.625w = 2.125
   \]
   ⟹ \( x + 1.0w = 2.0 \) ⟹ \( x = 2 - w \).

### Final General Solution

\[
\begin{aligned}
x &= 2 - w \\
y &= -0.5 - 1.5w \\
z &= w - 3 \\
w &= w
\end{aligned}
\]

### Conclusion:
The solution involves one parameter \( w \), which is free. Now, you can express this in the form:

\[
x = 2 + (-w), \quad y = -0.5 + (-1.5w), \quad z = -3 + w, \quad w = w
\]

Do you want more details or have any questions?

Here are 5 related questions for further practice:
1. How would the solution change if one equation was dependent on the others?
2. Can you solve the system using matrix inversion instead of Gaussian elimination?
3. How would you check the consistency of the system before solving?
4. How does the rank of the matrix affect the number of free variables?
5. What are the implications if a row reduces entirely to zeros?

**Tip:** Always verify if a system is consistent (no contradictions) before solving for free variables.

Solve the system of equations represented by the following augmented matrix: [-8 8 -1 5 | -17, -7 6 -1 3 | -14, -3 4 0 3 | -8].

Learn how to solve a system of linear equations using Gaussian Elimination. This problem covers key concepts in Linear Algebra, such as augmented matrices and back-substitution. Follow along with detailed steps to solve the system and express the solution with free variables.

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