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We are tasked with finding the solutions to the homogeneous system \( A \mathbf{x} = 0 \), where \( A \) is the matrix given by:

\[
A = \begin{pmatrix}
1 & 5 & -9 & 6 \\
0 & 1 & -1 & 6
\end{pmatrix}
\]

### Step 1: Analyze the Row-Echelon Form of the Matrix
The matrix \( A \) is already in row echelon form, so we can move to solving the system by identifying the pivot and free variables.

- **Pivot variables**: These are the variables corresponding to the leading entries (non-zero entries) in each row.
  - From the first row: \( x_1 \) is a pivot variable.
  - From the second row: \( x_2 \) is a pivot variable.

- **Free variables**: The variables corresponding to the columns without pivots.
  - \( x_3 \) and \( x_4 \) are free variables.

### Step 2: Solve the System Using Back-Substitution

From the second row equation:
\[
x_2 - x_3 + 6x_4 = 0 \quad \Rightarrow \quad x_2 = x_3 - 6x_4
\]

From the first row equation:
\[
x_1 + 5x_2 - 9x_3 + 6x_4 = 0
\]
Substituting \( x_2 = x_3 - 6x_4 \):
\[
x_1 + 5(x_3 - 6x_4) - 9x_3 + 6x_4 = 0
\]
Simplify:
\[
x_1 + 5x_3 - 30x_4 - 9x_3 + 6x_4 = 0 \quad \Rightarrow \quad x_1 - 4x_3 - 24x_4 = 0
\]
Thus:
\[
x_1 = 4x_3 + 24x_4
\]

### Step 3: Write the Parametric Vector Form of the Solution

Let the free variables \( x_3 \) and \( x_4 \) be parameters:
\[
x_1 = 4x_3 + 24x_4, \quad x_2 = x_3 - 6x_4, \quad x_3 = x_3, \quad x_4 = x_4
\]

The solution can be written as:
\[
\mathbf{x} = x_3 \begin{pmatrix} 4 \\ 1 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 24 \\ -6 \\ 0 \\ 1 \end{pmatrix}
\]

This represents the general solution in parametric vector form.

### Summary of Solution:
- **Free variables**: \( x_3 \), \( x_4 \)
- **Parametric solution**: 
\[
\mathbf{x} = x_3 \begin{pmatrix} 4 \\ 1 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 24 \\ -6 \\ 0 \\ 1 \end{pmatrix}
\]

Would you like further details or have any specific questions?

### Related questions:
1. How do you interpret free variables in a system of equations?
2. What are the steps for row-reducing a matrix?
3. How can you identify if a system has infinitely many solutions?
4. What is the significance of the parametric vector form in solving systems of equations?
5. How does the rank of a matrix affect the number of free variables in a system?

### Tip:
Always identify pivot and free variables after putting the matrix into row echelon form to quickly determine the solution structure of the system.

Find all the solutions of Ax = 0, where A is row equivalent to the given matrix. Identify the free variables and write the solution in parametric vector form.

Discover how to find the solutions of a homogeneous system Ax = 0 using row echelon form. Learn to identify free variables and express the solution in parametric vector form, suitable for Linear Algebra students.

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