Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, and/or z.)
− 
1
5
x	 + 	y	 − 	
4
5
z	 = 	0
− 
4
5
x	 − 	
1
5
y	 + 	z	 = 	0
x	 − 	
4
5
y	 − 	
1
5
z	 = 	0


We are given the following system of equations:

\[
\begin{aligned}
- \frac{1}{5}x + y - \frac{4}{5}z &= 0 \\
- \frac{4}{5}x - \frac{1}{5}y + z &= 0 \\
x - \frac{4}{5}y - \frac{1}{5}z &= 0
\end{aligned}
\]

We need to use the **Gauss-Jordan row reduction method** to solve this system.

### Step 1: Write the augmented matrix
The system of equations is represented by the augmented matrix:

\[
\left[\begin{array}{ccc|c}
-\frac{1}{5} & 1 & -\frac{4}{5} & 0 \\
-\frac{4}{5} & -\frac{1}{5} & 1 & 0 \\
1 & -\frac{4}{5} & -\frac{1}{5} & 0
\end{array}\right]
\]

### Step 2: Apply Gauss-Jordan row reduction
We will perform row operations to reduce this matrix to row-echelon form, then to reduced row-echelon form.

Let's begin row-reduction. I will compute the steps to solve this system.It seems there was an issue during the row-reduction process due to division by zero, which suggests that the matrix might contain linearly dependent rows or some other structural issue that leads to indeterminacy or inconsistency.

I'll re-examine the matrix and handle it carefully to avoid division by zero or other complications.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the manual Gauss-Jordan steps in detail instead? Let me know if you have any questions.

Here are 5 related questions you might find interesting:
1. How do you determine if a system of equations has no solution or infinitely many solutions?
2. What are the differences between row echelon form and reduced row echelon form?
3. How do you interpret the result of a singular matrix in a system of equations?
4. How does the number of free variables relate to the number of solutions in a linear system?
5. Can every linear system be solved using Gauss-Jordan elimination?

**Tip:** In Gauss-Jordan elimination, always check for leading zeros to avoid division by zero errors.

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, and/or z.)

- (1/5)x + y - (4/5)z = 0
- (4/5)x - (1/5)y + z = 0
x - (4/5)y - (1/5)z = 0

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method. This step-by-step guide explains the row reduction process and how to handle cases with no solution or infinitely many solutions. Ideal for students in Grades 11-12 or college-level linear algebra courses.

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