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 and/or y.)
5x	 + 	3y	 = 	8
3x	 + 	5y	 = 	8
(x, y) = 

To solve the given system of equations using Gauss-Jordan elimination, we first represent the equations as an augmented matrix:

\[
\begin{bmatrix}
5 & 3 & | & 8 \\
3 & 5 & | & 8
\end{bmatrix}
\]

### Step 1: Row Reduction

1. **Make the leading coefficient of the first row 1**:
   We can do this by dividing the first row by 5.

\[
\begin{bmatrix}
1 & \frac{3}{5} & | & \frac{8}{5} \\
3 & 5 & | & 8
\end{bmatrix}
\]

2. **Eliminate the first column of the second row**:
   We replace the second row by subtracting 3 times the first row from it.

\[
\text{Row 2} \leftarrow \text{Row 2} - 3 \times \text{Row 1}
\]

Calculating Row 2:

\[
3 - 3 \cdot 1 = 0
\]
\[
5 - 3 \cdot \frac{3}{5} = 5 - \frac{9}{5} = \frac{25}{5} - \frac{9}{5} = \frac{16}{5}
\]
\[
8 - 3 \cdot \frac{8}{5} = 8 - \frac{24}{5} = \frac{40}{5} - \frac{24}{5} = \frac{16}{5}
\]

So we have:

\[
\begin{bmatrix}
1 & \frac{3}{5} & | & \frac{8}{5} \\
0 & \frac{16}{5} & | & \frac{16}{5}
\end{bmatrix}
\]

3. **Make the leading coefficient of the second row 1**:
   Divide the second row by \(\frac{16}{5}\).

\[
\begin{bmatrix}
1 & \frac{3}{5} & | & \frac{8}{5} \\
0 & 1 & | & 1
\end{bmatrix}
\]

4. **Eliminate the second column of the first row**:
   We replace Row 1 by subtracting \(\frac{3}{5}\) times Row 2 from it.

\[
\text{Row 1} \leftarrow \text{Row 1} - \frac{3}{5} \times \text{Row 2}
\]

Calculating Row 1:

\[
1 - \frac{3}{5} \cdot 0 = 1
\]
\[
\frac{3}{5} - \frac{3}{5} \cdot 1 = 0
\]
\[
\frac{8}{5} - \frac{3}{5} \cdot 1 = \frac{8}{5} - \frac{3}{5} = \frac{5}{5} = 1
\]

Thus, the final augmented matrix is:

\[
\begin{bmatrix}
1 & 0 & | & 1 \\
0 & 1 & | & 1
\end{bmatrix}
\]

### Step 2: Solution

From the final matrix, we can see:

\[
x = 1
\]
\[
y = 1
\]

### Final Answer

The solution to the system of equations is:

\[
(x, y) = (1, 1)
\]

If you need any further details or have questions, feel free to ask! Here are some related questions:

1. What are other methods to solve systems of equations?
2. Can you demonstrate a scenario with no solution using a similar approach?
3. How does the Gauss-Jordan elimination method compare to substitution?
4. What would the solution look like if the system were dependent?
5. How do we interpret the geometric representation of this solution?

**Tip:** Practice solving systems of equations with different methods to enhance your understanding of their applications.

Use Gauss-Jordan row reduction to solve the given system of equations: 5x + 3y = 8, 3x + 5y = 8.

Explore the step-by-step solution of the system of equations 5x + 3y = 8 and 3x + 5y = 8 using Gauss-Jordan elimination. Understand key algebraic concepts and methods suitable for high school students.

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