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
15x	 − 	9y	 = 	24
(x, y) = 

 
  



Let's solve the system of equations using Gauss-Jordan row reduction.

The given system is:

\[
\begin{aligned}
1. \ 5x - 3y &= 8 \\
2. \ 15x - 9y &= 24
\end{aligned}
\]

We can represent this system as an augmented matrix:

\[
\begin{pmatrix}
5 & -3 & | & 8 \\
15 & -9 & | & 24
\end{pmatrix}
\]

### Step 1: Make the leading coefficient of the first row a 1.
To do this, we divide the first row by 5:

\[
\begin{pmatrix}
1 & -\frac{3}{5} & | & \frac{8}{5} \\
15 & -9 & | & 24
\end{pmatrix}
\]

### Step 2: Eliminate the first element in the second row.
We want to eliminate the 15 in the second row. Subtract 15 times the first row from the second row:

\[
R_2 \rightarrow R_2 - 15R_1
\]

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

### Step 3: Analyze the matrix.
The second row is now all zeros, which indicates that the system has **dependent equations**. This means there are infinitely many solutions.

To express the solution in terms of a parameter, let's set \( y = t \) (where \( t \) is a free parameter). 

From the first equation:

\[
x - \frac{3}{5}y = \frac{8}{5}
\]

Substitute \( y = t \):

\[
x - \frac{3}{5}t = \frac{8}{5}
\]

Solve for \( x \):

\[
x = \frac{8}{5} + \frac{3}{5}t
\]

\[
x = \frac{8 + 3t}{5}
\]

### Final solution:
The solution to the system is:

\[
(x, y) = \left( \frac{8 + 3t}{5}, t \right)
\]

where \( t \) is any real number.

---

Do you want more details on any specific step or have any questions? 

Here are five related questions you might want to explore:
1. How does Gauss-Jordan elimination compare to Gaussian elimination?
2. What is the significance of dependent systems in linear algebra?
3. Can we solve larger systems using Gauss-Jordan elimination?
4. How can we interpret the geometric meaning of dependent solutions?
5. What happens if the system has no solution?

**Tip:** Always check for row of zeros in the augmented matrix — it helps identify dependent or inconsistent systems.

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
15x - 9y = 24
(x, y) = 

Learn how to solve a system of linear equations using Gauss-Jordan row reduction. This detailed step-by-step solution explores the system 5x - 3y = 8 and 15x - 9y = 24, revealing the dependent nature of the equations and expressing the solution in parametric form. Suitable for students in Grades 10-12.

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