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To solve the system of equations using Gauss-Jordan elimination, let's convert the system into an augmented matrix:

The system of equations is:
\[
\begin{aligned}
6x + 7y &= 15 \\
5x + 2y &= 24
\end{aligned}
\]

The corresponding augmented matrix is:
\[
\begin{pmatrix}
6 & 7 & | & 15 \\
5 & 2 & | & 24
\end{pmatrix}
\]

### Step 1: Make the leading coefficient in the first row a 1
To achieve this, divide the first row by 6:
\[
\begin{pmatrix}
1 & \frac{7}{6} & | & \frac{15}{6} \\
5 & 2 & | & 24
\end{pmatrix}
=
\begin{pmatrix}
1 & \frac{7}{6} & | & 2.5 \\
5 & 2 & | & 24
\end{pmatrix}
\]

### Step 2: Eliminate the 5 in the second row, first column
We perform the row operation: \(\text{Row 2} = \text{Row 2} - 5 \times \text{Row 1}\):
\[
\text{Row 2} = (5 \ 2 \ | \ 24) - 5 \times (1 \ \frac{7}{6} \ | \ 2.5)
\]
\[
= (5 - 5 \cdot 1, 2 - 5 \cdot \frac{7}{6}, 24 - 5 \cdot 2.5)
\]
\[
= (0, 2 - \frac{35}{6}, 24 - 12.5)
\]
\[
= (0, \frac{-23}{6}, 11.5)
\]
The new matrix is:
\[
\begin{pmatrix}
1 & \frac{7}{6} & | & 2.5 \\
0 & \frac{-23}{6} & | & 11.5
\end{pmatrix}
\]

### Step 3: Make the leading coefficient of the second row a 1
We divide the second row by \(\frac{-23}{6}\):
\[
\begin{pmatrix}
1 & \frac{7}{6} & | & 2.5 \\
0 & 1 & | & -3
\end{pmatrix}
\]

### Step 4: Eliminate the \(\frac{7}{6}\) in the first row, second column
We perform the row operation: \(\text{Row 1} = \text{Row 1} - \frac{7}{6} \times \text{Row 2}\):
\[
\text{Row 1} = (1 \ \frac{7}{6} \ | \ 2.5) - \frac{7}{6} \times (0 \ 1 \ | -3)
\]
\[
= (1 - 0, \frac{7}{6} - \frac{7}{6} \cdot 1, 2.5 - \frac{7}{6} \cdot -3)
\]
\[
= (1, 0, 2.5 + 3.5)
\]
\[
= (1, 0, 6)
\]
The final matrix is:
\[
\begin{pmatrix}
1 & 0 & | & 6 \\
0 & 1 & | & -3
\end{pmatrix}
\]

### Solution:
The solution to the system is:
\[
x = 6, \quad y = -3
\]
Therefore, the solution in the format \((x, y)\) is:
\[
(6, -3)
\]

Would you like a further breakdown of any step? Here are some related questions to practice:

1. What are the main differences between Gaussian elimination and Gauss-Jordan elimination?
2. How do you verify that the solution of a system of equations is correct?
3. Can every system of equations be solved using Gauss-Jordan elimination?
4. What is the purpose of converting a system of equations into row echelon form?
5. What would happen if the system of equations had no solution or infinitely many solutions?

**Tip:** It's important to keep track of your row operations and check the results of each step to avoid calculation errors.

Use Gauss-Jordan elimination to solve the following system of equations: 6x + 7y = 15 and 5x + 2y = 24.

Learn how to solve the system of equations 6x + 7y = 15 and 5x + 2y = 24 using Gauss-Jordan elimination. This detailed step-by-step guide covers the transformation of the system into an augmented matrix, row operations, and solving for the variables. Suitable for high school students in Grades 9-12.

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