Use the​ Gauss-Jordan method to solve the following system of equations.
5x
minus
4y
equals
9
10x
minus
8y
equals
1

To solve the system of equations using the Gauss-Jordan method, we first write the augmented matrix for the system of equations:

\[
\begin{align*}
5x - 4y &= 9 \\
10x - 8y &= 1
\end{align*}
\]

The augmented matrix is:

\[
\begin{pmatrix}
5 & -4 & | & 9 \\
10 & -8 & | & 1
\end{pmatrix}
\]

**Step 1: Make the element in the first row, first column a 1.**

Divide the first row by 5:

\[
\begin{pmatrix}
1 & -\frac{4}{5} & | & \frac{9}{5} \\
10 & -8 & | & 1
\end{pmatrix}
\]

**Step 2: Make the elements below the leading 1 in the first column zero.**

Subtract 10 times the first row from the second row:

\[
\text{Row 2} = \text{Row 2} - 10 \cdot \text{Row 1}
\]

\[
\begin{pmatrix}
1 & -\frac{4}{5} & | & \frac{9}{5} \\
0 & 0 & | & -17
\end{pmatrix}
\]

**Step 3: Make the element in the second row, second column a 1.**

This is impossible because the second row is \((0, 0, |, -17)\), which implies the system is inconsistent. Therefore, there are no solutions to this system of equations.

If you have any questions or need further details, feel free to ask!

Here are 5 questions you might find interesting:
1. How do you solve a system of equations using the substitution method?
2. What are the differences between the Gauss-Jordan method and Gaussian elimination?
3. How do you interpret the results of an inconsistent system of equations?
4. Can you explain the concept of row operations in solving linear systems?
5. How do you determine if a system of equations has no solution, one solution, or infinitely many solutions?

**Tip:** When using the Gauss-Jordan method, always aim to transform the augmented matrix into reduced row echelon form to easily identify the solutions.

Learn how to solve a system of linear equations using the Gauss-Jordan method. Detailed step-by-step solution for the equations 5x - 4y = 9 and 10x - 8y = 1, including transformation into augmented matrix and identification of solutions. Suitable for advanced high school students and learners of linear algebra.

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