https://file.aiquickdraw.com/mathbot/file/1728178852244l4aceqmk.jpg

To solve the system of equations given in the image:

\[
-5x - y = 10
\]
\[
10x - 3y = -45
\]

we will use the Gauss-Jordan elimination method by writing the system as an augmented matrix and solving it step by step.

### Step 1: Write the system as an augmented matrix
The system can be written in matrix form as follows:

\[
\left[
\begin{array}{ccc|c}
-5 & -1 & 0 & 10 \\
10 & -3 & 0 & -45
\end{array}
\right]
\]

### Step 2: Perform row operations to solve for \(x\) and \(y\)
We will now use row operations to transform the matrix into reduced row-echelon form (RREF).

1. **Make the first element of the first row equal to 1** by dividing the first row by \(-5\):
   \[
   R1 \to \frac{R1}{-5}
   \]
   Resulting in:
   \[
   \left[
   \begin{array}{ccc|c}
   1 & \frac{1}{5} & 0 & -2 \\
   10 & -3 & 0 & -45
   \end{array}
   \right]
   \]

2. **Eliminate the first element in the second row** by subtracting 10 times the first row from the second row:
   \[
   R2 \to R2 - 10R1
   \]
   Resulting in:
   \[
   \left[
   \begin{array}{ccc|c}
   1 & \frac{1}{5} & 0 & -2 \\
   0 & -5 & 0 & -25
   \end{array}
   \right]
   \]

3. **Make the second element of the second row equal to 1** by dividing the second row by \(-5\):
   \[
   R2 \to \frac{R2}{-5}
   \]
   Resulting in:
   \[
   \left[
   \begin{array}{ccc|c}
   1 & \frac{1}{5} & 0 & -2 \\
   0 & 1 & 0 & 5
   \end{array}
   \right]
   \]

4. **Eliminate the second element in the first row** by subtracting \(\frac{1}{5}\) times the second row from the first row:
   \[
   R1 \to R1 - \frac{1}{5} R2
   \]
   Resulting in:
   \[
   \left[
   \begin{array}{ccc|c}
   1 & 0 & 0 & -3 \\
   0 & 1 & 0 & 5
   \end{array}
   \right]
   \]

Now the system is in reduced row-echelon form, meaning we have:

\[
x = -3 \quad \text{and} \quad y = 5
\]

### Final Solution:
The solution to the system of equations is:

\[
(x, y) = (-3, 5)
\]

Would you like further clarification on any steps? Here are some related questions to explore:

1. How is Gauss-Jordan elimination different from Gaussian elimination?
2. What are the possible outcomes when solving a system of linear equations using matrices?
3. Can you explain the significance of row-echelon form and reduced row-echelon form in linear algebra?
4. How would the augmented matrix change if the system had no solution?
5. What happens when you perform row operations that lead to a zero row in the matrix?

**Tip**: When using Gauss-Jordan elimination, always ensure that your leading coefficients (pivots) become 1 as you proceed with row operations!

Solve the system of equations using the Gauss-Jordan method:

-5x - y = 10
10x - 3y = -45

Write the system of equations as an augmented matrix and solve for x and y.

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method. This problem involves converting the system into an augmented matrix and performing row operations to reach reduced row-echelon form. Suitable for advanced high school or early undergraduate students studying linear algebra.

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