Solve using Matrix Gauss Jordan Elimination -2x+6y=-22   x+2y=-9

We are given the system of equations:

\[
-2x + 6y = -22
\]
\[
x + 2y = -9
\]

We will solve this using **Gauss-Jordan Elimination** by converting the system into an augmented matrix, performing row operations to reach **reduced row echelon form (RREF)**, and then interpreting the result.

### Step 1: Set up the augmented matrix.
The augmented matrix for the given system is:

\[
\begin{bmatrix}
-2 & 6 & | & -22 \\
1 & 2 & | & -9
\end{bmatrix}
\]

### Step 2: Perform row operations.
We need to use row operations to convert this matrix into the reduced row echelon form.

#### 1. Make the first pivot (in the top-left corner) a 1.
We can achieve this by dividing the entire first row by -2:

\[
R1 \to \frac{R1}{-2} \quad \Rightarrow \quad \begin{bmatrix} 1 & -3 & | & 11 \\ 1 & 2 & | & -9 \end{bmatrix}
\]

#### 2. Eliminate the 1 below the pivot (in the second row, first column).
We can subtract the first row from the second row:

\[
R2 \to R2 - R1 \quad \Rightarrow \quad \begin{bmatrix} 1 & -3 & | & 11 \\ 0 & 5 & | & -20 \end{bmatrix}
\]

#### 3. Make the second pivot (in the second row, second column) a 1.
We can do this by dividing the entire second row by 5:

\[
R2 \to \frac{R2}{5} \quad \Rightarrow \quad \begin{bmatrix} 1 & -3 & | & 11 \\ 0 & 1 & | & -4 \end{bmatrix}
\]

#### 4. Eliminate the -3 above the pivot (in the first row, second column).
We add 3 times the second row to the first row:

\[
R1 \to R1 + 3 \times R2 \quad \Rightarrow \quad \begin{bmatrix} 1 & 0 & | & -1 \\ 0 & 1 & | & -4 \end{bmatrix}
\]

### Step 3: Interpret the result.
The matrix is now in reduced row echelon form, which corresponds to the system of equations:

\[
x = -1
\]
\[
y = -4
\]

Thus, the solution to the system is:

\[
x = -1, \quad y = -4
\]

Would you like further details or clarifications? 

Here are 5 related questions to explore:
1. How is Gauss-Jordan elimination different from Gaussian elimination?
2. Can you apply this method to a 3x3 system of equations?
3. What are the benefits of using matrix methods over substitution or elimination?
4. How can you verify the solution of a system of equations after solving it?
5. Can all systems of linear equations be solved using Gauss-Jordan elimination?

**Tip**: Always check the final solution by plugging the values of variables back into the original equations to verify correctness.

Learn how to solve the system of equations -2x + 6y = -22 and x + 2y = -9 using the Gauss-Jordan elimination method. This step-by-step guide covers important linear algebra concepts like augmented matrices, row operations, and reduced row echelon form (RREF). Suitable for high school students in Grades 10-12.

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