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*.)

*x* + *y* =  0     3*x* − *y*  =  9         *x*  −  *y*  =  −1(*x*, *y*) =

We can solve the given system of equations using the Gauss-Jordan row reduction method. The system of equations is:

\[
\begin{aligned}
x + y &= 0 \quad (1) \\
3x - y &= 9 \quad (2) \\
x - y &= -1 \quad (3)
\end{aligned}
\]

We first convert the system of equations into an augmented matrix:

\[
\left[
\begin{array}{ccc|c}
1 & 1 & 0 & 0 \\
3 & -1 & 0 & 9 \\
1 & -1 & 0 & -1
\end{array}
\right]
\]

Now, we will perform row reduction to get this matrix into reduced row-echelon form (RREF).

### Step 1: Row operations

- Leave \( R_1 \) unchanged.
- Replace \( R_2 \) with \( R_2 - 3R_1 \).
- Replace \( R_3 \) with \( R_3 - R_1 \).

This gives us the new matrix:

\[
\left[
\begin{array}{ccc|c}
1 & 1 & 0 & 0 \\
0 & -4 & 0 & 9 \\
0 & -2 & 0 & -1
\end{array}
\right]
\]

### Step 2: Eliminate below the pivot

- Divide \( R_2 \) by \(-4\) to make the pivot \(1\).
- Replace \( R_3 \) with \( R_3 - \left(\frac{1}{2}\right)R_2 \).

The matrix becomes:

\[
\left[
\begin{array}{ccc|c}
1 & 1 & 0 & 0 \\
0 & 1 & 0 & -\frac{9}{4} \\
0 & 0 & 0 & \frac{7}{2}
\end{array}
\right]
\]

### Step 3: Check consistency

The last row \( \left[ 0 \ 0 \ 0 \ \frac{7}{2} \right] \) implies the equation \( 0 = \frac{7}{2} \), which is a contradiction.

### Conclusion:

Since this leads to an inconsistency, the system has **no solution**.

---

Would you like more details or have any questions? Here are five related questions to explore further:

1. How can we interpret inconsistent systems in terms of geometry?
2. What is the difference between row-echelon form and reduced row-echelon form?
3. Can Gaussian elimination always determine if a system is consistent or inconsistent?
4. How does Gauss-Jordan elimination compare with other methods for solving systems of equations?
5. What happens when a system has infinitely many solutions?

**Tip**: Always check for inconsistencies early in the row-reduction process to avoid unnecessary steps.

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.)
x + y = 0, 3x - y = 9, x - y = -1

Learn how to solve a system of equations using Gauss-Jordan elimination. This problem involves reducing a system of linear equations to row-echelon form and checking for inconsistencies, demonstrating a key concept in linear algebra. Suitable for students in Grades 10-12.

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