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Let's begin by solving the system of equations using the Gauss-Jordan method:

The system of equations is:

1. \( y = x - 3 \)
2. \( y = 4 + z \)
3. \( z = -1 - x \)

### Step 1: Convert the system into an augmented matrix.
Rewriting the system in standard form:

\[
\begin{aligned}
x - y & = 3 & \text{(from equation 1)} \\
y - z & = 4 & \text{(from equation 2)} \\
x + z & = -1 & \text{(from equation 3)}
\end{aligned}
\]

This corresponds to the augmented matrix:
\[
\begin{pmatrix}
1 & -1 & 0 & | & 3 \\
0 & 1 & -1 & | & 4 \\
1 & 0 & 1 & | & -1
\end{pmatrix}
\]

### Step 2: Perform row operations to achieve reduced row-echelon form (RREF).
1. Subtract Row 1 from Row 3 to eliminate the \(x\)-term in Row 3:

\[
R_3 = R_3 - R_1 \quad \Rightarrow \quad 
\begin{pmatrix}
1 & -1 & 0 & | & 3 \\
0 & 1 & -1 & | & 4 \\
0 & 1 & 1 & | & -4
\end{pmatrix}
\]

2. Subtract Row 2 from Row 3 to simplify:

\[
R_3 = R_3 - R_2 \quad \Rightarrow \quad 
\begin{pmatrix}
1 & -1 & 0 & | & 3 \\
0 & 1 & -1 & | & 4 \\
0 & 0 & 2 & | & -8
\end{pmatrix}
\]

3. Divide Row 3 by 2 to normalize:

\[
R_3 = \frac{R_3}{2} \quad \Rightarrow \quad 
\begin{pmatrix}
1 & -1 & 0 & | & 3 \\
0 & 1 & -1 & | & 4 \\
0 & 0 & 1 & | & -4
\end{pmatrix}
\]

4. Use Row 3 to eliminate the \(z\)-terms from Row 2:

\[
R_2 = R_2 + R_3 \quad \Rightarrow \quad 
\begin{pmatrix}
1 & -1 & 0 & | & 3 \\
0 & 1 & 0 & | & 0 \\
0 & 0 & 1 & | & -4
\end{pmatrix}
\]

5. Use Row 2 to eliminate the \(y\)-term in Row 1:

\[
R_1 = R_1 + R_2 \quad \Rightarrow \quad 
\begin{pmatrix}
1 & 0 & 0 & | & 3 \\
0 & 1 & 0 & | & 0 \\
0 & 0 & 1 & | & -4
\end{pmatrix}
\]

### Step 3: Read the solution from the matrix.
The resulting matrix gives the system:

\[
\begin{aligned}
x & = 3 \\
y & = 0 \\
z & = -4
\end{aligned}
\]

Thus, the solution is \( (x, y, z) = (3, 0, -4) \).

### Conclusion:
The correct choice is **A**: There is one solution, \( (3, 0, -4) \), in the order \(x, y, z\).

---

Would you like more details on any step or further clarification?

Here are five related questions you might find helpful:
1. How do you convert a system of equations into an augmented matrix?
2. What is the Gauss-Jordan elimination method, and how does it differ from Gaussian elimination?
3. Can a system of equations have no solution or infinitely many solutions?
4. How can you identify if a system is consistent or inconsistent?
5. How does the reduced row-echelon form help in solving systems of linear equations?

**Tip:** Always double-check your row operations during elimination to avoid small arithmetic errors that can lead to incorrect solutions.

Use the Gauss-Jordan method to solve the system of equations. y = x - 3, y = 4 + z, z = -1 - x

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method. The system involves three equations: y = x - 3, y = 4 + z, z = -1 - x. Suitable for students in Grades 10-12, this problem covers key linear algebra concepts, including systems of equations and row operations.

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