https://file.aiquickdraw.com/mathbot/file/1729574960887fwkjp9h5.jpg

The system of equations provided is:

\[
1. \, x - y + z = 4
\]
\[
2. \, 3x + y + 4z = 6
\]
\[
3. \, x + y + z = 1
\]

We need to determine whether the system has a unique solution, infinite solutions, no solution, or none of the above. Let's analyze this by solving the system.

### Step 1: Write the system in matrix form.
The system of equations can be written as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where:

\[
A = \begin{bmatrix} 
1 & -1 & 1 \\
3 & 1 & 4 \\
1 & 1 & 1 
\end{bmatrix}, \, \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \, \mathbf{b} = \begin{bmatrix} 4 \\ 6 \\ 1 \end{bmatrix}
\]

### Step 2: Perform row reduction (Gaussian elimination) to analyze the system.

We will attempt to reduce the augmented matrix \( [A | b] \):

\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & 4 \\
3 & 1 & 4 & 6 \\
1 & 1 & 1 & 1 \\
\end{array}
\right]
\]

- Subtract \( 3 \times \text{row 1} \) from row 2:
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & 4 \\
0 & 4 & 1 & -6 \\
1 & 1 & 1 & 1 \\
\end{array}
\right]
\]

- Subtract \( 1 \times \text{row 1} \) from row 3:
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & 4 \\
0 & 4 & 1 & -6 \\
0 & 2 & 0 & -3 \\
\end{array}
\right]
\]

- Next, divide row 2 by 4:
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & 4 \\
0 & 1 & \frac{1}{4} & -\frac{3}{2} \\
0 & 2 & 0 & -3 \\
\end{array}
\right]
\]

- Subtract \( 2 \times \text{row 2} \) from row 3:
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & 4 \\
0 & 1 & \frac{1}{4} & -\frac{3}{2} \\
0 & 0 & -\frac{1}{2} & 0 \\
\end{array}
\right]
\]

### Step 3: Solve the system.

From the third row, we get:
\[
-\frac{1}{2}z = 0 \implies z = 0
\]

Substituting \( z = 0 \) into the second row:
\[
y + \frac{1}{4}z = -\frac{3}{2} \implies y = -\frac{3}{2}
\]

Substituting \( y = -\frac{3}{2} \) and \( z = 0 \) into the first row:
\[
x - y + z = 4 \implies x + \frac{3}{2} = 4 \implies x = \frac{5}{2}
\]

### Step 4: Conclusion.

The system has a unique solution:
\[
x = \frac{5}{2}, \, y = -\frac{3}{2}, \, z = 0
\]

Thus, the correct answer is **B. Unique Solution**.

---

Would you like more details on any step? Here are five related questions to explore:

1. How do we determine if a system has infinite solutions?
2. What is Gaussian elimination and why is it useful in solving systems of linear equations?
3. How can matrix rank help in determining the nature of solutions to a system?
4. Could we solve this system using another method like substitution or Cramer's rule?
5. How do we interpret the geometric meaning of a system with a unique solution?

**Tip**: Always check for consistency in your system of equations by analyzing the augmented matrix carefully.

The equations x - y + z = 4, 3x + y + 4z = 6, and x + y + z = 1. What type of solution does this system have?

Learn how to determine the solution to a system of linear equations, including x - y + z = 4, 3x + y + 4z = 6, and x + y + z = 1. This detailed step-by-step guide covers Gaussian elimination and matrix operations to find the unique solution.

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