4x+3y+z=14
x-3y+2z=-6
11x-2y+3z=8
the solution set is {}

We can solve this system of three linear equations using either substitution, elimination, or matrix methods. In this case, I will use **Gaussian elimination** to find the solution set.

The system of equations is:

\[
\begin{aligned}
1. & \quad 4x + 3y + z = 14 \\
2. & \quad x - 3y + 2z = -6 \\
3. & \quad 11x - 2y + 3z = 8
\end{aligned}
\]

### Step 1: Write the system as an augmented matrix.
\[
\begin{pmatrix}
4 & 3 & 1 & | & 14 \\
1 & -3 & 2 & | & -6 \\
11 & -2 & 3 & | & 8 \\
\end{pmatrix}
\]

### Step 2: Perform row operations to achieve row echelon form.

1. Swap **Row 1** and **Row 2** to get a leading 1 in the first position.
\[
\begin{pmatrix}
1 & -3 & 2 & | & -6 \\
4 & 3 & 1 & | & 14 \\
11 & -2 & 3 & | & 8 \\
\end{pmatrix}
\]

2. Eliminate the first element of rows 2 and 3 by subtracting suitable multiples of Row 1.
   - Row 2 → Row 2 - 4(Row 1)
   - Row 3 → Row 3 - 11(Row 1)

\[
\begin{pmatrix}
1 & -3 & 2 & | & -6 \\
0 & 15 & -7 & | & 38 \\
0 & 31 & -19 & | & 74 \\
\end{pmatrix}
\]

### Step 3: Make the pivot in the second column a 1 by dividing Row 2 by 15.
\[
\begin{pmatrix}
1 & -3 & 2 & | & -6 \\
0 & 1 & -\frac{7}{15} & | & \frac{38}{15} \\
0 & 31 & -19 & | & 74 \\
\end{pmatrix}
\]

### Step 4: Eliminate the second column of Row 3 by subtracting 31 times Row 2 from Row 3.
\[
\begin{pmatrix}
1 & -3 & 2 & | & -6 \\
0 & 1 & -\frac{7}{15} & | & \frac{38}{15} \\
0 & 0 & \frac{14}{15} & | & \frac{32}{15} \\
\end{pmatrix}
\]

### Step 5: Solve for \(z\), \(y\), and \(x\).
- From the third row: 
\[
\frac{14}{15}z = \frac{32}{15} \quad \Rightarrow \quad z = \frac{32}{14} = \frac{16}{7}
\]
- Substitute \(z = \frac{16}{7}\) into Row 2 to solve for \(y\):
\[
y - \frac{7}{15} \times \frac{16}{7} = \frac{38}{15} \quad \Rightarrow \quad y = \frac{8}{3}
\]
- Substitute \(y = \frac{8}{3}\) and \(z = \frac{16}{7}\) into Row 1 to solve for \(x\):
\[
x - 3 \times \frac{8}{3} + 2 \times \frac{16}{7} = -6 \quad \Rightarrow \quad x = -\frac{17}{7}
\]

### Solution set:
\[
x = -\frac{17}{7}, \quad y = \frac{8}{3}, \quad z = \frac{16}{7}
\]

Would you like further details on any of the steps?

### 5 Related Questions:
1. How is Gaussian elimination different from Gauss-Jordan elimination?
2. What are the benefits of using matrix methods for solving systems of equations?
3. How can row operations affect the solution of a system of equations?
4. Can a system of linear equations have no solution or infinite solutions?
5. How does back-substitution work in solving systems of equations?

### Tip:
Always check the consistency of your system of equations before attempting to solve them, as some systems may not have a unique solution.

4x + 3y + z = 14; x - 3y + 2z = -6; 11x - 2y + 3z = 8. The solution set is {}

Explore how to solve a system of three linear equations using Gaussian elimination: 4x + 3y + z = 14, x - 3y + 2z = -6, and 11x - 2y + 3z = 8. The solution involves converting the equations into matrix form, performing row operations, and using back-substitution to find the unique solution set for x, y, and z. Ideal for high school students in Grades 10-12.

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