If system of equation
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + `lambda`z = `mu` has more than two solutions. Find `(mu -lambda )`

Answer

To solve the problem, we need to analyze the given system of equations and find the values of \( \mu \) and \( \lambda \) such that the system has more than two solutions. This typically occurs when the equations are dependent, leading to infinitely many solutions.

### Step-by-Step Solution:

1. **Write the System of Equations:**
The system of equations is given as:
\[ \begin{align*} 1. & \quad x + y + z = 6 \quad \text{(Equation 1)} \\ 2. & \quad x + 2y + 3z = 10 \quad \text{(Equation 2)} \\ 3. & \quad 3x + 2y + \lambda z = \mu \quad \text{(Equation 3)} \end{align*} \]

2. **Convert to Matrix Form:**
The augmented matrix form of the system can be represented as:
\[ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 1 & 2 & 3 & | & 10 \\ 3 & 2 & \lambda & | & \mu \end{bmatrix} \]

3. **Row Operations:**
We will perform row operations to simplify the matrix. Start by subtracting the first row from the second row:
\[ R_2 \leftarrow R_2 - R_1 \]
This gives us:
\[ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 2 & | & 4 \\ 3 & 2 & \lambda & | & \mu \end{bmatrix} \]

4. **Continue Row Operations:**
Now, we will subtract 3 times the first row from the third row:
\[ R_3 \leftarrow R_3 - 3R_1 \]
This results in:
\[ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 2 & | & 4 \\ 0 & -1 & \lambda - 3 & | & \mu - 18 \end{bmatrix} \]

5. **Add Rows:**
Next, we add the second row to the third row:
\[ R_3 \leftarrow R_3 + R_2 \]
This gives us:
\[ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 0 & 1 & 2 & | & 4 \\ 0 & 0 & \lambda - 1 & | & \mu - 14 \end{bmatrix} \]

6. **Condition for Infinite Solutions:**
For the system to have infinitely many solutions, the last row must be all zeros:
\[ \lambda - 1 = 0 \quad \text{and} \quad \mu - 14 = 0 \]
This leads to:
\[ \lambda = 1 \quad \text{and} \quad \mu = 14 \]

7. **Calculate \( \mu - \lambda \):**
Finally, we need to find \( \mu - \lambda \):
\[ \mu - \lambda = 14 - 1 = 13 \]

### Final Answer:
\[ \mu - \lambda = 13 \]