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The problem asks to "Investigate the value of \(\lambda\) and \(\mu\) for which the system of equations":

\[
\begin{aligned}
x + y + z &= 6 \\
x + 2y + 3z &= 10 \\
x + 2y + \lambda z &= \mu
\end{aligned}
\]

is consistent. 

### Approach:

1. **Consistency of the system**: A system of equations is consistent if there exists at least one solution that satisfies all the equations simultaneously. Therefore, we need to check under what conditions on \(\lambda\) and \(\mu\), the third equation fits with the solution of the first two.

2. **Step 1: Solve the first two equations**:
   \[
   x + y + z = 6 \quad \text{(Equation 1)}
   \]
   \[
   x + 2y + 3z = 10 \quad \text{(Equation 2)}
   \]
   Subtract Equation 1 from Equation 2 to eliminate \(x\):
   \[
   (x + 2y + 3z) - (x + y + z) = 10 - 6
   \]
   Simplifying:
   \[
   y + 2z = 4 \quad \text{(Equation 4)}
   \]
   This gives a relationship between \(y\) and \(z\): 
   \[
   y = 4 - 2z
   \]

3. **Step 2: Substitute this into Equation 1**:
   \[
   x + (4 - 2z) + z = 6
   \]
   Simplifying:
   \[
   x + 4 - z = 6
   \]
   \[
   x = z + 2 \quad \text{(Equation 5)}
   \]

   So, the general solution for the system (from the first two equations) is:
   \[
   x = z + 2, \quad y = 4 - 2z, \quad z = z
   \]

4. **Step 3: Use this in the third equation**:
   The third equation is:
   \[
   x + 2y + \lambda z = \mu
   \]
   Substituting \(x = z + 2\) and \(y = 4 - 2z\):
   \[
   (z + 2) + 2(4 - 2z) + \lambda z = \mu
   \]
   Simplifying:
   \[
   z + 2 + 8 - 4z + \lambda z = \mu
   \]
   \[
   (-3z + \lambda z) + 10 = \mu
   \]
   \[
   (\lambda - 3)z + 10 = \mu
   \]

5. **Step 4: Investigate for consistency**:
   For the system to be consistent, this equation should hold for some values of \(z\). There are two cases:

   - If \(\lambda = 3\), the term with \(z\) vanishes, and we are left with:
     \[
     10 = \mu
     \]
     So, for \(\lambda = 3\), \(\mu\) must be \(10\) for the system to be consistent.

   - If \(\lambda \neq 3\), the coefficient of \(z\) is non-zero, so \(\mu = (\lambda - 3)z + 10\). In this case, \(\mu\) can take any value depending on \(z\), meaning the system will be consistent for any \(z\), provided \(\mu = (\lambda - 3)z + 10\).

### Conclusion:
- For \(\lambda = 3\), the system is consistent if \(\mu = 10\).
- For \(\lambda \neq 3\), the system is consistent if \(\mu = (\lambda - 3)z + 10\).

Would you like further clarification or details?

### Follow-up Questions:
1. What is the condition for the system to be inconsistent?
2. How do we interpret solutions geometrically when \(\lambda = 3\)?
3. Can this system be extended to higher dimensions?
4. How does the value of \(\lambda\) affect the nature of the solutions?
5. What are the implications of having a unique solution versus infinitely many solutions?

**Tip**: Always check for both the specific and general cases when solving systems of equations to avoid missing any special conditions like \(\lambda = 3\).

Investigate the value of \(\lambda\) and \(\mu\) for which the system of equations: 1. x + y + z = 6 2. x + 2y + 3z = 10 3. x + 2y + \lambda z = \mu is consistent.

Explore the conditions under which a parametric system of linear equations remains consistent. This analysis uses algebraic techniques to investigate values of \(\lambda\) and \(\mu\) that ensure the system has at least one solution. This topic is suitable for advanced high school or early college students studying algebra or linear algebra.

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