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To solve this problem, we need to investigate under what conditions the system of linear equations has no solution. The system of equations given is:

\[
x + y + z = 3
\]
\[
2x + 4y + 6z = \lambda
\]
\[
x + 2y + \mu z = 6
\]

For this system to have no solution, the equations must either be inconsistent (i.e., lead to contradictory equations) or represent planes that do not intersect.

### Step 1: Analyze the system for dependency
1. The second equation can be written as:
   \[
   2x + 4y + 6z = \lambda
   \]
   Dividing through by 2, this becomes:
   \[
   x + 2y + 3z = \frac{\lambda}{2}
   \]
   This equation should be compared with the first and third equations.

### Step 2: Investigate the condition for inconsistency
For the system to be inconsistent, the equations should represent parallel planes that never intersect. This can occur if the coefficients of the variables in the equations are proportional, but the constants on the right-hand side are different.

The first equation is:
\[
x + y + z = 3
\]
The second equation (after simplification) is:
\[
x + 2y + 3z = \frac{\lambda}{2}
\]
The third equation is:
\[
x + 2y + \mu z = 6
\]

### Step 3: Check conditions for no solution
- If \(\lambda = 6\), the second and third equations become dependent, so the system may have no solution depending on \(\mu\).
- If \(\mu = 20\), this creates another dependency with the other equations, leading to a potential inconsistency.

Thus, the system has no solution when \(\lambda = 6\) and \(\mu = 20\), because the second and third equations would become contradictory.

### Conclusion:
The correct answer is:
\[
\lambda = 6, \mu = 20
\]

Let me know if you need more detailed steps or clarification.

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Here are five related questions:
1. What is the general method to determine if a system of equations has no solution?
2. How do you check if planes are parallel or coincident in three-dimensional space?
3. What happens if two equations in a system are dependent but have different constants?
4. How can Gaussian elimination be used to solve a system of equations?
5. What is the geometric interpretation of a system of linear equations in three variables?

**Tip**: Always check if the equations are linearly dependent by reducing them to simpler forms to avoid mistakes in solutions.

Investigate the value of λ and μ such that the system of equations x + y + z = 3; 2x + 4y + 6z = λ; x + 2y + μz = 6 has no solution if ____.

Explore how to find the values of λ and μ in a system of linear equations where no solution exists. This problem focuses on analyzing the linear dependency between equations and identifying conditions for inconsistency. Suitable for undergraduate linear algebra students.

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