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The system of equations given is:

\[
\begin{aligned}
1) & \quad x + y + z = 3, \\
2) & \quad 2x + 4y + 6z = 20, \\
3) & \quad 2x + 4y + \lambda z = \mu.
\end{aligned}
\]

We are asked to find the values of \(\lambda\) and \(\mu\) such that the system has **no solution**.

### Approach:

1. **Analyze consistency**: The system of linear equations will be **inconsistent** (i.e., have no solution) if one equation is not a linear combination of the others or leads to a contradictory result.

2. **Row reduce or use matrix representation**: We will check whether there is a contradiction in the system by comparing the coefficients of the variables in each equation. 

Let’s begin by simplifying the second equation, and then analyze the third.

### Simplifying the second equation:

\[
2x + 4y + 6z = 20 \implies x + 2y + 3z = 10.
\]

Now, we have the system:

\[
\begin{aligned}
1) & \quad x + y + z = 3, \\
2) & \quad x + 2y + 3z = 10, \\
3) & \quad 2x + 4y + \lambda z = \mu.
\end{aligned}
\]

### Subtract equation 1 from equation 2:

\[
(x + 2y + 3z) - (x + y + z) = 10 - 3,
\]

which simplifies to:

\[
y + 2z = 7. \quad \text{(Equation 4)}
\]

### Analyze the third equation:

Now, using the simplified system, we need to ensure that the third equation either leads to a contradiction or does not fit with the system for some values of \(\lambda\) and \(\mu\). Let's substitute \(y + 2z = 7\) into the third equation. This will help determine the values of \(\lambda\) and \(\mu\) where the system becomes inconsistent.

I'll proceed with the detailed calculation next.The solution shows that \(x\), \(y\), and \(z\) depend on both \(\lambda\) and \(\mu\) as follows:

\[
\begin{aligned}
x &= \frac{\mu}{\lambda - 6} + \frac{4 - 4\lambda}{\lambda - 6}, \\
y &= -\frac{2\mu}{\lambda - 6} + \frac{7\lambda - 2}{\lambda - 6}, \\
z &= \frac{\mu}{\lambda - 6} - \frac{20}{\lambda - 6}.
\end{aligned}
\]

### Condition for no solution:
For the system to have **no solution**, the denominator \(\lambda - 6\) must be zero, which leads to \(\lambda = 6\). If \(\lambda = 6\), the equations become inconsistent because the terms involving \(x\), \(y\), and \(z\) would lead to division by zero, causing contradictions.

Thus, the system of equations **has no solution if \(\lambda = 6\)**.

Would you like more details, or do you have any questions about this solution?

### Related questions:
1. How does setting \(\lambda = 6\) cause the system to become inconsistent?
2. What happens if \(\lambda \neq 6\)? Can we determine a specific value for \(\mu\)?
3. How does the solution change if we modify one of the original equations?
4. Can you explain why inconsistency occurs when \(\lambda = 6\)?
5. How would the solution behave if we changed the constants in the equations?

### Tip:
Whenever you encounter systems of equations, check for inconsistencies by simplifying and comparing equations to identify special conditions for no solutions.

Investigate the value of λ and μ such that the system of equations x + y + z = 3; 2x + 4y + 6z = 20; 2x + 4y + λz = μ has no solution.

Learn how to find values of λ and μ for which the system of equations x + y + z = 3, 2x + 4y + 6z = 20, and 2x + 4y + λz = μ has no solution. This problem covers key concepts in linear algebra and inconsistent systems of equations. Suitable for advanced high school or undergraduate students.

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