To solve the system of equations given by:
1. \(3x + 4y + 5z = \mu\)
2. \(x + 2y + 3z = 1\)
3. \(4x + 4y + 4z = \delta\)
and determine the values of \((\delta, \mu)\) for which the system is inconsistent, we will follow these steps:
### Step 1: Write the Augmented Matrix
We can represent the system of equations in matrix form as follows:
\[
\begin{bmatrix}
3 & 4 & 5 & | & \mu \\
1 & 2 & 3 & | & 1 \\
4 & 4 & 4 & | & \delta
\end{bmatrix}
\]
### Step 2: Apply Row Operations
We will perform row operations to simplify the augmented matrix.
1. First, we can swap the first row with the second row to make calculations easier:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 1 \\
3 & 4 & 5 & | & \mu \\
4 & 4 & 4 & | & \delta
\end{bmatrix}
\]
2. Next, we will eliminate the first variable \(x\) from the second and third rows. We can do this by performing the following operations:
- \(R_2 \leftarrow R_2 - 3R_1\)
- \(R_3 \leftarrow R_3 - 4R_1\)
After performing these operations, the augmented matrix becomes:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 1 \\
0 & -2 & -4 & | & \mu - 3 \\
0 & -4 & -8 & | & \delta - 4
\end{bmatrix}
\]
### Step 3: Further Row Operations
Next, we can simplify the second and third rows further. We can multiply the second row by \(-\frac{1}{2}\):
\[
\begin{bmatrix}
1 & 2 & 3 & | & 1 \\
0 & 1 & 2 & | & \frac{3 - \mu}{2} \\
0 & -4 & -8 & | & \delta - 4
\end{bmatrix}
\]
Now, we can eliminate the second variable \(y\) from the third row:
- \(R_3 \leftarrow R_3 + 4R_2\)
This gives us:
\[
\begin{bmatrix}
1 & 2 & 3 & | & 1 \\
0 & 1 & 2 & | & \frac{3 - \mu}{2} \\
0 & 0 & 0 & | & \delta - 4 + 4 \cdot \frac{3 - \mu}{2}
\end{bmatrix}
\]
### Step 4: Set Up the Inconsistency Condition
For the system to be inconsistent, the last row must represent a contradiction, which occurs when:
\[
\delta - 4 + 4 \cdot \frac{3 - \mu}{2} \neq 0
\]
Simplifying this gives:
\[
\delta - 4 + 6 - 2\mu \neq 0 \implies \delta - 2\mu + 2 \neq 0
\]
### Step 5: Find Values of \((\delta, \mu)\)
To find the values of \((\delta, \mu)\) that make the system inconsistent, we can set:
\[
\delta - 2\mu + 2 = 0
\]
This leads to:
\[
\delta = 2\mu - 2
\]
### Step 6: Check Options
We need to check which pairs \((\delta, \mu)\) satisfy this equation.
For example, if we check the option \((4, 3)\):
\[
\delta = 4, \mu = 3 \implies 4 = 2(3) - 2 \implies 4 = 6 - 2 \implies 4 = 4
\]
This satisfies the equation, hence \((\delta, \mu) = (4, 3)\) is a valid solution.
### Conclusion
The values of \((\delta, \mu)\) for which the system is inconsistent can be:
\[
\boxed{(4, 3)}
\]
