The corner points of the feasible region of a system of linear inequations are (0, 0), (5, 0), (6,5), (6, 8), (4, 10) and (0,8). If Z = 3x - 4y, then the minimum value of Z occurs at:

Options

A. (0,8)
B. (6,5)
C. (6,8)
D. (4, 10)

Answer

To find the minimum value of \( Z = 3x - 4y \) at the given corner points of the feasible region, we will evaluate \( Z \) at each corner point.

### Step-by-Step Solution:

1. **List the Corner Points**: The corner points of the feasible region are:
- \( (0, 0) \)
- \( (5, 0) \)
- \( (6, 5) \)
- \( (6, 8) \)
- \( (4, 10) \)
- \( (0, 8) \)

2. **Evaluate \( Z \) at Each Corner Point**:
- For the point \( (0, 0) \):
\[ Z = 3(0) - 4(0) = 0 \]
- For the point \( (5, 0) \):
\[ Z = 3(5) - 4(0) = 15 \]
- For the point \( (6, 5) \):
\[ Z = 3(6) - 4(5) = 18 - 20 = -2 \]
- For the point \( (6, 8) \):
\[ Z = 3(6) - 4(8) = 18 - 32 = -14 \]
- For the point \( (4, 10) \):
\[ Z = 3(4) - 4(10) = 12 - 40 = -28 \]
- For the point \( (0, 8) \):
\[ Z = 3(0) - 4(8) = 0 - 32 = -32 \]

3. **Compare the Values of \( Z \)**:
- \( Z(0, 0) = 0 \)
- \( Z(5, 0) = 15 \)
- \( Z(6, 5) = -2 \)
- \( Z(6, 8) = -14 \)
- \( Z(4, 10) = -28 \)
- \( Z(0, 8) = -32 \)

4. **Determine the Minimum Value**:
The minimum value of \( Z \) occurs at the point \( (0, 8) \) where \( Z = -32 \).

### Final Answer:
The minimum value of \( Z \) occurs at the point \( (0, 8) \) and the minimum value is \( -32 \).

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