IF THE MINIMUM VALUE OF AN OBJECTIVE FUNCTION Z=ax+by occurs at two point(3,4) and 4,3

Options

A. 15
B. `-15`
C. 21
D. can't be determine

Answer

To solve the problem, we need to find the minimum value of the objective function \( Z = ax + by \) given that it occurs at two points: (3, 4) and (4, 3).

### Step-by-Step Solution:

1. **Understanding the Objective Function**:
The objective function is given as \( Z = ax + by \). We need to find the coefficients \( a \) and \( b \) such that the minimum value occurs at the specified points.

2. **Substituting the Points into the Objective Function**:
We will substitute the points (3, 4) and (4, 3) into the objective function:
- For point (3, 4):
\[ Z_1 = a(3) + b(4) = 3a + 4b \]
- For point (4, 3):
\[ Z_2 = a(4) + b(3) = 4a + 3b \]

3. **Setting the Objective Function Values Equal**:
Since both points yield the minimum value of \( Z \), we can set the two expressions for \( Z \) equal to each other:
\[ 3a + 4b = 4a + 3b \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 3a + 4b - 4a - 3b = 0 \]
Simplifying this leads to:
\[ -a + b = 0 \quad \Rightarrow \quad b = a \]

5. **Finding the Minimum Value**:
Now that we have \( b = a \), we can substitute \( b \) back into either equation to find the minimum value. Let's use \( b = a \) in \( Z_1 \):
\[ Z = 3a + 4b = 3a + 4a = 7a \]
The minimum value of \( Z \) depends on the value of \( a \).

6. **Conclusion**:
The minimum value of the objective function \( Z \) occurs at both points (3, 4) and (4, 3) and is given by \( Z = 7a \), where \( a \) can be any real number.