Question

If \( Z = 7x + y \) subject to \[ 5x + y \geq 5, \quad x + y \geq 3, \quad x \geq 0, \quad y \geq 0, \quad \text{then the minimum value of } Z \text{ is} \]

• A. 2
• B. 5
• C. 6
• D. 3

Hint:

In linear programming, always graph the constraints to identify the feasible region and then evaluate the objective function at the vertices to find the maximum or minimum values.

Answer 1

The Correct Option is B

Step 1: Analyze the constraints.
We are given the linear programming problem with constraints. To find the minimum value of \( Z = 7x + y \), we need to graph the constraints and find the feasible region. The inequalities are: \[ 5x + y \geq 5, \quad x + y \geq 3, \quad x \geq 0, \quad y \geq 0 \]
Step 2: Find the intersection points.
By solving the system of equations derived from the constraints, we find the feasible region, and the point that minimizes \( Z \) occurs at \( x = 0, y = 5 \). Substituting into \( Z \), we get: \[ Z = 7(0) + 5 = 5 \]
Step 3: Conclusion.
Thus, the minimum value of \( Z \) is 5, making option (B) the correct answer.