Question

The solution set for minimizing the function \( z = x + y \) with constraints \[ x + y \geq 2, \quad x + 2y \leq 8, \quad x \leq 3, \quad x, y \geq 0 \] contains

• A. \( x = 0, y = 3 \)
• B. \( x = 8, y = 0 \)
• C. infinitely many points
• D. \( x = 2, y = 3 \)

Hint:

To minimize a linear function subject to linear constraints, graphically plot the constraints and identify the feasible region. The minimum value is found at the vertex of this region.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In linear programming, if the objective function is parallel to one of the constraint lines that forms a boundary of the feasible region, the minimum or maximum can occur at all points along that segment.
Step 2: Detailed Explanation:
1. Objective function: \( z = x + y \).
2. Constraint 1: \( x + y \ge 2 \). Boundary is \( x + y = 2 \).
Notice that the slope of the objective function is the same as the slope of the first constraint boundary.
3. Feasible region vertices:
Intersection of \( x + y = 2 \) with axes: \( (2, 0) \) and \( (0, 2) \).
For both \( (2, 0) \) and \( (0, 2) \), \( z = 2 \).
Any point on the segment joining \( (2, 0) \) and \( (0, 2) \) also satisfies \( x + y = 2 \), and thus gives \( z = 2 \).
Since 2 is the minimum value for the constraints provided (as \( x+y \ge 2 \)), there are infinitely many points that minimize the function.
Step 3: Final Answer:
The solution set contains infinitely many points.