In a linear programming problem, the inequalities, X + 2Y ≤ 12 and 3X + 4Y ≥ 28, hold as binding constraints. The problem's optimal solution would be:
A) X = 4, and Y = -4.
B) X = 4, and Y = 8.
C) X = 4, and Y = 4.
D) X = 8, and Y = 4.
E) X = 12, Y = 0.

Question

Quizplus · Accepted Answer

When a constraint holds as a binding constraint, it means that the optimal solution lies on the boundary of that constraint. Therefore, we can use the given inequalities to find the intersection point of their boundaries.

For X + 2Y ≤ 12, we can rewrite it as Y ≤ (-1/2)X + 6 and graph it as a straight line with intercepts at (0,6) and (12,0).

For 3X + 4Y  28, we can rewrite it as Y  (-3/4)X + 7 and graph it as a straight line with intercepts at (0,7) and (9.33,0).

The feasible region is the region that satisfies both inequalities, which is bounded by the x-axis, the line Y = (-1/2)X + 6, and the line Y = (-3/4)X + 7.

The optimal solution is the point on the boundary of a feasible region that maximizes the objective function. Since the objective function is not given in the problem, we cannot determine the exact solution. However, we can see that the only point on the boundary that satisfies both binding constraints is the intersection point of the line X + 2Y = 12 and 3X + 4Y = 28, which is X = 4 and Y = 4 (choice C). Therefore, C is the correct answer.

In a Linear Programming Problem, the Inequalities, X + 2Y