Question

Solve the LP problem. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. (Enter EMPTY if the region is empty. Enter UNBOUNDED if the function is unbounded.)

Maximize p = 4x + 2y subject to
−3x
+ y ≥ 5
x + 5y ≤ 7
x ≥ 0, y ≥ 0.

p=
(x,y)=( )

Answer 1

The optimal solution for the LP problem is p = 10 at the point (0, 5), where the feasible region is bounded and the objective function is not unbounded.

Step-by-step explanation:

To solve the given linear programming (LP) problem, we must find the maximum value of the objective function p = 4x + 2y given the constraints:

• −3x + y ≥ 5
• x + 5y ≤ 7
• x ≥ 0
• y ≥ 0

First, we rewrite the constraints as inequalities to plot on the coordinate plane:

• y ≥ 3x + 5
• y ≤ −(1/5)x + 7/5
• x ≥ 0
• y ≥ 0

After plotting these lines, we look for the feasible region where all inequalities intersect. The feasible region is bounded and non-empty, and since our objective function is linear and the feasible region is bounded on the plane, the function will not be unbounded.

Next, we find the corner points of the feasible region and evaluate the objective function at each of these points:

• (0, 7/5)
• (0, 5)
• (1, 2)

Calculating the value of p at each point:

• For (0, 7/5): p = 4(0) + 2(7/5) = 14/5
• For (0, 5): p = 4(0) + 2(5) = 10
• For (1, 2): p = 4(1) + 2(2) = 8

The maximum value of the objective function occurs at the point (0, 5), where p = 10.

Therefore the optimal solution is p = 10, with the coordinates of the solution as (x, y) = (0, 5).