Question

how many optimal solutions does the following problem have? max z = 5x1 7x2 s.t. 2x1 − x2 ≤ −1 −x1 2x2 ≤ −1 x1, x2 ≥ 0

Answer 1

The given problem is a linear programming problem with 2 variables and constraints. By graphing the feasible region and finding the corner points, we can determine the number of optimal solutions to the problem, which in this case is 3.

Step-by-step explanation:

The given problem is a linear programming problem, and its solution can be found graphically. The objective function is max z = 5x1 + 7x2 subject to the constraints:

1. 2x1 - x2 ≤ -1
2. -x1 + 2x2 ≤ -1
3. x1 ≥ 0 and x2 ≥ 0

To find the number of optimal solutions, we need to determine the region of feasible solutions. Plotting the feasible region on a graph and identifying the corner points will help us find the optimal solutions.

After drawing the graph, we find that the feasible region has 3 corner points: (0, 0), (0, -0.5), and (0.5, 0). These corner points represent the optimal solutions. Therefore, the given problem has 3 optimal solutions.