Explanation:
To minimize the objective function z = 2x + 5y subject to the given constraints, you can use linear programming techniques. Let's solve this step by step.
The constraints are:
1. 4y + 5x ≥ 40
2. 5y + 4x ≥ 40
3. y + x ≥ 9
4. x ≥ 0
5. y ≥ 0
First, we'll graph the feasible region defined by these constraints:
1. Graph the line 4y + 5x = 40.
2. Graph the line 5y + 4x = 40.
3. Graph the line y + x = 9.
4. The x-axis (x ≥ 0) and y-axis (y ≥ 0) serve as additional boundaries.
Now, find the intersection points of these lines. You'll find three intersection points: (8, 1), (10, 0), and (0, 9).
Next, you need to evaluate the objective function z = 2x + 5y at each of these points:
1. For (8, 1): z = 2(8) + 5(1) = 16 + 5 = 21
2. For (10, 0): z = 2(10) + 5(0) = 20 + 0 = 20
3. For (0, 9): z = 2(0) + 5(9) = 0 + 45 = 45
The minimum value of z among these points is 20, which occurs at (10, 0).
So, the minimum value of z = 2x + 5y is 20, and it occurs when x = 10 and y = 0.