What point in the feasible region maximizes the objective function?
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{y - 5<_ 4x; y + x<_ 10; x>_0, y>_3
Maximum for
P = 7x - 5y
To maximize the objective function P = 7x - 5y, we need to find the point in the feasible region that gives the highest value for P.
The feasible region is defined by the following inequalities:
y - 5 ≤ 4x
y + x ≤ 10
x ≥ 0
y ≥ 3
To find the feasible region, we can solve these inequalities one by one.
1. Solve y - 5 ≤ 4x:
Rearrange the inequality to y ≤ 4x + 5.
Graph this inequality on a coordinate plane.
2. Solve y + x ≤ 10:
Rearrange the inequality to y ≤ -x + 10.
Graph this inequality on the same coordinate plane.
3. Solve x ≥ 0:
This inequality represents the non-negative x-axis. Graph it on the same coordinate plane.
4. Solve y ≥ 3:
This inequality represents the line y = 3 and the shaded region above it. Graph it on the same coordinate plane.
Now, we can determine the overlapping shaded region of all the inequalities. The feasible region is where all the shaded areas overlap.
Next, we can find the corner points or vertices of the feasible region by finding the intersection points of the lines/inequalities.
By visually examining the graph or by solving the simultaneous equations, we can find that the three vertices of the feasible region are (0, 3), (2, 6), and (5, 5).
Now, we substitute each of these vertices into the objective function P = 7x - 5y to find the maximum value.
For (0, 3):
P = 7(0) - 5(3) = -15
For (2, 6):
P = 7(2) - 5(6) = -14
For (5, 5):
P = 7(5) - 5(5) = 10
So, the maximum value of the objective function P is achieved at the point (5, 5) in the feasible region.