Maximize the objective function P = 3x + 5y under these constraints: x + y ≤ 6, x - y ≥ 2, x ≥ 0, y ≥ 0
I’m so damn confused and couldn’t start even a sentence. I’ve a test tomorrow. pls help.
Given objective function,
P = 3x + 5y
Given constraints,
x + y ≤ 6, x - y ≥ 2, x ≥ 0, y ≥ 0
Writing inequalities as equation,
x + y = 6 --- (i)
x - y = 2 --- (ii)
x = 0, y = 0 --- (iii)
From (i),
x + y = 6
| x | 0 | 6 |
|---|---|---|
| y | 6 | 6 |
Taking origin (0, 0) as the testing point in inequality of equation (i),
0 + 0 ≤ 6 [True]
So, its region is towards the testing point (shown using arrow in the graph).
From (ii),
x - y = 2
| x | 0 | 2 |
|---|---|---|
| y | -2 | 0 |
Taking origin (0, 0) as the testing point in inequality of equation (ii),
0 + 0 ≥ 2 [False]
So, its region is away from the testing point.
From (iii),
x = 0 and y = 0 represents the line on x-axis and y-axis respectively. Their inequality x ≥ 0 and y ≥ 0 shows that they lie in positive direction.
Now, plotting in graph,
From graph,
A(2, 0), B(6, 0) and C(4, 2) represents the common region.
Under the objective function, P = 3x + 5y,
| Point | x | y | P = 3x + 5y | Value | Remarks |
|---|---|---|---|---|---|
| A(2, 0) | 2 | 0 | 3(2) + 5(0) | 6 | Minimum |
| B(6, 0) | 6 | 0 | 3(6) + 5(0) | 18 | |
| C(4, 2) | 4 | 2 | 3(4) + 5(2) | 22 | Maximum |
Hence, the objective function P = 3x + 5y has maximum value 22 at C(4, 2).