# 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 y 0 6 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 y 0 2 -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).