Q. 115.0( 2 Votes )

# Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:

2x + 4y ≤ 8

3x + y ≤ 6

x + y ≤ 4

x ≥ 0, y ≥ 0 [CBSE 2015]

Answer :

Given,

Objective function is: z = 2x + 5y

Constraints are:

2x + 4y ≤ 8

3x + y ≤ 6

x + y ≤ 4

x ≥ 0, y ≥ 0

the maximum value of z can only be obtained at the corner points of the feasible region. So we need to check the value of z at all corner points of the feasible region.

So, first, we will be finding out the feasible region by drawing the regions defined by constraints.

For plotting feasible region, we will be using the fundamentals of a straight line to get the feasible region as shown in the figure.

Clearly ABDC represents the feasible region and corner points are determined by solving:

3x+y = 6 and 2x + 4y = 8

x = 0 and 2x+4y = 8

y = 0 and 3x+y = 6

& x = 0 and y = 0

∴ value of objective function z at point A =

Value of Z at point B = 2× 2 + 0 = 4

**Value of Z at point C = 2× 0 + 5× 2 = 10**

Value of Z at point B = 2×0 + 0 = 0

Clearly Z is maximum at point **C(0,2)**

And the maximum value of Z = 10

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