Question

Maximize
z = 22x1 + 30x2,
subject to
x1 + 3x2 ≤ 6
x1 − 3x2 ≤ −3
x1 ≥ 0, x2 ≥ 0.

Answer 1

Given :

x₁ + 3x₂ ≤ 6

x₁ − 3x₂ ≤ −3

x₁≥ 0, x₂ ≥ 0.

To Find : Maximize z = 22x₁ + 30x₂

Solution:

Plot x₁  on x axis

Plot x₂  on y axis

and  draw Equation and then find half plane using inequality

x + 3y  = 6    plot points   (0 , 2 )  , ( 6 , 0)  and draw line

0 + 0 < 6  Hence region containing origin

x - 3y  = - 3   plot  points    (0 , 1)  , ( -3 , 0)  and draw line

0 - 0  > - 3 hence area not containing origin

x , y ≥ 0 hence 1st Quadrant only

Boundary points are

( 0, 1) , ( 1.5 , 1.5 ) , ( 0 , 2)

z = 22x₁ + 30x₂

0   , 1          22(0) + 30(1)  = 30

1.5   , 1.5    22(1.5) + 30(1.5)  = 78

0 , 2          22(0) + 30(2) = 60

78 is max at   x₁ = 1.5  ,   x₂ = 1.5

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