Determine graphically the solution set for each system of ...



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Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded

24. 2x - y > 4

4x – 2y < -2

[pic]

No solution, because there is no intersection between areas.

28. x + y > 20

x + 2y > 40

x > 0, y > 0

[pic]

Blue area denotes the intersection or the solution of systems of inequalities.

Region is unbounded.

34. 6x + 7y < 84

12x – 11y < 18

6x – 7y < 28

x > 0, y > 0

[pic]

Region is bounded.

Solve each linear programming problem by the method of corners.

14.

Maximize P = 2x + 5y

Subject to 2x + y < 16

2x + 3y < 24

y < 6

x > 0, y > 0

[pic]

Evaluate P at each corner.

P(0,6) = 2*0+5*6 = 30

P(3,6) = 2*3+5*6 = 36

P(6,4) = 2*6+5*4 = 32

P(8,0) = 2*8+5*0 = 16

So function is max at (3, 6) and max P = 36.

20.

Maximize C = 2x + 5y

Subject to 4x + y > 40

2x + y > 30

x + 3y > 30

x > 0, y > 0

[pic]

Region is unbounded. So, 2x+5y increases continuously. No optimal solution for maximum.

28.

Find the maximum and minimum of P = 4x + 3y Subject to

3x + 5y > 20

3x + y < 16

-2x + y < 1

x > 0, y > 0

• [pic]

Solve for P at each corner point.

P(15/13, 43/13) = (4*15/13) + (3*43/13) = 14.5385

P(3, 7) = 4*3+3*7 = 33

P(5, 1) = 4*5+3*1 = 23

So Maximum is P = 33 at x = 3 and y = 7

Minimum is P = 14.5385 at x = 15/13 and y = 43/13[pic]

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