Solve the following LPP graphically: Maximize Z = 9x + 13y subject to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0 Solution: Convert the constraints into equations and find the intercept made - Mathematics and Statistics

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Solve the following LPP graphically:

Maximize Z = 9x + 13y subject to constraints

2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0

Solution: Convert the constraints into equations and find the intercept made by each one of it.

Inequation Equation X intercept Y intercept Region
2x + 3y ≤ 18 2x + 3y = 18 (9, 0) (0, ___) Towards origin
2x + y ≤ 10 2x + y = 10 ( ___, 0) (0, 10) Towards origin
x ≥ 0, y ≥ 0 x = 0, y = 0 X axis Y axis ______

The feasible region is OAPC, where O(0, 0), A(0, 6),

P( ___, ___ ), C(5, 0)

The optimal solution is in the following table:

Point Coordinates Z = 9x + 13y Values Remark
O (0, 0) 9(0) + 13(0) 0  
A (0, 6) 9(0) + 13(6) ______  
P ( ___,___ ) 9( ___ ) + 13( ___ ) ______ ______
C (5, 0) 9(5) + 13(0) ______  

∴ Z is maximum at __( ___, ___ ) with the value ___.

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Solution

Convert the constraints into equations and find the intercept made by each one of it.

Inequation Equation X intercept Y intercept Region
2x + 3y ≤ 18 2x + 3y = 18 (9, 0) ( 0, 6) Towards origin
2x + y ≤ 10 2x + y = 10 (5, 0) (0, 10) Towards origin
x ≥ 0, y ≥ 0 x = 0, y = 0 X axis Y axis 1st quadrant

The feasible region is OAPC, where O(0, 0), A(0, 6),

P(3, 4), C(5, 0)

The optimal solution is in the following table:

Point Coordinates Z = 9x + 13y Values Remark
O (0, 0) 9(0) + 13(0) 0  
A (0, 6) 9(0) + 13(6) 78  
P (3, 4) 9(3) + 13(4) 79 maximum
C (5, 0) 9(5) + 13(0) 45  

∴ Z is maximum at P(3, 4) with the value 79.

  Is there an error in this question or solution?
Chapter 2.6: Linear Programming - Q.5 (E)

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Solution: Convert the constraints into equations and find the intercept made by each one of it.

Inequations Equations X intercept Y intercept Region
5x + y ≥ 10 5x + y = 10 ( ___, 0) (0, 10) Away from origin
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x + 4y ≥ 12 x + 4y = 12 (12, 0) (0, 3) Away from origin
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∵ Origin has not satisfied the inequations.

∴ Solution of the inequations is away from origin.

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In the figure, ABCD represents

The set of the feasible solution where

A(12, 0), B( ___, ___ ), C ( ___, ___ ) and D(0, 10).

The coordinates of B are obtained by solving equations

x + 4y = 12 and x + y = 6

The coordinates of C are obtained by solving equations

5x + y = 10 and x + y = 6

Hence the optimum solution lies at the extreme points.

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Point Coordinates Z = 4x + 5y Values Remark
A (12, 0) 4(12) + 5(0) 48  
B ( ___, ___ ) 4( ___) + 5(___ ) ______ ______
C ( ___, ___ ) 4( ___) + 5(___ ) ______  
D (0, 10) 4(0) + 5(10) 50  

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