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Minimum and maximum z = 5x + 2y subject to the following constraints:

x-2y ≤ 2

3x+2y ≤ 12

-3x+2y ≤ 3

x ≥ 0,y ≥ 0

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#### Solution

x-2y ≤ 2

3x+2y ≤ 12

-3x+2y ≤ 3

x ≥ 0,y ≥ 0

Converting the inequations into equations, we obtain the lines

x-2y = 2 .....(i)

3x+2y = 12......(ii)

-3x+2y = 3.......(iii)

x = 0,y =0

From the graph, we get the corner points as

A(0, 5), B(3.5, 0.75), C(2, 0), D(1.5, 3.75), O(0, 0)

The values of the objective function are:

Point (x, y) | Values of the objective function Z = 5x + 2y |

A(0, 5) | 5 × 0 + 2 × 5 = 10 |

B(3.5, 0.75) | 5 × 3.5 + 2 × 0.75 = 19 (Maximum) |

C(2, 0) | 5 × 2 + 2 × 0= 10 |

D(1.5, 3.75) | 5 × 1.5 + 2 × 3.75 = 15 |

O(0, 0) | 5 × 0 + 2 × 0 = 0 (Minimum) |

The maximum value of Z is 19 and its minimum value is 0.

Concept: Graphical Method of Solving Linear Programming Problems

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