#### Question

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.

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#### Reference Material

Solution for question: Minimum and Maximum Z = 5x + 2y Subject to the Following Constraints: concept: Graphical Method of Solving Linear Programming Problems. For the courses CBSE (Science), CBSE (Arts), PUC Karnataka Science, CBSE (Commerce), HSC Science (Computer Science), HSC Science (Electronics), HSC Arts, HSC Science (General) , ISC (Arts), ISC (Science), ISC (Commerce)