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Graph

**Solve the following problem :**

Minimize Z = 2x + 3y Subject to x – y ≤ 1, x + y ≥ 3, x ≥ 0, y ≥ 0

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#### Solution

To find the graphical solution, construct the table as follows:

Inequation | Equation | Double intercept form | Points (x_{1}, x_{2}) |
Region |

x – y ≤ 1 | x – y = 1 | `x/(1) + y/(-1)` = 1 | A (1, 0) B (0, –1) |
0 – 0 ≤ 1 |

x + y ≥ 1 | x + y = 3 | `x/(3) + y/(3)` = 1 | C (3, 0) D (0, 3) |
0 + 0 ≥ 3 ∴ 0 ≥ 3 ∴ non-origin side |

x ≥ 0 | x = 0 | – | R.H.S. of Y- axis | |

y ≥ | y = 0 | above X-axis |

The shaded portion Y' DE is the feasible region.

Whose vertices are D(0, 3) and E

E is the point of intersection of lines

x – y = 1 …(i)

x + y = 3 …(ii)

∴ By (i) + (ii), we get

x – y = 1

x + y = 3

2x = 4

∴ x = `(4)/(2)` = 2

Substituting x = 2 in (i), we get

2 – y = 1

∴ y = 1

∴ E(2, 1)

Here, the objective function is Z = 2x + 3y

Now, we will find minimum value of Z as follows:

Feasible points | The value of Z = 2x + 3y |

D(0, 3) | Z = 2(0) + 3(3) = 9 |

E(2, 1) | Z = 2(2) + 3(1) = 4 + 3 = 7 |

∴ Z has minimum value 7 at E(2, 1)

∴ Z is minimum, when x = 2, y = 1.

Concept: Mathematical Formulation of Linear Programming Problem

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