Using graphical method, find the maximum value of

$2x+y$

subject to

$4x + 3y \leq 12 \\ 4x + y \leq 8 \\ 4x - y \leq 8 \\ x,y \geq 0$.

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$4x + 3y \leq 12$

x | 0 | 3 |

y | 4 | 0 |

$4x + y \leq 8$

x | 0 | 2 |

y | 8 | 0 |

$4x - y \leq 8$

x | 0 | 2 |

y | -8 | 0 |

The value of the objective function at each of these extreme points is as follows

Extreme Point Coordinates ( x,y) | Objective function valuez=2x+y |

O(0,0) | 2(0)+1(0)=0 |

A(0,4) | 2(0)+1(4)=4 |

B($\frac{3}{2}$,2) | 2($\frac{3}{2}$ )+1(2)=5 |

C(2,0) | 2(2)+1(0)=4 |

The maximum value of the objective function z=5 occurs at the extreme point ($\frac{3}{2}$,2).

Hence, the optimal solution to the given LP problem is **$x=\frac{3}{2},y=2$ and $max\ z=5$.**

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