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  4. The corner points of the feasible region determined by the following system of linear inequalities 2 x+y ≤ 10, x+3 y ≤ 15, x, y ≥ 0 are (0,0)(5,0) (3,4) and (0,5). Let Z=p x+q y, where p, q>0. Condition on p and q, so that the maximum of Z occurs at both (3,4) and (0,5), is

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Q. The corner points of the feasible region determined by the following system of linear inequalities $2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0$ are $(0,0)(5,0)$ $(3,4)$ and $(0,5)$. Let $Z=p x+q y$, where $p, q>0$. Condition on $p$ and $q$, so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$, is

Linear Programming

Solution:

The maximum value of $Z$ is unique.
It is given that the maximum value of $Z$ occurs at two points $(3,4)$ and $(0,5)$. Value of $Z$ at $(3,4)=$ Value of $Z$ at $(0,5)$
$ \Rightarrow p(3)+q(4)=p(0)+q(5), 3 p+4 q=5 q \Rightarrow 3 p=q $