Solve the following LP problems using the Simplex Method
maximize $P = 10 x_{1} + 5 x_{2}$
subject to $4 x_{1} + x_{2} \leq 28$
$2 x_{1} + 3 x_{2} \leq 24$
$x_{1}, x_{2} \geq 0$

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Solution

$p =$ $10 x_{1}$ + $5 x_{2}$

from these two equtions
$4 x_{1} + x_{2} = 28$ -->1
$2 x_{1} + 3 x_{2} = 24$ -->2

From $1$ and $2$
we get $(6, 4)$
from the graph the corner points are $(0, 28), (6, 4), (12, 0)$

therefore we will get maximum profit at (6,4)

$p = 10 \times 6 + 5 \times 4$
$p = 80$
therefore the optimal solution is $80$

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For the linear programming problem:

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Subject to:

- 2 x_{1} + 3 x_{2} \leq 9

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The above problem has

Consider the primal problem in LP.

Maximize Z = 4 X_{1} + 3 X_{2}

Subject to

X_{1} + X_{2} \leq 8 2 X_{1} + X_{2} \leq 10 X_{1}; X_{2} \geq 0

together with its dual (LD) Then

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