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Find all basic solutions to the following problem

Find all basic solutions to the following problem

Max. $Z = x_1 +3x_2 + 3x_3, $

$\text{subject to} \hspace{0.2cm} x_1 + 2 x_2 +3 x_3=4; 2 x_1 +3x_2 +5 x_3=7$

$x_1, x_2, x_3 ≥ 0$

1 Answer
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Since there are three variables and two constraints, a basic solution can be obtained by putting 3 - 2 = 1 variable equals to zero & there will be $^3C_2= 3$ basic solutions.

No. of basic solutions Non – basic variables =0 Basic variables Equations and the values of the basic variables Is the solution feasible Is the solution degenerate? Value of z Is the solution optimal ?
1. $x_1$=0 $x_2 , x_3$ $2x_2 +3x_3 = 4,\\3x_2 +5x_3=7,\\x_2 = -1, x_3=2$ No No 0 No
2. $x_2$=0 $x_1,x_3$ $x_1 +3x_3=4,\\2x_1 +5x_3=7,\\x_1=1, x_3=1$ Yes No 4 No
3. $x_3$=0 $x_1,x_2$ $x_1 + 2x_2=4,\\2x_1+3x_2=7,\\x_1=2, x_2 =1$ Yes No 5 Yes
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