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I have one exercise asking me to show that $x*=(3/2,1)$ is primal feasible for the linear programming problem: $$max 3x_1 +2x_2$$ $$2x_1+x_2 \leq 4$$ $$2x_1+3x_2 \leq 6$$ I can see that it fulfill the constraints. But is there any other method to show so? And also is it optimal solution?

EDIT:

The question now is: Is $x^*$ optimal?

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  • $\begingroup$ Hint: Where there's a primal, there's a dual. $\endgroup$ – Sean Roberson Apr 19 '17 at 20:05
  • $\begingroup$ yeah I know, the dual feasible is y=(5/4,1/4).. $\endgroup$ – MZ97 Apr 19 '17 at 20:24
  • $\begingroup$ Also help :)))) $\endgroup$ – MZ97 Apr 20 '17 at 21:12
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As this is an LP in two variables, you can plot the feasible region and check if this lies in the region. For optimality, we know that optimal solutions to convex LPs lie on the boundary, so if this point lies on one of the two lines (generally, hyperplanes), you can almost be sure you have an optimal solution.

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  • $\begingroup$ Is there any mathmatical way to find it out? I've tried that :) $\endgroup$ – MZ97 Apr 20 '17 at 21:18
  • $\begingroup$ Not really. Really, feasibility just asks if a point satisfies all constraints. Nothing complicated about that. $\endgroup$ – Sean Roberson Apr 20 '17 at 21:30

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