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The title is self-explanatory . :-)

thanks,

Actually my description is not comprehensive : I want to draw the graphical solution of an operation research problem

maximize $z = 7x_1 + 9x_2$.

  • under $x_1 + x_2 \leq 8$.
  • $x_2\leq 4$.
  • $2x_1 + 3x_2 \leq 19$.
  • $x_1\geq 0$.
  • $x_2\geq 0$.
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    @Arturo : The five conditions have to hold at the same time.2011-02-10

2 Answers 2

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Sivaram has answered your title problem: you simply plot the points $(0,0)$, $(b,-a)$ and $(-b,a)$ or any two of them. Then you draw the straight line through these points, extending it if necessary.

Let's extend it in two ways to your optimisation problem. First, how to draw $ax+by=c$ where $c$ is not $0$. Plot the points $(\frac{c}{a},0)$ and $(0,\frac{c}{b})$; this also works if one of $a$ or $b$ is $0$. Then you draw the straight line through these points, extending it if necessary. That, together with Sivaram's advice, is enough to draw your five constraints. This will give you a feasible region which is a simple convex polygon, in this case with five vertices.

The second extension is to draw a line parallel to $ax+by=0$ which passes through the point $(d,e)$. The answer is to plot the points $(d,e)$, $(d+b,e-a)$ and $(d-b,e+a)$ or any two of them. Then you draw the straight line through these points, extending it if necessary. You can do this with your function to be maximised and each of the five points. One or more of the five resulting lines will be further to the top-right than the others. The point(s) in your feasible region this passes though will give where $z$ is maximised.

If after all this you do not find that the maximum possible value of $z$ is $62$ then either you or I have made a mistake.

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To draw any straight line, all you need is two points. In the case of $ax+by = 0$, the line clearly passes through $(0,0)$. Further, the line passes through couple of other points which are easy to recognize namely $(b,-a)$ and $(-b,a)$. So just mark these points on the sheet and draw it. If $a=0$, then the line is nothing but the $X$ axis while if $b=0$, then the line is nothing but the $Y$ axis.

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    Could you explain what exactly you mean?2011-02-09