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February 15th, 2013, 05:57 PM  #1 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Linear programming
Yesterday I studied linear programming. It includes problems of the type: Maximise Z=ax+by subject to the constraints a1x+b1y greater than or equal to A>(1) and a2x+b2y greater than or equal to B.>(2) I studied that the value for x and y was any of the corner points you get when you draw the inequalities (1) and (2). Can anyone give a proof for this? Also someone please teach how to do this ungraphically? 
February 15th, 2013, 11:47 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,933 Thanks: 2207 
What do you mean by "ungraphically"? What you see by using a graph can be expressed algebraically, but the algebra is easier to produce after seeing the graph.

February 16th, 2013, 12:17 AM  #3 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Linear programming
Skip, I meant algebraically by using the word "ungraphically". That was a new word that I discovered. I thought there wouldn't be confusion. But now I understand, sorry.......... I need the solution algebraically, i.e., without drawing the graph "ungraphically"; can you help?..... 
February 16th, 2013, 01:21 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,933 Thanks: 2207 
Would you accept what is obviously just an algebraic version of what is easily seen by graphing? For example, graphical approach: consider just the first quadrant algebraic approach: x > 0 and y > 0 etc. 
February 16th, 2013, 06:01 AM  #5 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Linear programming
When they told me not to graph something, I graphed it anyway. IMHO, you don't want to eliminate pictures as part of the learning process and anyone who asks students to do so isn't doing them any favors. Especially in calculus regarding volumes of revolution, I would tell the students 'draw the picture' and would get annoyed the few times they would respond 'my professor wants me to do it without the graph'. 
February 16th, 2013, 05:08 PM  #6  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Quote:
Whatever you can find graphically, you can find with algebra, right  
February 16th, 2013, 10:19 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,933 Thanks: 2207 
Whatever you can find graphically, you can find with algebra. Whether anyone will believe you devised the algebra without drawing a graph is another matter!

February 17th, 2013, 01:02 AM  #8 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Linear programming
So, you can make the algebra only by looking at the graph? Convince me.

February 17th, 2013, 07:26 PM  #9 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Linear programming
Ok, at least explain algebraically why the solution turns to be one of the corner points you get you when you draw the graph......

February 18th, 2013, 03:31 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,933 Thanks: 2207 
How can an extremity of a finite polygon be anything other than a corner?


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