Question:Find a point in every region defined by a system of linear equations

Question:Find a point in every region defined by a system of linear equations

Maple
How do I find a point in every region defined by a system of linear equations? I have a system of eight linear equations of the form Ai.x+Bi.y=0 (i=1..8) Ai and Bi are numerical values. If I plot all eight equations on one graph then I get numerous bounded regions defined by three or more of the equations and numerous unbounded regions defined by two or more of the equations. (It is possible – though unlikely - for two of the solutions to be parallel or collinear. ) My aim is to find one point – any point does - within every bounded region and one in every unbounded region. It is easy to do this by inspection. However, I need to do this automatically. The system is dynamic so the Ai and Bi values change and I do not know in advance how many regions there are. I can see that it is straightforward to find all – (8-1)! I think - solutions of pairs of equations (intersecting points) and I can see that it will be possible to write an algorithm to find the regions but before I do two questions: 1) Is there a particular branch of linear programming which deals with this sort of problem? 2) Or even better are there commands in maple to directly deal with this? Any suggestions most welcome. Brian
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