document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1098453>Question 1098453</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the graphic method to solve this problem\r<BR>\n" );
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document.write( "Minimize the objective function:\r<BR>\n" );
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document.write( "3x+2y\r<BR>\n" );
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document.write( "Subject to the constraints:\r<BR>\n" );
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document.write( "2x+y>=30<BR>\n" );
document.write( "2x+5y>=50<BR>\n" );
document.write( "x+y=>20<BR>\n" );
document.write( "x>=0<BR>\n" );
document.write( "y>=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #712842 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=712842\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>The solution you got from tutor ikleyn was very well done and very thorough.<br><BR>\n" );
document.write( "However, there is a shortcut for getting to the final answer, without having to evaluate the objective function at all the vertices of the feasibility region.  In fact, with this shortcut it is not even necessary to find all the vertices of the feasibility region.<br><BR>\n" );
document.write( "I will refer to her response in describing the shortcut.<br><BR>\n" );
document.write( "The slopes of the three constraint lines are -2 (red), -1 (blue), and -2/5 (green).<br><BR>\n" );
document.write( "The objective function, evaluated at some particular point, is 3x+2y equal to some number.  All lines with equations of that form have slopes of -3/2.<br><BR>\n" );
document.write( "We need to find where a line with slope -3/2 touches a single vertex of the feasibility region, and does not pass through any portion of the feasibility region.<br><BR>\n" );
document.write( "If you think about that, it is going to occur at the vertex where the slope of the objective function is between the slopes of the two intersecting constraint functions.<br><BR>\n" );
document.write( "Since the slope of the objective function, -3/2, is between -2 and -1 (the intersection of the red and blue lines), that vertex is where the minimum value of the objective function is going to occur.<br><BR>\n" );
document.write( "So the vertex where the minimum value of the objective function occurs is the intersection of the red and blue lines.<br><BR>\n" );
document.write( "The only vertex of the feasibility region for which you need to determine the coordinates is that one; and then the minimum value of the objective function is the objective function evaluated at those coordinates.<br><br><BR>\n" );
document.write( "So the streamlined process for solving this kind of problem is<BR>\n" );
document.write( "(1) Find the slopes of the constraint lines and of the objective function;<BR>\n" );
document.write( "(2) Find the two constraint lines for which the slope of the objective function is between the slopes of the two constraint lines;<BR>\n" );
document.write( "(3) Find the coordinates of the intersection of those two constraint lines; and<BR>\n" );
document.write( "(4) Evaluate the objective function at that point.<BR>\n" );
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