document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206461>Question 1206461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Minimize Z=2X+Y<BR>\n" );
document.write( "3Y+4X≥24<BR>\n" );
document.write( "3Y+2X≥16<BR>\n" );
document.write( "Subject to2Y+2X≥14<BR>\n" );
document.write( "X≥0<BR>\n" );
document.write( "Y≥0	\r<BR>\n" );
document.write( "\n" );
document.write( "X=<BR>\n" );
document.write( "Y=	 <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843982 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=843982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The response from the other tutor shows a solution using the standard method shown in most references.<br><BR>\n" );
document.write( "There is a refinement of that standard method that reduces the amount of effort needed to solve the problem.<br><BR>\n" );
document.write( "Here is a graph of the constraint boundary lines:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C10%2C-2%2C10%2C%28-4%2F3%29x%2B8%2C%28-2%2F3%29x%2B16%2F3%2C-x%2B7%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "You can determine where the objective function is minimized by comparing the slope of the objective function to the slopes of the constraint boundary lines.<br><BR>\n" );
document.write( "The slopes of the constraint boundary lines are -4/3, -2/3, and -1; the slope of the objective function is -2.<br><BR>\n" );
document.write( "The objective function will be minimized where a line with slope -2 just touches the feasibility region.<br><BR>\n" );
document.write( "Imagine lines with slopes of -2 added to the graph above of the constraint boundary lines.  Since the slope of the objective function is more negative than the slopes of all of the constraint boundary lines, the function will be minimized at the corner of the feasibility region where the constraint boundary line has a slope closest to -2.  That slope is -4/3, which is the red line in the graph.<br><BR>\n" );
document.write( "Here is a graph showing the constraint boundary lines along with a line with slope -2 that just touches the feasibility region:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C10%2C-2%2C10%2C%28-4%2F3%29x%2B8%2C%28-2%2F3%29x%2B16%2F3%2C-x%2B7%2C-2x%2B8%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "All other lines with slope -2 would either not pass through any part of the feasibility region, or they would pass through the middle of the feasibility region, where the objective function would not be minimized.<br><BR>\n" );
document.write( "So the objective function is minimized at (0,8).  At that point the value of the objective function is 2x+y = 2(0)+8 = 8.<br><BR>\n" );
document.write( "ANSWER: The objective function has a minimum value of 8 at (0,8).<br>\r<BR>\n" );
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