document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1044529>Question 1044529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use linear programming to maximize z=10x+15y subject to\r<BR>\n" );
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document.write( "x+4y < 360\r<BR>\n" );
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document.write( "2x+y<300\r<BR>\n" );
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document.write( "x>0, y>0\r<BR>\n" );
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document.write( "all signs are equal to </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #659828 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=659828\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The constraints are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B4y+%3C=+360%2C%0D%0A2x%2By%3C=300+%2C%0D%0Ax%3E=0%2C+y%3E=0%29 ALIGN=MIDDLE  ALT=\"system%28x%2B4y+%3C=+360%2C%0D%0A2x%2By%3C=300+%2C%0D%0Ax%3E=0%2C+y%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . That is the quadrilateral limited by the lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%2Cred%28x=0%29%2Cred%28y=0%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%2Cred%28x=0%29%2Cred%28y=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-50%2C450%2C-100%2C400%2C%0D%0Agrid%281%29%2Cblue%28line%28-100%2C115%2C760%2C-100%29%29%2C%0D%0Agreen%28line%28-50%2C400%2C200%2C-100%29%29%2C%0D%0Alocate%28152%2C25%2CA%29%2Clocate%282%2C325%2CP%29%2C%0D%0Alocate%28-16%2C96%2CC%29%2Clocate%28360%2C30%2CQ%29%2C%0D%0Alocate%28122%2C85%2CB%29%2Clocate%284%2C25%2CO%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Sketching a graph is very helpful.<BR>\n" );
document.write( "It allows you to \"see\" the feasible region you are working with.<BR>\n" );
document.write( "In this case, it is the region below both blue an green lines,<BR>\n" );
document.write( "above the x-axis, and<BR>\n" );
document.write( "to the right of the y-axis.<BR>\n" );
document.write( "The feasible region is obviously the quadrilateral <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OABC ALIGN=MIDDLE  ALT=\"OABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is easy to see that shrewdly selected test point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%281%2C1%29 ALIGN=MIDDLE  ALT=\"I%281%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "within <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OABC ALIGN=MIDDLE  ALT=\"OABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meets the requirement of all the constraints,<BR>\n" );
document.write( "and if you move away from that point crossing over any boundary line,<BR>\n" );
document.write( "you cease to meet all requirements.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x=0%29 ALIGN=MIDDLE  ALT=\"red%28x=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the y-axis.<BR>\n" );
document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y=0%29 ALIGN=MIDDLE  ALT=\"red%28y=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the x-axis.<BR>\n" );
document.write( "Those two lines intersect at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the origin.<BR>\n" );
document.write( " <BR>\n" );
document.write( "To graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%2By=300%29 ALIGN=MIDDLE  ALT=\"green%282x%2By=300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , I connected with a straight line its x- and y-intercept.<BR>\n" );
document.write( "I found   those intercepts by solving<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28green%282x%2By=300%29%2Cred%28y=0%29%29 ALIGN=MIDDLE  ALT=\"system%28green%282x%2By=300%29%2Cred%28y=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x=300%2Cx=0%29 ALIGN=MIDDLE  ALT=\"system%282x=300%2Cx=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=150%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=150%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28150%2C0%29 ALIGN=MIDDLE  ALT=\"A%28150%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28green%282x%2By=300%29%2Cred%28x=0%29%29 ALIGN=MIDDLE  ALT=\"system%28green%282x%2By=300%29%2Cred%28x=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=300%2Cx=0%29 ALIGN=MIDDLE  ALT=\"system%28y=300%2Cx=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%2C300%29 ALIGN=MIDDLE  ALT=\"P%280%2C300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "To graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28x%2B4y=360%29%29 ALIGN=MIDDLE  ALT=\"blue%28x%2B4y=360%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , I did the same.<BR>\n" );
document.write( "I found its x- and y-intercepts by<BR>\n" );
document.write( "solving<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cred%28y=0%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cred%28y=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=360%2Cred%28y=0%29%29 ALIGN=MIDDLE  ALT=\"system%28x=360%2Cred%28y=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=0%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28360%2C0%29 ALIGN=MIDDLE  ALT=\"Q%28360%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cred%28y=0%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cred%28y=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284y=360%2Cred%28x=0%29%29 ALIGN=MIDDLE  ALT=\"system%284y=360%2Cred%28x=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%29%2Cy=90%29 ALIGN=MIDDLE  ALT=\"system%28x=0%29%2Cy=90%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%280%2C90%29 ALIGN=MIDDLE  ALT=\"C%280%2C90%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "I also had to find the point where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28x%2B4y=360%29%29 ALIGN=MIDDLE  ALT=\"blue%28x%2B4y=360%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%2By=300%29 ALIGN=MIDDLE  ALT=\"green%282x%2By=300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  intersect.<BR>\n" );
document.write( "I did it by solving<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28120%2C60%29 ALIGN=MIDDLE  ALT=\"B%28120%2C60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are many ways to salve that. Here is one:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2Cgreen%282x%2By=300%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2Cgreen%282x%2By=300%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2C2%28360-4y%29%2By=300%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2C2%28360-4y%29%2By=300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2C720-8y%2By=300%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2C720-8y%2By=300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2C720-7y=300%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2C720-7y=300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2C720-300=7y%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2C720-300=7y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2C420=7y%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2C420=7y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x=360-4y%29%2Cy=60%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x=360-4y%29%2Cy=60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=360-4%2A60%2Cy=60%29 ALIGN=MIDDLE  ALT=\"system%28x=360-4%2A60%2Cy=60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=360-240%2Cy=60%29 ALIGN=MIDDLE  ALT=\"system%28x=360-240%2Cy=60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=120%2Cy=60%29 ALIGN=MIDDLE  ALT=\"system%28x=120%2Cy=60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Finally, we need to maximize <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=10x%2B15y ALIGN=MIDDLE  ALT=\"z=10x%2B15y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , within that feasible region.<BR>\n" );
document.write( "Because the boundaries and the function are linear,<BR>\n" );
document.write( "the maximum will be<BR>\n" );
document.write( "either at one of the vertices of the feasible region,<BR>\n" );
document.write( "or all along one of the edges of that region.<BR>\n" );
document.write( " <BR>\n" );
document.write( "It could have been obvious to us from the start that the maximum would be at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28120%2C60%29 ALIGN=MIDDLE  ALT=\"B%28120%2C60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see note below),<BR>\n" );
document.write( "but the recipe usually taught to solve this kind of problems involves calculating the value of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28x%2Cy%29=10x%2B15y ALIGN=MIDDLE  ALT=\"z%28x%2Cy%29=10x%2B15y\"   DISABLED_style= \"max-width:90%; height:auto;\" > at every vertex.<BR>\n" );
document.write( "The idea may be the educational value of<BR>\n" );
document.write( "encouraging practice,<BR>\n" );
document.write( "valuing hard work, and<BR>\n" );
document.write( "helping you visualize the situation.<BR>\n" );
document.write( "So, we worked hard to get to this point, and we continue along the expected path.<BR>\n" );
document.write( " <BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%280%2C0%29=10%2A0%2B15%2A0=0 ALIGN=MIDDLE  ALT=\"z%280%2C0%29=10%2A0%2B15%2A0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , obviously no maximum.<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28150%2C0%29 ALIGN=MIDDLE  ALT=\"A%28150%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28150%2C0%29=10%2A150%2B15%2A0=1500 ALIGN=MIDDLE  ALT=\"z%28150%2C0%29=10%2A150%2B15%2A0=1500\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28120%2C60%29 ALIGN=MIDDLE  ALT=\"B%28120%2C60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28120%2C60%29=10%2A120%2B15%2A60=1200%2B900=2100 ALIGN=MIDDLE  ALT=\"z%28120%2C60%29=10%2A120%2B15%2A60=1200%2B900=2100\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%280%2C90%29 ALIGN=MIDDLE  ALT=\"C%280%2C90%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%280%2C90%29=10%2A0%2B15%2A90=1350 ALIGN=MIDDLE  ALT=\"z%280%2C90%29=10%2A0%2B15%2A90=1350\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The maximum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28z=2100%29 ALIGN=MIDDLE  ALT=\"highlight%28z=2100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28B%28120%2C60%29%29 ALIGN=MIDDLE  ALT=\"highlight%28B%28120%2C60%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "Versions of this type of problem are very popular.<BR>\n" );
document.write( "You have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%3E=0%2Cy%3E=0%29 ALIGN=MIDDLE  ALT=\"system%28x%3E=0%2Cy%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , telling you the feasible region is in quadrant I.<BR>\n" );
document.write( "There are two boundary line constraints: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28Lx%2BMy%3C=N%2CPx%2BQy%3C=R%29 ALIGN=MIDDLE  ALT=\"system%28Lx%2BMy%3C=N%2CPx%2BQy%3C=R%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with all positive coefficients,<BR>\n" );
document.write( "telling you that the feasible region is below two lines with positive x- and y-intercepts.<BR>\n" );
document.write( "and the function to maximize, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=Rx%2BSy ALIGN=MIDDLE  ALT=\"z=Rx%2BSy\"   DISABLED_style= \"max-width:90%; height:auto;\" > , has positive coefficients.<BR>\n" );
document.write( "To make you work harder, hey make the maximum be at the intersection of the two lines,<BR>\n" );
document.write( "by making the slope of the lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Z=constant ALIGN=MIDDLE  ALT=\"Z=constant\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-R%2FS ALIGN=MIDDLE  ALT=\"-R%2FS\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "be in between the slopes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-M%2FL ALIGN=MIDDLE  ALT=\"-M%2FL\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-Q%2FP ALIGN=MIDDLE  ALT=\"-Q%2FP\"   DISABLED_style= \"max-width:90%; height:auto;\" > , of the two slanted boundary lines.<BR>\n" );
document.write( "So, in this case, we could immediately see that<BR>\n" );
document.write( "the boundary line slopes are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F1=-2 ALIGN=MIDDLE  ALT=\"-2%2F1=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with the slope of any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=constant ALIGN=MIDDLE  ALT=\"z=constant\"   DISABLED_style= \"max-width:90%; height:auto;\" > line,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2F15=-2%2F3 ALIGN=MIDDLE  ALT=\"-10%2F15=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in between them <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C-2%2F3%3C-1%2F4 ALIGN=MIDDLE  ALT=\"-2%3C-2%2F3%3C-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells us that the maximum is at the intersection of the two slanted boundary lines,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28x%2B4y=360%29%2Cgreen%282x%2By=300%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28120%2C60%29 ALIGN=MIDDLE  ALT=\"B%28120%2C60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the maximum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28120%2C60%29=10%2A120%2B15%2A60=1200%2B900=2100 ALIGN=MIDDLE  ALT=\"z%28120%2C60%29=10%2A120%2B15%2A60=1200%2B900=2100\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );