document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=877011>Question 877011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the maximum and minimum value of F = 9x + 40y subject to the constraints<BR>\n" );
document.write( "a) y-x&#8805;1<BR>\n" );
document.write( "b) 2&#8804;x&#8804;5<BR>\n" );
document.write( "c) y-x<3\r<BR>\n" );
document.write( "\n" );
document.write( "Can you please help me out ? Thanks so much in advance:) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #529128 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=529128\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE SOLUTION:<BR>\n" );
document.write( "The region limited by the constraints <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y-x%3E=1%29%2Cgreen%282%3C=x%3C=5%29%2Cblue%28y-x%3C=3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y-x%3E=1%29%2Cgreen%282%3C=x%3C=5%29%2Cblue%28y-x%3C=3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (it needs to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x%3C=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x%3C=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to have a solution)<BR>\n" );
document.write( "is the parallelogram below, with vertices at (2,3), (2,5), (5,6), and (5,8).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28450%2C450%2C-4%2C8%2C-1%2C9%2C%0D%0Agrid%281%29%2Cred%28line%28-6%2C-5%2C14%2C15%29%29%2C%0D%0Agreen%28line%282%2C-6%2C2%2C14%29%29%2Cgreen%28line%285%2C-6%2C5%2C14%29%29%2C%0D%0Ablue%28line%28-6%2C-3%2C14%2C17%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The maximum is at (5,8), where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A5%2B40%2A8=45%2B320=highlight%28365%29 ALIGN=MIDDLE  ALT=\"F=9%2A5%2B40%2A8=45%2B320=highlight%28365%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The minimum is at (2,3), where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A2%2B40%2A3=18%2B120=highlight%28138%29 ALIGN=MIDDLE  ALT=\"F=9%2A2%2B40%2A3=18%2B120=highlight%28138%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You would know how much you have to \"show your work,\"<BR>\n" );
document.write( "I will just over-explain how to get to the solution, because I do not know how much explanation you want/need.<BR>\n" );
document.write( " <BR>\n" );
document.write( "HOW TO GET TO THE SOLUTION:<BR>\n" );
document.write( "I like to make a graph/sketch, because it helps me understand the problem, and avoid mistakes.<BR>\n" );
document.write( "Graphing the \"feasible region\" limited by those constraints is nice and probably expected.<BR>\n" );
document.write( "However, this region was so easy to imagine and calculate that graphing was not absolutely needed.<BR>\n" );
document.write( "The constraints set the limits/borders of the feasible region as portions of the lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y-x=1%29%2Cgreen%282=x%29%2Cgreen%28x=5%29%2Cblue%28y-x=3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y-x=1%29%2Cgreen%282=x%29%2Cgreen%28x=5%29%2Cblue%28y-x=3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We should realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282=x%29 ALIGN=MIDDLE  ALT=\"green%282=x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28x=5%29 ALIGN=MIDDLE  ALT=\"green%28x=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are vertical (and parallel) lines,<BR>\n" );
document.write( "and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y-x=1%29 ALIGN=MIDDLE  ALT=\"red%28y-x=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are slanted (and parallel) lines.<BR>\n" );
document.write( "The feasible region has four parallel sides; it is a parallelogram.<BR>\n" );
document.write( "By plotting all four of those lines on the same graph, I find the corners graphically.<BR>\n" );
document.write( "Otherwise, I would have to solve fours systems of two equations,<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y-x=1%29%2Cgreen%282=x%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y-x=1%29%2Cgreen%282=x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "to find the corners where those lines intersect.<BR>\n" );
document.write( "How to graph the vertical lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282=x%29 ALIGN=MIDDLE  ALT=\"green%282=x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28x=5%29 ALIGN=MIDDLE  ALT=\"green%28x=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is obvious.<BR>\n" );
document.write( "The region limited by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282%3C=x%3C=5%29 ALIGN=MIDDLE  ALT=\"green%282%3C=x%3C=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is obviously the space between those vertical lines, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at least <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Graphing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y-x=1%29 ALIGN=MIDDLE  ALT=\"red%28y-x=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a tiny bit more difficult, but still easy.<BR>\n" );
document.write( "To graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y-x=1%29 ALIGN=MIDDLE  ALT=\"red%28y-x=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can transform the equations into the equivalent slope-intercept form,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y-x=1%29 ALIGN=MIDDLE  ALT=\"red%28y-x=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y=x%2B1%29 ALIGN=MIDDLE  ALT=\"red%28y=x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=x%2B3%29 ALIGN=MIDDLE  ALT=\"blue%28y=x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=x%2B3%29 ALIGN=MIDDLE  ALT=\"blue%28y=x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=1%2Ax%2B3%29 ALIGN=MIDDLE  ALT=\"blue%28y=1%2Ax%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > tells us that<BR>\n" );
document.write( "the blue line has a y-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meaning that as x increases by 1, y increases by 1,<BR>\n" );
document.write( "so the blue line crosses the y-axis at (0,3), and from there goes to (1,4), (2,5), (3,8), and so on.<BR>\n" );
document.write( "Otherwise, we could find the x- and y-intercepts by respectively setting<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is true for all points on the x-axis) and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is true for all points on the y-axis).<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%280-x=3%29 ALIGN=MIDDLE  ALT=\"blue%280-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28x=-3%29 ALIGN=MIDDLE  ALT=\"blue%28x=-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we know that the x-intercept for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-3,0),<BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-0=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-0=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=3%29 ALIGN=MIDDLE  ALT=\"blue%28y=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we know that the y-intercept for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (0,3).<BR>\n" );
document.write( "With those intercepts we can connect (-3,0) and (0,3) with a straight line to get the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-x=3%29 ALIGN=MIDDLE  ALT=\"blue%28y-x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The constraints <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y-x%3E=1%29%2Cblue%28y-x%3C=3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y-x%3E=1%29%2Cblue%28y-x%3C=3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y%3E=x%2B1%29%2Cblue%28y%3C=x%2B3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y%3E=x%2B1%29%2Cblue%28y%3C=x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > determine the region between the red and blue lines.<BR>\n" );
document.write( "We know that because<BR>\n" );
document.write( "we notice that (0,0) with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y-x%3E=1%29%2Cblue%28y-x%3C=3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y-x%3E=1%29%2Cblue%28y-x%3C=3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or because we realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%28y%3E=x%2B1%29%2Cblue%28y%3C=x%2B3%29%29 ALIGN=MIDDLE  ALT=\"system%28red%28y%3E=x%2B1%29%2Cblue%28y%3C=x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the space above the red line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y=x%2B1%29 ALIGN=MIDDLE  ALT=\"red%28y=x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but below the blue line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=x%2B3%29 ALIGN=MIDDLE  ALT=\"blue%28y=x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Once we have determined that the feasible region is the parallelogram bordered by those lines,<BR>\n" );
document.write( "we determine the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+9x+%2B+40y ALIGN=MIDDLE  ALT=\"F+=+9x+%2B+40y\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the corners of the feasible region.<BR>\n" );
document.write( "In any problem of this kind, a linear function of x and y, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+9x+%2B+40y ALIGN=MIDDLE  ALT=\"F+=+9x+%2B+40y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "will be maximum at one corner or at two corners and a border line connecting those two corners.<BR>\n" );
document.write( "The same goes for the minimum.<BR>\n" );
document.write( "In this case, it was obvious that<BR>\n" );
document.write( "the maximum values for x and y in the feasible region, at (5,8), would give the maximum value to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+9x+%2B+40y ALIGN=MIDDLE  ALT=\"F+=+9x+%2B+40y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that the minimum values for x and y, at (2,3), would give the minimum value to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+9x+%2B+40y ALIGN=MIDDLE  ALT=\"F+=+9x+%2B+40y\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In general, you would have to show the calculations for all corners:<BR>\n" );
document.write( "at (2,3), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A2%2B40%2A3=18%2B120=138 ALIGN=MIDDLE  ALT=\"F=9%2A2%2B40%2A3=18%2B120=138\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at (2,5), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A2%2B40%2A5=18%2B200=218 ALIGN=MIDDLE  ALT=\"F=9%2A2%2B40%2A5=18%2B200=218\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at (5,6), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A5%2B40%2A6=45%2B240=285 ALIGN=MIDDLE  ALT=\"F=9%2A5%2B40%2A6=45%2B240=285\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at (5,8), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A5%2B40%2A8=45%2B320=365 ALIGN=MIDDLE  ALT=\"F=9%2A5%2B40%2A8=45%2B320=365\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then you would compare the values to find that<BR>\n" );
document.write( "the greatest (largest) value, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A5%2B40%2A8=45%2B320=highlight%28365%29 ALIGN=MIDDLE  ALT=\"F=9%2A5%2B40%2A8=45%2B320=highlight%28365%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is the maximum,<BR>\n" );
document.write( "and the smallest value, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=9%2A2%2B40%2A3=18%2B120=highlight%28138%29 ALIGN=MIDDLE  ALT=\"F=9%2A2%2B40%2A3=18%2B120=highlight%28138%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is the minimum. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );