document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=548268>Question 548268</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Maximization <BR>\n" );
document.write( "Maximize p = 7x - 4y subject to<BR>\n" );
document.write( "y >= 2x - 8<BR>\n" );
document.write( "y <= 10 - x<BR>\n" );
document.write( "y <= (x + 8) / 2<BR>\n" );
document.write( "x >= 1<BR>\n" );
document.write( "y >= 2\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #356861 by </B><A HREF=/tutors/aboutme.mpl?userid=mathie123><B>mathie123(224)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathie123><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=356861\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I will first look at the equations (I rewrote (3) a bit... if you don't see what I did let me know):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+2x+-+8 ALIGN=MIDDLE  ALT=\"y+%3E=+2x+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >          (1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+10+-+x ALIGN=MIDDLE  ALT=\"y+%3C=+10+-+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >          (2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+%281%2F2%29%2Ax+%2B+4 ALIGN=MIDDLE  ALT=\"y+%3C=+%281%2F2%29%2Ax+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (3)\r<BR>\n" );
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document.write( "(2)-(3) leaves:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=6-%283%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"0%3C=6-%283%2F2%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29%2Ax%3C=6 ALIGN=MIDDLE  ALT=\"%283%2F2%29%2Ax%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%3C=12 ALIGN=MIDDLE  ALT=\"3x%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=4 ALIGN=MIDDLE  ALT=\"x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Equation (1) can be rewritten as <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%3C=-2x%2B8 ALIGN=MIDDLE  ALT=\"-y%3C=-2x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >           (4)\r<BR>\n" );
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document.write( "Equation (4) +equation (2) gives<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=18-3x ALIGN=MIDDLE  ALT=\"0%3C=18-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%3C=18 ALIGN=MIDDLE  ALT=\"3x%3C=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=6 ALIGN=MIDDLE  ALT=\"x%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (Note: we already know x<=4... so this is no surprise...)\r<BR>\n" );
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document.write( "Equation (4)+ Equation(3) gives<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=%28-3%2F2%29%2Ax%2B12 ALIGN=MIDDLE  ALT=\"0%3C=%28-3%2F2%29%2Ax%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29%2Ax%3C=12 ALIGN=MIDDLE  ALT=\"%283%2F2%29%2Ax%3C=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%3C=24 ALIGN=MIDDLE  ALT=\"3x%3C=24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=8 ALIGN=MIDDLE  ALT=\"x%3C=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (Again... no surprise... but it's good to check as we could have gotten a different value)\r<BR>\n" );
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document.write( "So we know now that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3C=x%3C=4 ALIGN=MIDDLE  ALT=\"1%3C=x%3C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.(So the maximum value of x is 4)   \r<BR>\n" );
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document.write( "We have to do the same thing for y by eliminating x<BR>\n" );
document.write( "Just a reminder:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+2x+-+8 ALIGN=MIDDLE  ALT=\"y+%3E=+2x+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >          (1)  which is really just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%3C=-2x%2B8 ALIGN=MIDDLE  ALT=\"-y%3C=-2x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >           (4)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+-x%2B10 ALIGN=MIDDLE  ALT=\"y+%3C=+-x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >           (2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+%281%2F2%29%2Ax+%2B+4 ALIGN=MIDDLE  ALT=\"y+%3C=+%281%2F2%29%2Ax+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (3)\r<BR>\n" );
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document.write( "Let's first compare (4) and (2) <BR>\n" );
document.write( "We need to eliminate x so I will multiply equation (2) by 2 to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%3C=-2x%2B20 ALIGN=MIDDLE  ALT=\"2y%3C=-2x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >       (5)<BR>\n" );
document.write( "So adding (4) and (5) we get <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=+28 ALIGN=MIDDLE  ALT=\"y%3C=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's compare (1) and (3) <BR>\n" );
document.write( "We will first multiply equation (3) by -4 to get ((WHEN MULTIPLYING OR DIVIDING BY A NEGATIVE NUMBER, YOU MUST CHANGE INEQUALITY SIGN))<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4y%3E=-2x-16 ALIGN=MIDDLE  ALT=\"-4y%3E=-2x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >        (6) <BR>\n" );
document.write( "Now adding (6) to (1) we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3y%3E=-24 ALIGN=MIDDLE  ALT=\"-3y%3E=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=8 ALIGN=MIDDLE  ALT=\"y%3C=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >   \r<BR>\n" );
document.write( "\n" );
document.write( "One more comparison...  (2) and (3) <BR>\n" );
document.write( "We will multiply (3) by -2 (and switch the sign) to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y%3E=-x-8 ALIGN=MIDDLE  ALT=\"-2y%3E=-x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >        (7)<BR>\n" );
document.write( "Now we want to compare (7) with (2) but first we must rearrange (2) so the inequality signs match <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+-x%2B10 ALIGN=MIDDLE  ALT=\"y+%3C=+-x%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >           (2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%3E=x-10 ALIGN=MIDDLE  ALT=\"-y%3E=x-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >            (8)<BR>\n" );
document.write( "And finally adding (7) and (8) we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3y%3E=-18 ALIGN=MIDDLE  ALT=\"-3y%3E=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=6 ALIGN=MIDDLE  ALT=\"y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >     (This is a surprise!)    <BR>\n" );
document.write( "So if all these must be true we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%3C=y%3C=6 ALIGN=MIDDLE  ALT=\"2%3C=y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the maximum bound on y is 6.  \r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "To maximize p we must sub in both our maximum x and maximum y into the formula p=7x-4y... I will let you do this part:) \r<BR>\n" );
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document.write( "Phew!    Hopefully this helps! \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );