document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178973>Question 1178973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the linear programming problem.\r<BR>\n" );
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document.write( "Maximize P = 15x + 65y<BR>\n" );
document.write( "Subject to x + 2y ≤ 16<BR>\n" );
document.write( "            x ≥ 0<BR>\n" );
document.write( "            y ≥ 0\r<BR>\n" );
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document.write( "What is the maximum value of P?\r<BR>\n" );
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document.write( "Please explain this homework question to me step-by-step  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #808450 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=808450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i solved graphically and i also solved using a simplex method tool.\r<BR>\n" );
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document.write( "the simplex method tool can be found at <a  rel=nofollow HREF= \"https://www.zweigmedia.com/RealWorld/simplex.html\" target = \"_blank\">https://www.zweigmedia.com/RealWorld/simplex.html</a>\r<BR>\n" );
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document.write( "the graphing tool can be found at <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a>\r<BR>\n" );
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document.write( "the results from the simplex method tool are shown below.\r<BR>\n" );
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document.write( "the results from the graphing tool are shown below.\r<BR>\n" );
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document.write( "the simplex method tool is fairly straight forward and it give you the optimal solution only.\r<BR>\n" );
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document.write( "that optimal solution is at the (0,8) where the objective function of 15x + 65y is equal to 15 * 0 + 65 * 8 = 520.\r<BR>\n" );
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document.write( "the graph tool requires a little more work.\r<BR>\n" );
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document.write( "using the desmos.com calculator, you graph the opposite of the constraint inequalities.<BR>\n" );
document.write( "the area of the graph that is not shaded is the region of feasibility.<BR>\n" );
document.write( "the maximum / minimum value of the solution is at the corner points of the graph.\r<BR>\n" );
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document.write( "those corner points are (0,8), (16,0) and (0,0).<BR>\n" );
document.write( "the objective function is evaluated at those corner points.<BR>\n" );
document.write( "the objective function is p = 15x + 65y.<BR>\n" );
document.write( "at (0,8), 15x + 65y = 15 * 0 + 65 * 8 = 520<BR>\n" );
document.write( "at (16,0), 15x + 65y = 15 * 16 + 65 * 0 = 240<BR>\n" );
document.write( "at (0,0), 15x + 65y = 15 * 0 + 65 * 0 = 0\r<BR>\n" );
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document.write( "both methods give you a maximum value of the objective function at (x,y) = 0,8).\r<BR>\n" );
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document.write( "the objective function is p = 5x + 65y\r<BR>\n" );
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document.write( "the contraint functions are:\r<BR>\n" );
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document.write( "x >= 0<BR>\n" );
document.write( "y >= 0<BR>\n" );
document.write( "x + 2y <= 16\r<BR>\n" );
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document.write( "you will be graphing the opposite of the constraint functions.<BR>\n" );
document.write( "specifically, you would graph.\r<BR>\n" );
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document.write( "x <= 0<BR>\n" );
document.write( "y <= 0<BR>\n" );
document.write( "x + 2y >= 16\r<BR>\n" );
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document.write( "the reason for graphing the opposite of the inequalities is because, with the desmos.com software, the region of feasibility shows up clearly.<BR>\n" );
document.write( "if you tried to graph the inequalities themselves, the feasible region would be much harder to spot.\r<BR>\n" );
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