document.write( "Question 810841: My daughter needs serious help with Linear Programming using Graphing Calculator.  She has a problem she has been trying to solve for over a month, I have tried to help her and searched endlessly for help.  Hopefully, the link below works. Everything comes up fine except for the very last part. We keep getting either an Error message or 137 instead of 78, as the problem shows.  Thanks in advance for any help you can provide. One person said the 78 may not belong there at all and that the problem should end after the 136. Any thoughts?\r
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Algebra.Com's Answer #488499 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You don't need a calculator except to check and do\r\n" );
document.write( "basic calculations.  This is a graphing problem to\r\n" );
document.write( "do by hand on graph paper.  \r\n" );
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document.write( "Maximize P = 13x + 2y  \r\n" );
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document.write( "subject to these constraints:\r\n" );
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document.write( "-3x + 2y ≦ 8\r\n" );
document.write( "-8x +  y ≧ -48\r\n" );
document.write( "x ≧ 0, y ≧ 0\r\n" );
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document.write( "[Usually P stands for profit and x and y are variables\r\n" );
document.write( "for how many items to make or buy wholesale to sell\r\n" );
document.write( "retail.  In practical applications there are many\r\n" );
document.write( "variables and many inequalities.]\r\n" );
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document.write( "Those last two inequalities tell us that we don't\r\n" );
document.write( "need any negative numbers for our graph.  So we just\r\n" );
document.write( "need the upper right side of the graph, the first \r\n" );
document.write( "quadrant.\r\n" );
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document.write( "We plot the boundary lines on graph paper:\r\n" );
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document.write( "The boundary lines are the inequalities \r\n" );
document.write( "with = signs instead of inequality symbols.\r\n" );
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document.write( "First we draw the boundary line of the first inequality,\r\n" );
document.write( "which is: \r\n" );
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document.write( "-3x + 2y = 8 \r\n" );
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document.write( "We get two points, say the y-intercept (0,4) and (4,10) \r\n" );
document.write( "and draw a red line through them, like this:\r\n" );
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document.write( "Next we check to see whether the solutions are \r\n" );
document.write( "above that line or below it.\r\n" );
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document.write( "We substitute the origin (0,0) in the original inequality\r\n" );
document.write( "to see if the origin is a solution to the inequality.  If\r\n" );
document.write( "it is then ALL the poins on the same side of the boundary\r\n" );
document.write( "line that the origin is on.  If it is not a solution, then\r\n" );
document.write( "the solutions are on the side of the boundary line that the \r\n" );
document.write( "origin is NOT on.\r\n" );
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document.write( "So we test (0,0) by substituting x=0 and y=0 in the first\r\n" );
document.write( "inequality to see if it is a solution.  \r\n" );
document.write( "           -3x + 2y ≦ 8\r\n" );
document.write( "       -3(0) + 2(0) ≦ 8\r\n" );
document.write( "                  0 ≦ 8  That's true so the solutions are\r\n" );
document.write( "below the red boundary line. \r\n" );
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document.write( "Next we draw the boundary line of the second inequality,\r\n" );
document.write( "which is: \r\n" );
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document.write( "-8x + y = -48\r\n" );
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document.write( "by getting two points, the x-intercept (6,0) and (7,8).\r\n" );
document.write( "We draw a green line through them, like this:\r\n" );
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document.write( "Next we check to see whether the solutions are \r\n" );
document.write( "left of right of it.\r\n" );
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document.write( "As with the first inequality, we test (0,0) by substituting\r\n" );
document.write( "x=0 and y=0 in  the original inequality to see if it is a\r\n" );
document.write( "solution.  -8x + y ≧ -48\r\n" );
document.write( "       -8(0) + (0) ≧ -48\r\n" );
document.write( "                 0 ≧ -48  That's true so all the solutions \r\n" );
document.write( "are on the side of the green line that the origin is on,\r\n" );
document.write( "that is, left of the green line:\r\n" );
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document.write( "So now we know that the set of feasible solutions are the\r\n" );
document.write( "points inside the 4 sided figure, below the red line and\r\n" );
document.write( "left of the green line.   Next we find all the  corner \r\n" );
document.write( "points of that 4-sided figure.  We plotted two of them \r\n" );
document.write( "already, the y-intercept of the first line (0,4) and the \r\n" );
document.write( "x-intercept of the second line (6,0).\r\n" );
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document.write( "And we can see that (0,0) is a corner point.\r\n" );
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document.write( "We have one other corner point to find, the point where the \r\n" );
document.write( "red line crosses the green line, so we solve the system of\r\n" );
document.write( "equations:\r\n" );
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document.write( "-3x + 2y =   8\r\n" );
document.write( "-8x +  y = -48\r\n" );
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document.write( "by either the substitution or the elimination (addition) mthod,\r\n" );
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document.write( "Solve the second for y,  y=-48+8x and substitute in\r\n" );
document.write( "       -3x + 2y = 8\r\n" );
document.write( "-3x + 2(-48+8x) = 8\r\n" );
document.write( " -3x - 96 + 16x = 8\r\n" );
document.write( "       13x - 96 = 8\r\n" );
document.write( "            13x = 104\r\n" );
document.write( "              x = 8\r\n" );
document.write( "Substitute in y = -48 + 8x\r\n" );
document.write( "              y = -48 + 8(8)\r\n" );
document.write( "              y = -48 + 64\r\n" );
document.write( "              y = 16\r\n" );
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document.write( "So the corner point where the red and green lines cross\r\n" );
document.write( "is (8,16)\r\n" );
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document.write( "Now we label all four corner points:\r\n" );
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document.write( "Even though all the points in and on the feasible region are\r\n" );
document.write( "possible solutions, the maximum and minimum points will always\r\n" );
document.write( "be one of the corner points of the feasible region.  So we make\r\n" );
document.write( "a table of corner points to find the value of the objective\r\n" );
document.write( "function, the equation for P, that we are to maximize:\r\n" );
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document.write( "Corner point  Objective function       Value of P\r\n" );
document.write( "                P = 13x + 2y\r\n" );
document.write( "  (0,0)         P = 13(0)+2(0) = 0          0   <-- minimum\r\n" );
document.write( "  (0,4)         P = 13(0)+2(4) = 8          8\r\n" );
document.write( " (8,16)         P = 13(8)+2(16) = 136     136   <-- maximum\r\n" );
document.write( "  (6,0)         P = 13(6)+2(0) = 78        78        \r\n" );
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document.write( "Answer:\r\n" );
document.write( "The best strategy is to choose x=8 and y=16 for a maximum\r\n" );
document.write( "profit of $136.    \r\n" );
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document.write( "Edwin

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