document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=977746>Question 977746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can u please help me solve this ..... Using the following system of inequalities find the maximum value of f(x,y) =3x + 9y          x>0    y>0.      3x + 2y=<18.        6x + 7y <42 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #599318 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=599318\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Can u please help me solve this ..... Using the following system of inequalities find the maximum value of \r<BR>\n" );
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document.write( "f(x,y) =3x + 9y          \r<BR>\n" );
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document.write( "x>0    y>0.      \r<BR>\n" );
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document.write( "3x + 2y=<18.        \r<BR>\n" );
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document.write( "6x + 7y <42<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "We add non-negative slack variables to the left side to \"take \r\n" );
document.write( "up the slack\" between the left side and the right side and make \r\n" );
document.write( "the inequalities into equations:\r\n" );
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document.write( "3x + 2y + s1 = 18.        \r\n" );
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document.write( "6x + 7y + s2 = 42.\r\n" );
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document.write( "We take the equation for what we want to maximize f(x,y) = 3x + 9y\r\n" );
document.write( "and get 0 on the right side:\r\n" );
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document.write( "-3x - 9y + f(x,y) = 0\r\n" );
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document.write( "We set the three equations up like this:\r\n" );
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document.write( "  x    y   s1   s2   f(x,y)    k\r\n" );
document.write( "--------------------------------    \r\n" );
document.write( " 3x + 2y + s1               = 18\r\n" );
document.write( " 6x + 7y      + s2          = 42\r\n" );
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document.write( "-3x - 9y           + f(x,y) =  0\r\n" );
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document.write( "Then we make the tableau matrix by taking off the \r\n" );
document.write( "variables and leave only the coefficients, and putting\r\n" );
document.write( "in partitions like this:\r\n" );
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document.write( "  x    y | s1   s2  |f(x,y)|   k\r\n" );
document.write( "--------------------------------    \r\n" );
document.write( "  3    2 |  1    0  |   0  |  18\r\n" );
document.write( "  6    7 |  0    1  |   0  |  42\r\n" );
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document.write( " -3   -9 |  0    0  |   1  |   0\r\n" );
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document.write( "The pivot column is the second one because -9 is the most negative \r\n" );
document.write( "number on the bottom row.\r\n" );
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document.write( "[That really isn't necessary to take the most negative number on the bottom row\r\n" );
document.write( "to indicate the pivot column, because any negative number on the bottom row will\r\n" );
document.write( "work, but it usually shortens the process to take the lowest negative number]:\r\n" );
document.write( "                                 ___          ___\r\n" );
document.write( "To find the pivot row we divide 2)18 = 9 and 7)42 = 6\r\n" );
document.write( "Since 6 is smaller and it came from row 2, the pivot row is row 2.\r\n" );
document.write( "So 7 is the pivot element (common to both the pivot column and row) \r\n" );
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document.write( "So we pivot on 7.  That is, we make the 7 into a 1 and use it to make\r\n" );
document.write( "to other numbers on the pivot column 0. \r\n" );
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document.write( "First we get a 1 where the 7 is by dividing that pivot row through by 7\r\n" );
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document.write( "  x    y | s1   s2  |f(x,y)|   k\r\n" );
document.write( "--------------------------------    \r\n" );
document.write( "  3    2 |  1    0  |   0  |  18\r\n" );
document.write( "6/7    1 |  0  1/7  |   0  |   6\r\n" );
document.write( "--------------------------------\r\n" );
document.write( " -3   -9 |  0    0  |   1  |   0\r\n" );
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document.write( "We get a 0 where the 2 is by multiplying the pivot row by -2\r\n" );
document.write( "and adding it to the top row:\r\n" );
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document.write( "     We take the pivot row  -> 6/7    1 |  0  1/7  |   0  |   6\r\n" );
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document.write( "     multiply it by -2  -->  -12/7   -2 |  0 -2/7  |   0  | -12\r\n" );
document.write( "     add the top row     -->     3    2 |  1    0  |   0  |  18\r\n" );
document.write( "     ---------------------------------------------------------------\r\n" );
document.write( "     get -->                   9/7    0 |  1 -2/7  |   0  |   6\r\n" );
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document.write( "     Replace the top row by that \r\n" );
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document.write( "  x    y | s1   s2  |f(x,y)|   k\r\n" );
document.write( "--------------------------------    \r\n" );
document.write( "9/7    0 |  1 -2/7  |   0  |   6\r\n" );
document.write( "6/7    1 |  0  1/7  |   0  |   6\r\n" );
document.write( "--------------------------------\r\n" );
document.write( " -3   -9 |  0    0  |   1  |   0\r\n" );
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document.write( "---\r\n" );
document.write( "We get a 0 where the -9 is by multiplying the pivot row by 9\r\n" );
document.write( "and adding it to the bottom row:\r\n" );
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document.write( "     We take that pivot row -> 6/7    1 |  0  1/7  |   0  |   6\r\n" );
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document.write( "     multiply it by  9  -->   54/7    9 |  0  9/7  |   0  |  54\r\n" );
document.write( "     add the bottom row -->     -3   -9 |  0    0  |   1  |   0\r\n" );
document.write( "     ----------------------------------------------------------\r\n" );
document.write( "     get -->                  33/7    0 |  0  9/7  |   1  |  54\r\n" );
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document.write( "     Replace the bottom row by that: \r\n" );
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document.write( "   x    y | s1   s2  |f(x,y)|   k\r\n" );
document.write( " --------------------------------    \r\n" );
document.write( " 9/7    0 |  1 -2/7  |   0  |   6\r\n" );
document.write( " 6/7    1 |  0  1/7  |   0  |   6\r\n" );
document.write( " --------------------------------\r\n" );
document.write( "33/7    0 |  0  9/7  |   1  |  54\r\n" );
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document.write( "There are no negative numbers left on the bottom row, so we write the\r\n" );
document.write( "bottom row back as an equations with the variables:\r\n" );
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document.write( "33/7x + 0y + 0s1 + 9/7s2 + 1f(x,y) = 54\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "33/7x + 9/7s2 + f(x,y) = 54\r\n" );
document.write( "                f(x,y) = 54 - 33/7x - 9/7s2\r\n" );
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document.write( "So f(x,y) will be a maximum when we don't subtract anything from it,\r\n" );
document.write( "and that will only be when x and s2 are both 0, since they can't be \r\n" );
document.write( "negative.\r\n" );
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document.write( "So the maximum possible value for f(x,y) is 54.\r\n" );
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document.write( "Often a problem will ask for what values of x and y gives this maximum\r\n" );
document.write( "value.\r\n" );
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document.write( "If you were asked that you would know that it was when x=0 and s2 = 0.\r\n" );
document.write( "However we want to know what y would have to be.\r\n" );
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document.write( "So we substitute x=0 and s2=0 in\r\n" );
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document.write( "   6x + 7y + s2 = 42\r\n" );
document.write( "6(0) + 7y + (0) = 42\r\n" );
document.write( "             7y = 42\r\n" );
document.write( "              y = 6\r\n" );
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document.write( "So the maximum possible value for f(x,y) is 54 and that's when x=0 and y=6. \r\n" );
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document.write( "Edwin</PRE> <BR>\n" );
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