document.write( "Question 1198985: ---------------
\n" ); document.write( "Maximize f=7x1+5x2 subject to\r
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Algebra.Com's Answer #832694 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"f=7x%5B1%5D%2B5x%5B2%5D\"$ subject to $\"8x%5B1%5D%2B6x%5B2%5D=65\"$ and $\"7x%5B1%5D%2B9x%5B2%5D=70\"$
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document.write( "In these linear programming problems we assume that no variables are ever\r\n" );
document.write( "negative, but they can be 0.  So all variables here are assumed to be\r\n" );
document.write( "non-negative.  , .\r\n" );
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document.write( "Introduce non-negative slack variables s₁ and s₂ to turn the\r\n" );
document.write( "inequalities into equations:\r\n" );
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document.write( "Rewrite  as \r\n" );
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document.write( "Rewrite the system in matrix form using the five variables:\r\n" );
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document.write( "Next, we will perform the pivot operation. That is a method to make\r\n" );
document.write( "one element in a column (the pivot column) 1, then use the 1 to\r\n" );
document.write( "make all the other elements in the pivot column 0.\r\n" );
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document.write( "We will find the pivot indicator, the pivot column, the pivot row,\r\n" );
document.write( "and the pivot element.\r\n" );
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document.write( "The pivot indicator is the most negative number on the bottom row.\r\n" );
document.write( "In this case -7 is the most negative number on the bottom row, so\r\n" );
document.write( "the pivot indicator is -7.\r\n" );
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document.write( "The pivot column is the column which the pivot indicator is in, so\r\n" );
document.write( "the pivot column is the 1st column.\r\n" );
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document.write( "Next we determine the pivot element. To do that, we divide each positive\r\n" );
document.write( "number in the pivot column above the pivot element into the number at the\r\n" );
document.write( "far right, in the column indicated by an \"=\".\r\n" );
document.write( "   8.125   10\r\n" );
document.write( "8)65     7)70\r\n" );
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document.write( "8.125 < 10, so 8.125 is the smaller of them, and since it was the 8 that we used\r\n" );
document.write( "to get the smaller quotient, that means 8 is pivot element.\r\n" );
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document.write( "The pivot row is the row that the pivot element is on, the 1st row.\r\n" );
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document.write( "Now we make the pivot element 1 by dividing the first row through by 8.\r\n" );
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document.write( "Now we make the 7 in the 1st column into a 0 by multiplying the 1st row by -7\r\n" );
document.write( "and adding it to row 2:\r\n" );
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document.write( "Now we make the -7 in the 1st column into a 0 by multiplying the 1st row by 7\r\n" );
document.write( "and adding it to row 3:\r\n" );
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document.write( "Now there are no negative numbers on the bottom row, so this is the final\r\n" );
document.write( "matrix.  Now we convert the matrix back into a system of equations:\r\n" );
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document.write( "The bottom equation contains f that we wish to maximize:\r\n" );
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document.write( "Solve it for f\r\n" );
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document.write( "We wish to maximize f.  It will be the largest it can possibly be when\r\n" );
document.write( "the two terms subtracted from 455/8 are as small as they can be.  They cannot\r\n" );
document.write( "be negative, so the smallest they can be is 0.  So if we make both x₂ and s₁\r\n" );
document.write( "equal to 0, we can keep the whole 455/8 for f and not subtract anything from\r\n" );
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document.write( "So we choose x₂ = 0 and s₁ = 0. Substituting:\r\n" );
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document.write( "So the answer is that f has a maximum value of 455/8, when x₁=65/8 and x₂=0.\r\n" );
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document.write( "Edwin

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