document.write( "Question 341432: Maximize 5x + 3y\r
\n" ); document.write( "\n" ); document.write( "Subject to the constraints
\n" ); document.write( "5x + 2y <= 40
\n" ); document.write( "3x + 6y <= 48
\n" ); document.write( "x <= 7
\n" ); document.write( "2x - y >= 3
\n" ); document.write( "x,y >= 0\r
\n" ); document.write( "\n" ); document.write( "Graph the following equation
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Algebra.Com's Answer #244486 by Fombitz(32388) $\"\"$ $\"About$

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Determine the feasible region by plotting the constraint equations.
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\n" ); document.write( "Find the intersection points.
\n" ); document.write( "Between the x-axis and $\"y=2x-3\"$
\n" ); document.write( " $\"2x-3=0\"$
\n" ); document.write( " $\"2x=3\"$
\n" ); document.write( " $\"x=3%2F2\"$
\n" ); document.write( "( $\"3%2F2\"$ , $\"0\"$ )
\n" ); document.write( "The other intersection point is ( $\"7\"$ , $\"0\"$ ).
\n" ); document.write( "Find the intersection points of $\"5x%2B2y=40\"$ and $\"x=7\"$
\n" ); document.write( " $\"35%2B2y=40\"$
\n" ); document.write( " $\"2y=5\"$
\n" ); document.write( " $\"y=5%2F2\"$
\n" ); document.write( "( $\"7\"$ , $\"5%2F2\"$ )\r
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\r
\n" ); document.write( "\n" ); document.write( "Find the intersection point between
\n" ); document.write( "1. $\"5x%2B2y=40\"$
\n" ); document.write( "2. $\"3x%2B6y=48\"$
\n" ); document.write( "From eq. 2,
\n" ); document.write( " $\"x%2B2y=16\"$
\n" ); document.write( " $\"2y=16-x\"$
\n" ); document.write( "Substitute into eq. 1,
\n" ); document.write( " $\"5x%2B16-x=40\"$
\n" ); document.write( " $\"4x=24\"$
\n" ); document.write( " $\"x=6\"$
\n" ); document.write( "Then from eq. 2,
\n" ); document.write( " $\"2y=16-6\"$
\n" ); document.write( " $\"y=5\"$
\n" ); document.write( "( $\"6\"$ , $\"5\"$ )
\n" ); document.write( "Finally find the intersection point between $\"y=2x-3\"$ and $\"3x%2B6y=48\"$
\n" ); document.write( " $\"2y=4x-6\"$
\n" ); document.write( "Substitute,
\n" ); document.write( " $\"x%2B2y=16\"$
\n" ); document.write( " $\"x%2B4x-6=16\"$
\n" ); document.write( " $\"5x=22\"$
\n" ); document.write( " $\"x=22%2F5\"$
\n" ); document.write( "Then
\n" ); document.write( " $\"y=44%2F5-15%2F5\"$
\n" ); document.write( " $\"y=29%2F5\"$ \r
\n" ); document.write( "\n" ); document.write( "( $\"22%2F5\"$ , $\"29%2F5\"$ )
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\n" ); document.write( ".
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\n" ); document.write( "The maximum and minimum of the function occurs at one of these vertices.
\n" ); document.write( "( $\"3%2F2\"$ , $\"0\"$ ) : $\"F=5x%2B3y=5%283%2F2%29=15%2F2\"$
\n" ); document.write( "( $\"7\"$ , $\"0\"$ ): $\"F=5x%2B3y=5%287%29=15%2F2\"$ \r
\n" ); document.write( "\n" ); document.write( "( $\"7\"$ , $\"5%2F2\"$ ): $\"F=5x%2B3y=5%287%29%2B3%285%2F2%29=70%2F2%2B15%2F2=85%2F2\"$ \r
\n" ); document.write( "\n" ); document.write( "( $\"6\"$ , $\"5\"$ ): $\"F=5x%2B3y=5%286%29%2B3%285%29=45\"$ \r
\n" ); document.write( "\n" ); document.write( "( $\"22%2F5\"$ , $\"29%2F5\"$ ): $\"F=5x%2B3y=5%2822%2F5%29%2B3%2829%2F5%29=110%2F5%2B87%2F5=197%2F5\"$
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\n" ); document.write( "The maximum value is $\"45\"$ and occurs at ( $\"6\"$ , $\"5\"$ ).\r
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