document.write( "Question 823732: Maximize p = 5x + 3y subject to
\n" ); document.write( "2x + 3y <= 12
\n" ); document.write( "3x + y >= 16
\n" ); document.write( "x + y >= 3
\n" ); document.write( "2x + y >= 6 \n" ); document.write( "

Algebra.Com's Answer #495932 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2x+%2B+3y+=+12\"$ represents a line on the x-y plane that divides the at plane into two halves.
\n" ); document.write( " $\"2x+%2B+3y+%3C=+12\"$ represents the half of the x-y plane that contains (0,0), the origin,
\n" ); document.write( "because with $\"x=y=0\"$ ,
\n" ); document.write( " $\"2x+%2B+3y=2%2A0%2B3%2A0=0%2B0=0+%3C=+12\"$ .
\n" ); document.write( "Graphing the line for $\"2x+%2B+3y+=+12\"$ is easy.
\n" ); document.write( "We just need 2 points, and we can find the x- and y-intercepts very easily:
\n" ); document.write( " $\"x=0\"$ --> $\"2%2A0%2B3y=12\"$ --> $\"2%2A0%2B3y=12\"$ --> $\"y=12%2F3\"$ --> $\"y=4\"$
\n" ); document.write( "gives us the y-intercept at (0,4).
\n" ); document.write( " $\"y=0\"$ --> $\"2x%2B3%2A0=12\"$ --> $\"2x=12\"$ --> $\"x=12%2F2\"$ --> $\"x=6\"$
\n" ); document.write( "gives us the y-intercept at (0,4).
\n" ); document.write( "We can graph that $\"2x+%2B+3y+=+12\"$ line, which is part of the solution to $\"2x+%2B+3y+%3C=+12\"$ ,
\n" ); document.write( "and we can indicate with a little arrow which half of the plane is the whole solution:
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\n" ); document.write( "We can do the same for the inequalities
\n" ); document.write( " $\"blue%283x+%2B+y+%3E=+16%29\"$ ,
\n" ); document.write( " $\"red%28x+%2B+y+%3E=+3%29\"$ , and
\n" ); document.write( " $\"green%282x+%2B+y+%3E=+6%29\"$
\n" ); document.write( "We can easily see that (0,0) the origin is not part of the solution to any of those 3 inequalities,
\n" ); document.write( "We can easily find the intercepts and graph the boundary lines for
\n" ); document.write( " $\"red%28x+%2B+y+=+3%29\"$ , which has intercepts at (0,3) and (3,0) , and
\n" ); document.write( " $\"green%282x+%2B+y+=+6%29\"$ , which has intercepts at (0,6) and (3,0) .
\n" ); document.write( "The line $\"blue%283x+%2B+y+=+16%29\"$ with intercepts at (0,16) and (16/3,0) looks a little more cumbersome for graphing, but transforming the equation,
\n" ); document.write( " $\"blue%283x+%2B+y+=+16%29\"$ <--> $\"blue%28+y+=-3x+%2B+16%29\"$ ,
\n" ); document.write( "makes it easier to find points more amenable to graphing, such as
\n" ); document.write( "(6,-2) from $\"x=6\"$ --> $\"y=-3%2A6%2B16=-18%2B16=-2\"$ and
\n" ); document.write( "(3,7) from $\"x=3\"$ --> $\"y=-3%2A3%2B16=-9%2B16=7\"$ .
\n" ); document.write( "For each inequality, the solution is the boundary line plus side of the boundary line that does not contain the origin.
\n" ); document.write( "We can add to the graph the 3 lines above, with little arrows showing which half of the plane is the solution to the inequality
\n" ); document.write( "
\n" ); document.write( "The graph of the 4 inequalities looks like this:
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, or better yet

\n" ); document.write( "
\n" ); document.write( "The points that satisfy all 3 inequalities form that quadrilateral in the middle of the last graph.
\n" ); document.write( "It would be good to know the coordinates of its vertices.
\n" ); document.write( "We know that (3,0) is one of them because it was the x-intercept for the red and green lines.
\n" ); document.write( "Point (1.5,3) seems to be the intersection of the black and green lines.
\n" ); document.write( "Substituting the coordinates into $\"2x+%2B+3y+=+12\"$ and $\"green%282x+%2B+y+=+6%29\"$ , we find that they satisfy both equations, so (1.5,3) is the intersection of the black and green lines. (That was solving ta system of linear equations by graphing.
\n" ); document.write( "The other two vertices are not so easy.
\n" ); document.write( " $\"system%282x%2B3y=12%2Cblue%283x%2By=16%29%29\"$ --> $\"system%282x%2B3%28-3x%2B16%29=12%2Cblue%28y=-3x%2B16%29%29\"$ --> $\"system%282x-9x%2B48=12%2Cblue%28y=-3x%2B16%29%29\"$ --> $\"system%28-7x=12-48%2Cblue%28y=-3x%2B16%29%29\"$ --> $\"system%28x=36%2F7%2Cy=-3%2836%2F7%29%2B16%29\"$ --> $\"system%28x=136%2F7%2Cy=%28-108%2B112%29%2F7%29\"$ --> $\"system%28x=32%2F7%2Cy=4%2F7%29\"$
\n" ); document.write( "gives us the intersection of the black and blue lines.
\n" ); document.write( " $\"system%28red%28x%2By=3%29%2Cblue%283x%2By=16%29%29\"$ --> $\"system%28red%28y=3-x%29%2C3x%2B3-x=16%29\"$ --> $\"system%28red%28y=3-x%29%2C2x%2B3=16%29%29\"$ --> $\"system%28red%28y=3-x%29%2C2x=16-3%29%29\"$ --> $\"system%28red%28y=3-x%29%2C2x=13%29%29\"$ --> $\"system%28y=3-13%2F2%2Cx=13%2F2%29%29\"$ --> $\"system%28y=-7%2F2%2Cx=13%2F2%29%29\"$ --> $\"system%28y=-3.5%2Cx=6.5%29%29\"$ gives us the intersection of the black and blue lines.
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\n" ); document.write( "Now we have to find at which vertex or vertices the function $\"p%28x%2Cy%29=5x%2B3y\"$ has the greatest value.
\n" ); document.write( " $\"p%283%2C0%29=5%2A3%2B3%2A0=15\"$
\n" ); document.write( " $\"p%281.5%2C3%29=5%2A1.5%2B3%2A3=7.5%2B9=16.5\"$
\n" ); document.write( " $\"p%286.5%2C-3.5%29=5%2A6.5%2B3%2A%28-3.5%29=32.5-10.5=22\"$
\n" ); document.write( "

\n" ); document.write( "The maximum happens at $\"%22%28%22\"$ $\"36%2F7\"$ $\"%22%2C%22\"$ $\"4%2F7\"$ $\"%22%29%22\"$ or $\"%22%28%22\"$ $\"5%261%2F7\"$ $\"%22%2C%22\"$ $\"4%2F7\"$ $\"%22%29%22\"$ and that maximum is $\"highlight%2827%263%2F7%29\"$ which is approximately $\"highlight%2827.43%29\"$ . \n" ); document.write( "

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