Question 926184
First, map out the feasible region.
{{{x+y<=10}}}
{{{y<=-x+10}}}
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{{{graph(300,300,-3,12,-3,12,-x+10)}}}
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{{{graph(300,300,-3,12,-3,12,y<=-x+10)}}}
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{{{-x + y <=5}}}
{{{y<=x+5}}}
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{{{graph(300,300,-3,12,-3,12,-x+10,x+5)}}}
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{{{2x+4y<=32}}}
{{{x+2y<=16}}}
{{{2y<=-x+16}}}
{{{y<=-x/2+8}}}
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{{{graph(300,300,-3,12,-3,12,-x+10,x+5,-x/2+8)}}}
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Find the vertices of the intersection points.
{{{-x/2+8=x+5}}}
{{{-x+16=2x+10}}}
{{{3x=6}}}
{{{x=2}}}
Then,
{{{y=2+5}}}
{{{y=7}}}
(2,7)
and
{{{-x/2+8=-x+10}}}
{{{-x+16=-2x+20}}}
{{{x=4}}}
Then,
{{{y=-4+10}}}
{{{y=6}}}
(4,6)
Also include,
(0,5) and (10,0).
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{{{drawing(300,300,-3,12,-3,12,grid(1),circle(2,7,0.3),circle(4,6,0.3),circle(0,5,0.3),circle(10,0,0.3),graph(300,300,-3,12,-3,12,-x+10,x+5,-x/2+8))}}}
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Now check the function using these points:
(2,7)
{{{C = 3x + 4y=3(2)+4(7)=6+28=34}}}
(4,6)
{{{C = 3x + 4y=3(4)+4(6)=12+24=36}}}
(0,5)
{{{C = 3x + 4y=3(0)+4(5)=0+20=20}}}
(10,0)
{{{C = 3x + 4y=3(10)+4(0)=30+0=30}}}
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The maximum occurs at (2,7)