Question 923166
{{{x+2y<=40}}}
{{{2y<=-x+40}}}
{{{y<=-x/2+20}}}
Graph {{{y=-x/2+20}}}
{{{graph(300,300,-10,40,-10,40,-x/2+20)}}}
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{{{x+y<=30}}}
{{{y<=-x+30}}}
Graph {{{y=-x+30}}} also.
{{{graph(300,300,-10,40,-10,40,-x/2+20,-x+30)}}}
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{{{2x+3y<=66}}}
{{{3y<=-2x+66}}}
{{{y<=-(2/3)x+22}}}
Graph {{{y=-(2/3)x+22}}} also.
{{{graph(300,300,-10,40,-10,40,-x/2+20,-x+30,-(2/3)x+22)}}}
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Find the points of intersection.
{{{-x/2+20=-(2/3)x+22}}}
{{{(-1/2+2/3)x=2}}}
{{{(1/6)x=2}}}
{{{x=12}}}
Then,
{{{y=-12/2+20=-6+20=14}}}
(12,14)
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{{{-x+30=-(2/3)x+22}}}
{{{-(1/3)x=-8}}}
{{{x=24}}}
Then,
{{{y=-24+30=6}}}
(24,6)
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{{{drawing(300,300,-10,40,-10,40,grid(1),circle(0,0,1),circle(0,20,1),circle(30,0,1),circle(24,6,1),circle(12,14,1),graph(300,300,-10,40,-10,40,-x/2+20,-x+30,-(2/3)x+22))}}}
The extrema of Z occur at one of these vertices.
Not sure if you need the maximum or minimum.
(0,0) : {{{4x+y=4(0)+0=0}}}<-- Minimum
(0,20) :{{{4x+y=4(0)+20=20}}}
(12,14) :{{{4x+y=4(12)+14=62}}}
(24,6) :{{{4x+y=4(24)+6=102}}}
(30,0) : {{{4x+y=4(30)+0=120}}}<-- Maximum