Question 176608
1){{{0 <=  x <= 3}}}
{{{0 <= y <= 7}}}
{{{drawing( 300, 300, -1, 4, -1, 8,grid( 1 ),
circle( 0, 0, .1 ),
circle( 3, 0, .1 ),
circle( 0, 7, .1 ),
circle( 3, 7, .1 ),
green(line(0,0,3,0)),
green(line(3,0,3,7)),
green(line(3,7,0,7)),
green(line(0,7,0,0))
)}}}
{{{P(x,y)=2x+3y}}} 
{{{P(0,0)=2(0)+3(0)=0}}}
{{{P(3,0)=2(3)+3(0)=6}}}
{{{P(3,7)=2(3)+3(7)=27}}}
{{{P(0,7)=2(0)+3(7)=21}}}
Max. value=27 occuring at (3,7), min. value =0 occuring at (0,0).
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2)First find the feasible region bounded by the lines,
{{{2x+y<=30}}}
{{{y<=30-2x}}}
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{{{x+y<=20}}}
{{{y<=20-x}}}
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{{{x>=0}}}
{{{y>=0}}}
Let's look at the graph of the two lines. 
The vertices are then the x-intercept and y-intercept of the graphs. 
 {{{ graph( 300, 300, -2, 18, -2, 38, 20-x, 30-2x) }}}
Y-intercept: (0,20)
X-intercept: (15,0)
The final point is their intersection point.
{{{30-2x=20-x}}}
{{{x=10}}}
{{{y=10}}}
{{{drawing( 300, 300, -2, 18, -2, 38,grid( 1 ),
circle( 10, 10, .4 ),
circle( 0,20, .4 ),
circle( 15, 0, .4 ),
circle( 0, 0, .4 ),
green(line(0,0,15,0)),
green(line(15,0,10,10)),
green(line(10,10,0,20)),
green(line(0,20,0,0))) }}}
{{{C(x,y)=x+4y}}}
{{{C(0,0)=0+4(0)=0}}}
{{{C(15,0)=15+4(0)=15}}}
{{{C(10,10)=10+4(10)=50}}}
{{{C(0,20)=0+4(20)=80}}}
The minimum for C=x+4y is 0 and occurs at (0,0).