Question 163622
First let's graph the feasible region.
{{{x+y<=8}}}
{{{4x-3y<=12}}}
{{{x>=0}}}
{{{y>=0}}}
{{{ graph( 300, 300, -2, 10, -2, 5, 8-x, (4x-12)) }}} 
The feasible region is the triangle bounded by vertices (3,0),(8,0), and (36/7,20/7).
The final point is determined by equating
{{{y=8-x}}}
and 
{{{y=(4x-12)/3}}}
as shown here,
{{{8-x=(4x-12)/3}}}
{{{24-3x=4x-12}}}
{{{-7x=-36}}}
{{{x=36/7}}}
Then
{{{y=8-x}}}
{{{y=8-36/7}}}
{{{y=56/7-36/7}}}
{{{y=20/7}}}
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The max and min of F occur at the vertices of the feasible region.
Check all the points to determine.
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.
(3,0)
{{{F=7y-4x}}} 
{{{F=7(0)-4(3)}}}
{{{F=-12}}}
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(8,0)
{{{F=7y-4x}}} 
{{{F=7(0)-4(8)}}}
{{{F=-32}}}
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(36/7,20/7)
{{{F=7y-4x}}} 
{{{F=7(20/7)-4(36/7)}}}
{{{F=140/7-144/7}}}
{{{F=-4/7}}}
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The min value occurs at (8,0) where F=-32.