Question 1196602
given:

{{{z=4x+5y}}}.........1) objective function

Subject to

{{{2x+y >= 10}}}........c1
{{{x+2y >= 8}}}..........c2

where c1 and c2 are {{{constraints}}}
  

{{{x}}},{{{y}}}->decision variables
{{{x}}},{{{y}}}>= {{{0}}} ->{{{non-negative}}} constraints  


first graph c1 and c2 (just positive parts in first quadrant )

{{{2x+y >= 10}}}........c1

{{{x}}}|{{{y}}}
{{{0}}}|{{{10}}}
{{{5}}}|{{{0}}}


{{{x+2y >= 8}}}........c2
{{{x}}}|{{{y}}}
{{{0}}}|{{{4}}}
{{{8}}}|{{{0}}}

graph them


{{{ drawing( 600, 600, -15, 15, -15, 15,
circle(0,10,.2),locate(0.3,10,p(0,10)),
circle(5,0,.2),locate(5,0.3,p(5,0)),
green(line(0,10,5,0)),

circle(0,4,.2),locate(0.3,4,p(0,4)),
circle(8,0,.2),locate(8,0.3,p(8,0)),
blue(line(0,4,8,0)),
graph( 600, 600, -15, 15, -15, 15, 0)) }}}


now find the coordinates of the c1 and c2

{{{2x+y = 10}}}
{{{x+2y = 8}}}-> solve for {{{x}}} and you have {{{x=8-2y}}}

substitute in {{{2x+y = 10}}}

{{{2(8-2y)+y = 10}}}

{{{16-4y+y = 10}}}

{{{16-3y = 10}}}

{{{16-10 = 3y}}}

{{{6 = 3y}}}

{{{y=2}}}


then

{{{x=8-2*2}}}

{{{x=4}}}

the feasible region is above the lines around intersection point which is ({{{4}}},{{{2}}})

{{{ drawing( 600, 600, -15, 15, -15, 15,
circle(0,10,.2),locate(0.3,10,p(0,10)),
circle(5,0,.2),locate(5,0.3,p(5,0)),
green(line(0,10,5,0)),

circle(0,4,.2),locate(0.3,4,p(0,4)),
circle(8,0,.2),locate(8,0.3,p(8,0)),
blue(line(0,4,8,0)),

circle(4,2,.2),locate(4,2,p(4,2)), locate(3.5,6,feas-region),

graph( 600, 600, -15, 15, -15, 15, 0)) }}}


corner points of the feas-region are ({{{0}}},{{{10}}}),({{{4}}},{{{2}}}), and ({{{8}}},{{{0}}})


now check at which point objective function have greatest value


{{{z=4x+5y}}}.........1) check ({{{0}}},{{{10}}})


{{{z=4*0+5*10}}}

{{{z=50}}}


{{{z=4x+5y}}}.........1) check ({{{4}}},{{{2}}})


{{{z=4*4+5*2}}}

{{{z=26}}}


{{{z=4x+5y}}}.........1) check ({{{8}}},{{{0}}})


{{{z=4*8+5*0}}}

{{{z=32}}}


as you can see, the maximum is at point  ({{{0}}},{{{10}}})