How do you graph the solutions to the system #y+2x<=0, 6x-5y<=15, y>= -8#?

1 Answer
Cesareo R.
Dec 29, 2016

See below.

Explanation:

Calling #s_1 ge 0, s_2 ge 0, s_3 ge 0# slack variables, we have

#{(y+2x+s_1 = 0),(6x-5y+s_2=0),(y-s_3 = -8):}#

Solving for #x,y,s_1# we obtain

#x=1/6(5(s_3-5)-s_2), y=s_3-8,s_1=1/3(49+s_2-8s_3)#

To have #s_1 ge0# suffices #49+s_2-8s_3 ge 0# or
#s_2 ge 8s_3-49# and this is attained if #8s_3-49 ge 0#
so finally if #s_3 ge 49/8=>s_2 ge 0=>s_1 ge 0# so the system has solutions

Attached a plot showing in light blue the feasible region.

enter image source here

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