How do you graph the solutions to the system y+2x<=0, 6x-5y<=15, y>= -8y+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 0s1≥0,s2≥0,s3≥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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