Question

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

Answer 1

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.