minimize ` z=20x_1+20x_2` , subject to `x_1>=0,x_2>=0,x_1+2x_2>=8,3x_1+2x_2>=15,5x_1+2x_2>=20.`

Maximize Z= 6x_1+10x_2 subject to x_1+2x_2gt10,2x_1+2x_2gt12,3x_1+x_2gt8,x_1,x_2gt0

Solve the following LPP graphically: Maximize: Z=4x_1+3x_2 subject to x_1+x_2le50 , x_1+2x_2le80 , 2x_1+x_2ge20 , x_1,x_2ge0

Maximize : Z=20x_1+40x_2 "Subject to" :x_1+x_2 le 1 x_1,x_2 ge 0 .

Minimize: Z=6x_1+7x_2 "Subject to":x_1+2x_2 ge 4 x_1,x_2 ge 0

Solve the following LPP graphically. Minimize z=30x_1+45x_2 subject to 2x_1+6x_2ge4 , 5x_1+2x_2ge5 and x_1,x_2ge0

Minimize: Z=20x_1+10x_2 "Subject to": x_1+2x_2 le 40 3x_1 -x_2 ge 30 4x_1+3x_2 ge= 60 x_1,x_2 ge 0

Solve the following LPP graphically Optimize Z=5x_1+25x_2 subject to -0.5x_1+x_2 le 2, x_1+x_2 ge 2, -x_1+5x_2 ge 5, x_1, x_2 ge 0

Minimize Z= 5x_1+3x_2 ''Subject to ":x_1+x_2 le 2 2x_1+x_2 ge 4 x_1,x_2 ge 0

Maximize Z =4x_1+3x_2 "Subject to" : x_1 +x_2 le 50 x_1+2x_2 le 80 2x_1+x_2 ge 20,x_1,x_2 ge 0