Minimize 3x+2y subject to the constraints `x+y ge 5 and x+2 y ge 6, x ge0, y ge 0`

Minimize Z = x + 4y subject to constraints x + 3y ge 3 , 2x + y ge 2, x ge 0 , y ge 0

The linear programming problem minimiseZ=3x+2y subject to the constraints x+y ge 8 3x +5y le 15 x ge 0, y ge 0 has

Minimize z=-3x+4u subject to the constraints x+2y le 8 3x+2y le 12 x ge 0, y ge 0 by graphical method.

Minimize Z = 3x + 2y subject to the constraints : x + y ge 8, 3x + 5y le 15, x ge 0, y ge 0 .

Maximize Z = 6x + 8y subject to the constraints x + y le 6, 3x + y ge 6, x - y ge 0, x, y ge 0 graphically.

Minimize Z= 3x+y, subject to constriants 2x +3y le 6, x +y ge 1, x ge 0 , y ge 0 Then

The point at which the minimum value of z = 8x + 12y subject to the constraints 2x +y ge 8, x + 2y ge 10 x ge 0, y ge 0 is obtained is