`4x+3y le 60, y ge 2x, x ge 3, x, y ge 0`

x ge 3, y ge 2

x+y le 6, x+y ge 4

y+8 ge 2x

Show that the minimum of Z occurs at more than two points. Minimise and Maximise Z=5x+10y subject to x+2y le 120, x+y ge 60, x-2y ge 0, x, y ge 0

x+y le 9. y gt x, x ge 0

x-2y le 3, 3x+4y ge 12, x ge 0, y ge 1

In the question of maximum value of z = 800 x + 12000 y subject to constraints 9x +12 y le 180, 3x + 4y le 60 , x + 3y le 30 , x ge 30 , x ge 0, y ge 0 …….. is not a point of feasible region .

The maximum value of Z = x + 4y subject to the constraints 3x+6y le 6, 4x+8y ge 16, x ge 0, y ge 0 is ………

Solve the following linear programming problems graphically : Minimise : Z = 3x + 5y subject to constraints -2x+y le 4, x+y ge 3, x-2y le 2, x, y ge 0 .

3x+4y le 60, x +3y le 30, x ge 0, y ge0