Find the maximum and minimum values of z=2x+3y subject to the following constraints
maximum value = when x= and y=
minimum value = when x= and y=
First, sketch the region corresponding to the system of constraints. x≥0 is the area to the right of x=0 (the y-axis). y≤0 is the area under y=0 (the x-axis). x-y≤2 is the area above the line y=x-2. x+y≤1 is the area below the line y=1-x. The constraints from the region highlighted in blue stripes.
Next, find the vertices of the region
At (0,0): z=2(0)+3(0)=0
At (1, 0): z=2(1)+3(0)=2
At (3/2, -1/2): z=2(3/2)+3(-1/2)=3/2
At (0, -2): z=2(0)+3(-2)=-6
The maximum value of z is 2, and this occurs when x=1 and y=0.
The minimum value of z is -6, and this occurs when x=0 and y=-2.