Solve the following LPP by using graphical method.
Maximize : `Z=6x+4y,`
Subject to `x le 2, x + y le 2, -2 x +y le 1, x ge 0, y ge 0.` Also find maximum value of Z.

Explanation

Let E be the point of intersection of lines x=2 and `x+y=3`. Solving theses equations we get `E-=(2,1)`
Let F be the point of intersetion of lines `x+y=3 and -2x+y=1`
Solving theses equations.
`({:(x+y=3),(-2x+y=1),(+ --):})/(3x=2)`
`:. x=(2)/(3)`
`y=1+2x=1+2xx(2)/(3)=1+(4)/(3)=(7)/(3):. F-=((2)/(3),(7)/(3))`
`:.` Fesiable region is shaded bounded region OGEFCO.
The co-ordinates of the corner points are `0(0,0), G (2,0),,E(2,1)F((2)/(3),(7)/(3)),C(0,1)`
Now, `Z=6x+4y`
`:. Z(0)=0+0=0`
`Z(G)=6xx2+0=12`
`Z(E)=6xx2+4xx1=12+4=16`
`Z(F)=6xx(2)/(3)+4xx(7)/(3)=4+(28)/(3)=(40)/(3)`