Solve the following using graphical method :

Minimize :Z=3x+5y

`2x+3x>=12`

`-x+y<=3`

`x<=4,y>=3,x>=0,y>=0`

#### Solution

Min `Z = 3x + 5y`

S.t. `2x + 3y ≥ 12 ` …(i)

`– x + y ≤ 3 ` …(ii)

`x ≤ 4, y ≥ 3, x ≥ 0, y ≥ 0`

Taking eqn (i)

2x + 3y = 12

Putting x = 0, y = 4 Let the point is (0, 4)

Now putting y = 0, x = 6 Let the point is (6, 0)

Now taking eqn (ii)

– x + y = 3

Putting x = 0, y = 3 (0, 3)

Putting y = 0, x = – 3 (– 3, 0)

The graph is as follows

ABCDA be the feasible region bounded by these

lines Now we find the coordinates of A, B, C and D

for A, Solving the eqns.

`2x+3y=12 and -xy=3`

We get

`x=(+3)/5 and y=18/5`

coordinate of `A((+3)/5,18/5)`

Now

`Z=3xx(+3/5)+5xx18/5`

=`(+9)/5+90/5=90/5`

For B, Solving the eqns

`2x+3y=12 and y=3`

We get `x=3/2,y=3`

∴Coordinate of B `(3/2,3)`

Now` Z=3xx3/2+5xx3`

=`9/2+15=39/2 `

For C. Solving the eqn x = 4 and y = 3

∴ Coordinate of C (4, 3)

Now `Z=3xx4+5xx3`

=`12+1=27`

For D, Solving the eqn

– x + y = 3 and x = 4

We get x = 4, y = 7

Now Z = 3 × 4 + 5 × 7

= 12 + 35 = 47

Min Z = `39/2, "for" x=3/2,y=3`