7. Solve the following Linear Programming Problems graphically:
Maximise    Z = 3x + 2y
subject to the constraints: x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0, 

We are to minimise
Z = 3x + 2y
subject to the constraints
x + 2y ≤ 10
3x + y ≤ 15
x, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Let us draw the graph of x + 2y = 10.
For x = 0, 2y = 10 or y = 5
For y = 0, x = 10
∴  line meets OX in A(10, 0) and OY in L(0, 5).
Again we draw the graph of 3x + y = 15.
For x = 0, y = 15
For y = 0, 3 x = 15 or x = 5
∴  line meets OX in B(5, 0) and OY in M(0, 15).
Since feasible region is the region which satisfies all the constraints.
∴  OBCL is the feasible region.
The comer points O(0, 0), B(5, 0), C(4, 3), L(0, 5).
At O(0, 0), Z = 0 + 0 = 0
At B(5, 0), Z = 15 + 0 = 15
At C(4, 3), Z = 12 + 6 = 18
At L(0, 5), Z = 0 + 10 = 10
∴  maximum value = 18 at (4, 3).