6. Solve the following Linear Programming Problems graphically:
Minimise    Z = 5x + 3y
subject to the constraints: 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

We have to maximise
Z = 5x + 3 y
subject to the constraints
3x + 5 y ≤ 15
5x + 2 y ≤ 10
x ≥ 0, 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 3x + 5 y= 15
For x = 0, 5 y = 15 or y = 3
For y = 0, 3x = 15 or x = 5
∴ line meets OX in A(5, 0) and OY in L(0, 3).
Again we draw the graph of 5x + 2 y = 10
For x = 0, 2 y = 10 or y = 5
For y = 0, 5x = 10 or x = 2
∴ line meets OX in B(2, 0) and OY in M(0, 5).
Since feasible region is the region which satisfies all the constraints.

∴ OBCL is the feasible region and corner points are O(0, 0), B(2, 0),
    
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