Here, the given function,
P = 5x + 8y,
Subject of constraints,
2x + 3y β₯ 15
, 3x + 2y β₯ 15, x β₯ 0, y β₯ 0,
Graphing 2x + 3y β₯ 15:
The related equation of 2x + 3y β₯ 15 Β is 2x + 3y = 15,
For x = 0,
2(0) + 3y = 15 β 3y = 15 β y = 5,
For y = 0,
2x + 3(0) = 15 β 2x = 15 β x = 7.5,
Join the points (0, 5) and (7.5, 0),
'β₯' represents the solid line,
For (0, 0), 2(0) + 3(0) Β β₯ 15( False ),
i.e. shaded region will not contain the origin.
Graphing 3x + 2y β₯ 15:
The related equation of 3x + 2y β₯ 15 is 3x + 2y = 15,
For x = 0,
3(0) + 2y = 15 β 2y = 15 β y = 7.5,
For y = 0,
3x + 2(0) = 15 β 3x = 15 β x = 5,
Join the points (0, 7.5) and (5, 0),
'β₯' represents the solid line,
For (0, 0), 3(0) + 2(0) Β β₯ 15( False ),
i.e. shaded region will not contain the origin.
By graphing,
We found the feasible region,
Having boundary points (3, 3), (0, 7.5) and (7.5,0),
For (3,3), P = 5(3) + 8(3) = 15 + 24 = 39,
For (0, 7.5), P = 5(0) + 8(7.5) = 60,
For (7.5, 0), P = 5(7.5) + 8(0) = 37.5
Hence, min (P) = 37.5