Minimize ` z = 6 x + 2y`, subject to ` 5 x + 9y le 90, x + y ge 4, y le 8, x ge 0, y ge 0 `.

Explanation

First we draw the lines AB, CD and EF whose equations are ` 5x + 9y = 90, x + y = 4 and y = 8 ` respectively.

The feasible region in APFDCA which is shaded in the figure.
The vertices of the feasible region are ` A ( 18,0), P, F(0, 8), D ( 0, 4) and C(4, 0 ) `.
P is the point of intersection of the lines ` 5 x + 9 y = 90 and y = 8`.
Substituting ` y = 8 ` in ` 5x + 9y = 90`, we get,
` 5 x + 9 ( 8) = 90 `
` therefore 5 x = 18 " " therefore x = 3.6 " " therefore P ( 3.6, 8) `
The values of the objective function ` z = 6x + 2y ` at these vertices are
` z (A) = 6 ( 18) + 2 ( 0) = 108 `
` z (P) = 6 ( 3.6 ) + 2 ( 8) + 21. 6 + 16 = 37.6`
` z ( F) = 6 (0) + 2 ( 8) = 16`
` z (D) = 6 ( 0 ) + 2 ( 4) = 8 `
` z (C) = 6 ( 4) + 2 ( 0 ) = 24 `
` therefore ` z has minimum value 8, when x = 0 and y = 4.