Minimize ` z = 7x + y `, subject to
` 5x + y ge 5, x + y ge 3, x ge 0 , y ge 0 `.

Explanation

Shaded region YBECX is the feasible region whose vertices are B (0,5) , E and C (3, 0) .
Point E is the intersection of lines,
` 5x + y = 5` ….(1)
and `x + y = 3`

Subtracting equation (2) from (1),
`{:(" "5x+y=5),(" "x+y=3),(ul(-" "-" "-)),(" "4x=2):}`
` :." " x = 2/4 = 1/2`
Substituting ` x = 1/2 ` in equation (2),
` 1/2 + y = 3`
` :. " " y = 5/2.`
` :. " " E = (1/2, 5/2)`.
Now, ` Z = 7x + y.`
` Z (B) = 7*0 + 5 = 5`
` Z(E) = 7* 1/2 + 5/2 = 7/2 + 5/2 = 6`
` Z (C) = 7*3+0 = 21`
` :. ` Minimum value of Z is 5 at B (0, 5)