We are to minimise
z = 5x + 7y
subject to the constraints
2x + y ≥ 8
x + 2y ≥ 10
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
2x + y = 8
For x = 0, y = 8
For y = 0, 2 x = 8 or x = 4
∴ line meets OX in A (4, 0) and OY in L (0, 8).

Again we draw the graph of
x + 2y = 10.
For x = 0, 2y = 10 or y = 5
For y = 0, x = 10
∴ line meets OX in B (10, 0) and OY in M (0, 5)
Since feasible region is the region which satisfies all the constraints
∴ shaded region is the feasible region and comer points are B (10, 0), C (2, 4), L (0, 8).
At B (10, 0), z = 5 (10) + 7 (0) = 50 + 0 = 50
At C (2, 4), z = 5 (2) +7 (4) = 10 + 28 = 38
At L (0, 8), z = 5 (0) + 7 (8) = 0 + 56 = 56
∴ 38 is the smallest value of z at (2, 4)
Since feasible region is unbounded
∴ we are to check whether this value is minimum.
For this we draw the graph of
5x + 7y < 38    ...(1)
Since (1) has no common point with feasible region.
∴ minimum value = 38 at (2, 4).