First we draw the lines AB and CD whose equations are 5x + y = 5 and x + y = 3 respectively.
The feasible region is XCPBY which is shaded in the graph.
The vertices of the feasible region are C (3, 0), P and B (0, 5).
P is the point of the intersection of the lines
5x + y = 5
and x + y = 3
On subtracting, we get
4x = 2 ∴ x = \(\frac{1}{2}\)
Substituting x = \(\frac{1}{2}\) in x + y = 3, we get
\(\frac{1}{2}\) + y = 3
∴ y = \(\frac{5}{2}\) ∴ P = \(\left(\frac{1}{2},\frac{5}{2}\right)\)
The values of the objective function z = 7x + y at these vertices are
z(C) = 7(3) + 0 = 21
z(B) = 7(0) + 5 = 5
∴ z has minimum value 5, when x = 0 and y = 5.
