Converting the inequations into equations, we obtain the following equations:
x + y = 4, 3x + 8y = 24, 10x + 7y = 35, x = 0 and y = 0.
These equations represent straight lines in XOY-plane.
The lines x + y = 4 meets the coordinate axes at A1 (4, 0) and B1 (0, 4). Join these points to obtain the line x + y = 4.
The line 3x + 8y = 24 meets the coordinates axes at A2 (8, 0) and B2 (0, 3). Join these points to obtain the line 3x + 8y = 24
The line 10x + 7y = 35 cuts the coordinates axes at A3 (3.5, 0) and B3 (0, 5). These points are joined to obtain the line
10x + 7y = 35.
Also, x = 0 is the y-axis and y = 0 is the x-axis.
The feasible region of the LPP is shaded in fig. The coordinates of the corner points of the feasible region OA3PQB2 are O (0, 0), A3 (3.5, 0), and B2 (0, 3).
Now, we take a constant value, say 10 for Z. Putting Z = 10 in Z = 5x + 7y, we obtain the line 5x + 7y = 10. Thus line meets the coordinates axes atP1 (2, 0) and Q1 . Join these points by a dotted line. Now move this line parallel to itself in the increasing direction away from the origin. P2Q2and P3Q3 are such lines. Out of these lines locate a line farthest from the origin and has at least one common point to the feasible region OA3PQB2. Clearly, P3Q3 is such line and it passes through the vertex Q (8/5, 12/5) of the feasible region. Hence x = 8/5 and y = 12/5 gives the maximum value of Z. The maximum value of Z is given by