Converting the given in equations into equations
x + 2y = 8
3x + 2y = 12
Region represented by x + 2y ≤ 8 :
The line x + 2y = 8 meets the coordinate axis at
A(8, 0) and B(0, 4).
x + 2y = 8
A(8, 0),B(0,10)
Join the points A and B to obtain the line.
Clearly (0, 0) satisfies the in equation x + 2y = 8.
So the region containing the origin represents the solution set of the in equation.
Region represented by 3x + 2y ≤ 12 :
The line 3x + 2y = 12 meets the coordinate axis at C(4, 0) and D(0, 6). 3x + 2y = 12
C(4, 0); D(0, 6)
Join the points C and D to obtain the line.
Clearly (0, 0) satisfies the in equation 3x + 2y = 12.
So the region containing.
The origin represents the solution set of the in equations.
Region represented by x ≥ 0, y ≥ 0 :
Since every point in the first quadrant satisfies these in equations.
So the first quadrant is the region represented by the in equations x ≥ 0, y ≥ 0.
The shaded region OCEB represents the common region of the above in equations.
This region is the feasible region of the given linear programming problem.
The coordinates of the comer points of the feasible region are O(0, 0), C(4, 0), E(2, 3) and 5(0, 4).
The point E(2, 3) has been obtained by solving the equations of the corresponding intersecting lines simultaneously.
The values of the objective function on these points are given in the following table :
| Point |
x-coordinate |
y-coordinate |
Objective function Z = 3x + 4y |
| O |
0 |
0 |
Z0 = -3(0) + 4(0) = 0 |
| C |
4 |
0 |
Zc = -3(4) + 4(0) = -12 |
| E |
2 |
3 |
ZE = -3(2) + 4(3) = 6 |
| B |
0 |
4 |
ZB = -3(0) + 4(4) = 16 |
It is clear from the above table that the objective function has minimum value at comer point C(4, 0).
So the minimum value of given linear programming problem at x = 4 and y = 0 is – 12.
