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Question
Find the maximum and minimum value of 2x + y subject to the constraints:
x + 3y ≥ 6, x − 3y ≤ 3, 3x + 4y ≤ 24, − 3x + 2y ≤ 6, 5x + y ≥ 5, x, y ≥ 0.
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Solution
First, we will convert the given inequations into equations, we obtain the following equations:
x + y = 4, x + y = 3, x − 2y = 2, x = 0 and y = 0.
The line x + 3y = 6 meets the coordinate axis at \[A\left( 6, 0 \right)\] and B(0, 2). Join these points to obtain the line x + 3y = 6.
Clearly, (0, 0) does not satisfies the inequation x + 3y ≥ 6. So, the region in xy-plane that does not contains the origin represents the solution set of the given equation.
The line x − 3y = 3 meets the coordinate axis at C(3, 0) and D(0, −1). Join these points to obtain the line x − 3y = 3.
Clearly, (0, 0) satisfies the inequation x − 3y ≤ 3. So, the region in xy-plane that contains the origin represents the solution set of the given equation.
The line 3x + 4y = 24 meets the coordinate axis at E(8, 0) and F(0, 6). Join these points to obtain the line 3x + 4y = 24.
Clearly, (0, 0) satisfies the inequation 3x + 4y ≤ 24. So, the region in xy-plane that contains the origin represents the solution set of the given equation.
The line −3x + 2y = 6 meets the coordinate axis at G(−2, 0) and H(0, 3). Join these points to obtain the line −3x + 2y = 6.
Clearly, (0, 0) satisfies the inequation −3x + 2y ≤ 6. So, the region in xy-plane that contains the origin represents the solution set of the given equation.
The line 5x + y = 5 meets the coordinate axis at \[I\left( 1, 0 \right)\] and J(0, 5). Join these points to obtain the line 5x + y = 5.
Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations.
These lines are drawn using a suitable scale.
The corner points of the feasible region are P \[\left( \frac{4}{13}, \frac{45}{13} \right)\] k \[\left( \frac{4}{3}, 5 \right)\] ,L \[\left( \frac{84}{13}, \frac{15}{13} \right)\], M \[\left( \frac{9}{2}, \frac{1}{2} \right)\] , N \[\left( \frac{9}{14}, \frac{25}{14} \right)\] The values of Z at these corner points are as follows.
| Corner point | Z = 2x + y |
| P \[\left( \frac{4}{13}, \frac{45}{13} \right)\] |
2 x \[\frac{4}{13}\] + \[\frac{45}{13}\] = \[\frac{53}{13}\] |
| K \[\left( \frac{4}{3}, 5 \right)\] | 2 x \[\frac{4}{3}\] + 5 = \[\frac{23}{3}\] |
| L \[\left( \frac{84}{13}, \frac{15}{13} \right)\] | 2 × \[\frac{84}{13}\] + \[\frac{15}{13}\] = \[\frac{183}{13}\] |
| M \[\left( \frac{9}{2}, \frac{1}{2} \right)\] | 2 × \[\frac{9}{2}\] + \[\frac{1}{2}\] = \[\frac{19}{2}\] |
| N \[\left( \frac{9}{14}, \frac{25}{14} \right)\] | 2 × \[\frac{9}{14}\] + \[\frac{25}{14}\] = \[\frac{43}{14}\] |
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