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प्रश्न
Maximize Z = 5x + 3y
Subject to
\[3x + 5y \leq 15\]
\[5x + 2y \leq 10\]
\[ x, y \geq 0\]
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उत्तर
Region represented by 3x + 5y ≤ 15 :
The line 3x + 5y = 15 meets the coordinate axes at A(5,0) and B(0,3) respectively. By joining these points we obtain the line 3x + 5y = 15.
Clearly (0,0) satisfies the inequation 3x + 5y ≤ 15. So,the region containing the origin represents the solution set of the inequation 3x + 5y ≤ 15.
Region represented by 5x + 2y ≤ 10 :
The line 5x + 2y = 10 meets the coordinate axes at C(2,0) and D(0,5) respectively. By joining these points we obtain the line 5x + 2y = 10.
Clearly (0,0) satisfies the inequation 5x + 2y ≤ 10. So,the region containing the origin represents the solution set of the inequation 5x + 2y ≤ 10.
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 x ≥ 0, and y ≥ 0.The feasible region determined by the system of constraints, 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, and y ≥ 0, are as follows.
The values of Z at these corner points are as follows.
| Corner point | Z = 5x + 3y |
| O(0, 0) | 5 × 0 + 3 × 0 = 0 |
| C(2, 0) |
5 × 2 + 3 × 0 = 10
|
| \[E\left( \frac{20}{19}, \frac{45}{19} \right)\] |
5x \[\frac{20}{19}\] +3x \[\frac{45}{19}\] = \[\frac{235}{19}\] |
| B(0, 3) | 5 0 + 3 × 3 = 9 |
Therefore, the maximum value of Z is
Thus, the optimal value of Z is \[\frac{235}{19}\] .
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