Final answer:
To maximize P=2x+y under the given inequalities, solve the inequalities to find x=y and then use the condition 2x+y ≤ 9x to determine that x can be as large as 3. Therefore, the maximum value of P is 3, as P=3x when x=y.
Step-by-step explanation:
To find the maximum for P = 2x + y given the inequalities (3x + y) ≤ (7x + 2y) ≤ 9x ≥ 0 and y ≥ 0, we need to solve the system of inequalities step by step.
First, we can simplify the given inequalities:
- 3x + y ≤ 7x + 2y simplifies to 4x ≥ y (after subtracting 3x and y from both sides).
- 7x + 2y ≤ 9x simplifies to 2x ≥ 2y or x ≥ y (after dividing by 2).
- The condition x ≥ 0 and y ≥ 0 are already simplified.
Now, we need to find the maximum value of P = 2x + y given these conditions. Since x ≥ y and 4x ≥ y, the maximum value of 2x + y would be when x is as large as possible but still within the constraints. This occurs when x is at its maximum possible value, which is when x = y. We know x = y must be true for the maximum value of P because if y were less than x, then P would not be at its maximum, since we'd have 4x > y and we could increase y (and thus increase P) without violating any of the constraints.
The maximum value occurs when x = y, meaning we have the system:
- x = y
- 2x + y ≤ 9x
- x ≥ 0
- y ≥ 0
Plugging x = y into 2x + y, we get 3x. To maximize x, given 3x ≤ 9x, we see that x could be as large as 9x/3, which is 3x = 3(1), since x must be ≥ 0. Hence, the maximum value of P is 3.