To determine which combination of acres for peas ([tex]\(x\)[/tex]) and carrots ([tex]\(y\)[/tex]) is valid under the given constraints, we need to test the provided pairs [tex]\((x, y)\)[/tex] against the following conditions:
1. Both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] must be non-negative: [tex]\(0 \leq x \leq 300\)[/tex] and [tex]\(0 \leq y \leq 300\)[/tex].
2. The total number of acres should not exceed 300: [tex]\(x + y \leq 300\)[/tex].
Let's evaluate each given point:
1. Point [tex]\((-25, 100)\)[/tex]
- This point has [tex]\(x = -25\)[/tex] and [tex]\(y = 100\)[/tex].
- First, we notice that [tex]\(x = -25\)[/tex] is negative, which violates the condition that [tex]\(x\)[/tex] must be non-negative.
- Hence, [tex]\((-25, 100)\)[/tex] is not a valid point.
2. Point [tex]\((380, -160)\)[/tex]
- This point has [tex]\(x = 380\)[/tex] and [tex]\(y = -160\)[/tex].
- Here, [tex]\(x = 380\)[/tex] exceeds the maximum limit of 300, and [tex]\(y = -160\)[/tex] is negative.
- Therefore, [tex]\((380, -160)\)[/tex] does not meet the constraints.
3. Point [tex]\((150, 145)\)[/tex]
- This point has [tex]\(x = 150\)[/tex] and [tex]\(y = 145\)[/tex].
- Check if both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are within the range [0, 300]:
- [tex]\(0 \leq 150 \leq 300\)[/tex]
- [tex]\(0 \leq 145 \leq 300\)[/tex]
- Now, check the sum [tex]\(x + y\)[/tex]:
- [tex]\(150 + 145 = 295 \leq 300\)[/tex]
- Thus, [tex]\((150, 145)\)[/tex] satisfies all the constraints and is a valid point.
4. Point [tex]\((100, 250)\)[/tex]
- This point has [tex]\(x = 100\)[/tex] and [tex]\(y = 250\)[/tex].
- Check if both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are within the range [0, 300]:
- [tex]\(0 \leq 100 \leq 300\)[/tex]
- [tex]\(0 \leq 250 \leq 300\)[/tex]
- Now, check the sum [tex]\(x + y\)[/tex]:
- [tex]\(100 + 250 = 350\)[/tex]
- The sum exceeds 300, hence [tex]\((100, 250)\)[/tex] does not meet the constraints.
Therefore, the only valid point among the given options is [tex]\((150, 145)\)[/tex].
