Question

Which of the following points represents a solution to the system graphed here?

y>−54x+125
𝑦
>
−
5
4
𝑥
+
125


y≤40
𝑦
≤
40


y≥−x+120
𝑦
≥
−
𝑥
+
120


(1 point)
Responses

(100,0)
(
100
,
0
)
left parenthesis 100 comma 0 right parenthesis

(110,−5)
(
110
,
−
5
)
left parenthesis 110 comma negative 5 right parenthesis

(150,20)
(
150
,
20
)
left parenthesis 150 comma 20 right parenthesis

(120,50)

Answer 1

To determine which point represents a solution to the system of inequalities, we must analyze each inequality and check if the given points satisfy all the conditions:

1. **For \( y > -\frac{5}{4}x + 125 \)**:
- This means we are looking for points that lie above the line defined by this inequality.

2. **For \( y \leq 40 \)**:
- This means we are looking for points that are on or below the horizontal line where \( y = 40 \).

3. **For \( y \geq -x + 120 \)**:
- This means we are looking for points that lie above the line defined by \( y = -x + 120 \).

Now, let's check the candidate points:

1. **Point (100, 0)**:
- \( y > -\frac{5}{4}(100) + 125 \) → \( 0 > -125 + 125 \) → \( 0 > 0 \) (not satisfied)
- \( y \leq 40 \) → \( 0 \leq 40 \) (satisfied)
- \( y \geq -100 + 120 \) → \( 0 \geq 20 \) (not satisfied)

This point does not satisfy all inequalities.

2. **Point (110, -5)**:
- \( y > -\frac{5}{4}(110) + 125 \) → \( -5 > -137.5 + 125 \) → \( -5 > -12.5 \) (satisfied)
- \( y \leq 40 \) → \( -5 \leq 40 \) (satisfied)
- \( y \geq -110 + 120 \) → \( -5 \geq 10 \) (not satisfied)

This point does not satisfy all inequalities.

3. **Point (150, 20)**:
- \( y > -\frac{5}{4}(150) + 125 \) → \( 20 > -187.5 + 125 \) → \( 20 > -62.5 \) (satisfied)
- \( y \leq 40 \) → \( 20 \leq 40 \) (satisfied)
- \( y \geq -150 + 120 \) → \( 20 \geq -30 \) (satisfied)

This point satisfies all inequalities.

4. **Point (120, 50)**:
- \( y > -\frac{5}{4}(120) + 125 \) → \( 50 > -150 + 125 \) → \( 50 > -25 \) (satisfied)
- \( y \leq 40 \) → \( 50 \leq 40 \) (not satisfied)
- \( y \geq -120 + 120 \) → \( 50 \geq 0 \) (satisfied)

This point does not satisfy all inequalities.

Based on this analysis, the point \( (150, 20) \) is a solution to the system of inequalities.