Question

Which point would be a solution to the system of linear inequalities shown below?
y, is less than or equal to, minus, one fifth, x, minus, 8, y, is greater than or equal to, 5, x, plus, 5
y≤−
5
1
​
x−8y≥5x+5
Answer
Multiple Choice Answers
left parenthesis, minus, 5, comma, 3, right parenthesis(−5,3)
left parenthesis, minus, 10, comma, 5, right parenthesis(−10,5)
left parenthesis, minus, 10, comma, minus, 6, right parenthesis(−10,−6)
left parenthesis, minus, 5, comma, 2, right parenthesis(−5,2)

Answer 1

To determine which point is a solution to the system of inequalities, we need to evaluate each point against the inequalities given:

1. **Inequality 1:** \( y \leq -\frac{1}{5}x - 8 \)
2. **Inequality 2:** \( y \geq 5x + 5 \)

Let's evaluate each point:

### Point (-5, 3)

1. **Evaluate \( y \leq -\frac{1}{5}x - 8 \):**
\[
3 \leq -\frac{1}{5}(-5) - 8
\]
\[
3 \leq 1 - 8 \implies 3 \leq -7 \quad (\text{False})
\]

2. **Evaluate \( y \geq 5x + 5 \):**
\[
3 \geq 5(-5) + 5
\]
\[
3 \geq -25 + 5 \implies 3 \geq -20 \quad (\text{True})
\]

### Point (-10, 5)

1. **Evaluate \( y \leq -\frac{1}{5}x - 8 \):**
\[
5 \leq -\frac{1}{5}(-10) - 8
\]
\[
5 \leq 2 - 8 \implies 5 \leq -6 \quad (\text{False})
\]

2. **Evaluate \( y \geq 5x + 5 \):**
\[
5 \geq 5(-10) + 5
\]
\[
5 \geq -50 + 5 \implies 5 \geq -45 \quad (\text{True})
\]

### Point (-10, -6)

1. **Evaluate \( y \leq -\frac{1}{5}x - 8 \):**
\[
-6 \leq -\frac{1}{5}(-10) - 8
\]
\[
-6 \leq 2 - 8 \implies -6 \leq -6 \quad (\text{True})
\]

2. **Evaluate \( y \geq 5x + 5 \):**
\[
-6 \geq 5(-10) + 5
\]
\[
-6 \geq -50 + 5 \implies -6 \geq -45 \quad (\text{True})
\]

### Point (-5, 2)

1. **Evaluate \( y \leq -\frac{1}{5}x - 8 \):**
\[
2 \leq -\frac{1}{5}(-5) - 8
\]
\[
2 \leq 1 - 8 \implies 2 \leq -7 \quad (\text{False})
\]

2. **Evaluate \( y \geq 5x + 5 \):**
\[
2 \geq 5(-5) + 5
\]
\[
2 \geq -25 + 5 \implies 2 \geq -20 \quad (\text{True})
\]

### Summary of Results:
- Point (-5, 3): True for the second, false for the first.
- Point (-10, 5): True for the second, false for the first.
- Point (-10, -6): True for both inequalities.
- Point (-5, 2): True for the second, false for the first.

### Conclusion:
The only point that satisfies both inequalities is **(-10, -6)**. Thus, the correct answer is:
**(-10, -6)**.