To determine which point satisfies the system of inequalities, we need to substitute each option into both inequalities and check if they hold true. Here are the inequalities:

1. $y < x - 7$
2. $y < \frac{1}{5}x - 2$

Let's examine each answer choice by plugging in the coordinates.

Option Analysis

1. Point $(-10, -2)$:

   • For $y < x - 7$: $-2 < -10 - 7 \Rightarrow -2 < -17$ (True)
   • For $y < \frac{1}{5}x - 2$: $-2 < \frac{1}{5}(-10) - 2 \Rightarrow -2 < -2 - 2 = -4$ (True)

   This point satisfies both inequalities.

2. Point $(-5, 2)$:

   • For $y < x - 7$: $2 < -5 - 7 \Rightarrow 2 < -12$ (False)

   This point does not satisfy the first inequality, so it is not a solution.

3. Point $(10, 7)$:

   • For $y < x - 7$: $7 < 10 - 7 \Rightarrow 7 < 3$ (False)

   This point does not satisfy the first inequality, so it is not a solution.

4. Point $(10, -2)$:

   • For $y < x - 7$: $-2 < 10 - 7 \Rightarrow -2 < 3$ (True)
   • For $y < \frac{1}{5}x - 2$: $-2 < \frac{1}{5}(10) - 2 \Rightarrow -2 < 2 - 2 = 0$ (True)

   This point satisfies both inequalities.

Conclusion

Both (-10, -2) and (10, -2) satisfy the inequalities.

If the question requires only one answer, choose (-10, -2) as the primary correct answer.

Would you like further details or have any questions?

Here are some related questions that could extend your understanding:

1. How do we graph solutions to systems of inequalities?
2. What methods can be used to solve systems of linear inequalities?
3. How does changing the slope in an inequality affect the solution set?
4. What is the impact of shifting the intercept in an inequality on its graph?
5. How would the solution change if one of the inequalities were non-strict (using $\leq$ instead of $<$)?

Tip: When dealing with inequalities, always double-check both the boundary (equality) condition and ensure the inequality direction is correctly interpreted when substituting values.