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

1. First inequality: $y > -4x + 6$
2. Second inequality: $y > \frac{1}{3}x - 7$

Let's test each of the given points:

1. Point $(9, -7)$:

• First inequality: $-7 > -4(9) + 6$ simplifies to $-7 > -36 + 6$ or $-7 > -30$ (True)
• Second inequality: $-7 > \frac{1}{3}(9) - 7$ simplifies to $-7 > 3 - 7$ or $-7 > -4$ (False)

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

2. Point $(-12, -2)$:

• First inequality: $-2 > -4(-12) + 6$ simplifies to $-2 > 48 + 6$ or $-2 > 54$ (False)
• Second inequality: $-2 > \frac{1}{3}(-12) - 7$ simplifies to $-2 > -4 - 7$ or $-2 > -11$ (True)

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

3. Point $(12, 1)$:

• First inequality: $1 > -4(12) + 6$ simplifies to $1 > -48 + 6$ or $1 > -42$ (True)
• Second inequality: $1 > \frac{1}{3}(12) - 7$ simplifies to $1 > 4 - 7$ or $1 > -3$ (True)

This point satisfies both inequalities.

4. Point $(-12, -7)$:

• First inequality: $-7 > -4(-12) + 6$ simplifies to $-7 > 48 + 6$ or $-7 > 54$ (False)
• Second inequality: $-7 > \frac{1}{3}(-12) - 7$ simplifies to $-7 > -4 - 7$ or $-7 > -11$ (True)

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

Conclusion:

The only point that satisfies both inequalities is $(12, 1)$.

Answer: $(12, 1)$

Would you like more details or have any questions about this?

Here are five related questions to consider:

1. How do you graph the boundary lines for linear inequalities?
2. What does the solution region of a system of linear inequalities represent?
3. How can you check if a point is on the boundary line of an inequality?
4. What happens to the solution region when you change the inequality sign (e.g., from $>$ to $\geq$)?
5. How do you solve a system of linear inequalities with more than two inequalities?

Tip: When checking if a point satisfies an inequality, substitute the point into the inequality and simplify. If the resulting statement is true for both inequalities, the point is a solution to the system.