To determine which point is a solution to the system of linear inequalities, we need to test each of the given points in the two inequalities:

1. Inequality 1: $y > 2x + 2$
2. Inequality 2: $y > -\frac{5}{6}x + 6$

Testing Each Point:

(0, -3)

1. For $y > 2x + 2$: $-3 > 2(0) + 2 \implies -3 > 2 \quad \text{(False)}$ This point does not satisfy the first inequality.

(0, -10)

1. For $y > 2x + 2$: $-10 > 2(0) + 2 \implies -10 > 2 \quad \text{(False)}$ This point does not satisfy the first inequality.

(0, 9)

1. For $y > 2x + 2$: $9 > 2(0) + 2 \implies 9 > 2 \quad \text{(True)}$
2. For $y > -\frac{5}{6}x + 6$: $9 > -\frac{5}{6}(0) + 6 \implies 9 > 6 \quad \text{(True)}$ This point satisfies both inequalities.

(6, -4)

1. For $y > 2x + 2$: $-4 > 2(6) + 2 \implies -4 > 14 \quad \text{(False)}$ This point does not satisfy the first inequality.

Conclusion:

The only point that satisfies both inequalities is (0, 9).

Final Answer:

(0, 9)