Question: Find the coordinates of the image of $G (-3, 9)$ after the translation $(x, y) \to (x + 10, y - 7)$.
1. **State the problem:** We need to find the new coordinates of point $G(-3, 9)$ after applying the translation rule $(x, y) \to (x + 10, y - 7)$.
2. **Recall the translation formula:** A translation moves every point by adding a fixed amount to the $x$ and $y$ coordinates. Here, the rule is:
$$ (x, y) \to (x + 10, y - 7) $$
which means add $10$ to the $x$-coordinate and subtract $7$ from the $y$-coordinate.
3. **Apply the translation:**
$$ x' = -3 + 10 = 7 $$
$$ y' = 9 - 7 = 2 $$
4. **Intermediate step showing cancellation:**
$$ x' = \cancel{-3} + 10 = 7 $$
$$ y' = 9 - \cancel{7} = 2 $$
5. **Final answer:** The image of $G(-3, 9)$ after the translation is $\boxed{(7, 2)}$.
6. **Check answer choices:** The correct choice is $(7, 2)$.