Question: Find the coordinates of the image of $Z (-3, 1)$ after the translation $(x, y) \to (x - 2, y - 7)$
1. **State the problem:** We need to find the new coordinates of point $Z(-3, 1)$ after applying the translation rule $(x, y) \to (x - 2, y - 7)$.
2. **Recall the translation formula:** A translation moves every point by adding or subtracting values from the original coordinates. Here, the translation subtracts 2 from the $x$-coordinate and 7 from the $y$-coordinate.
3. **Apply the translation:**
$$
(x, y) = (-3, 1) \\
(x - 2, y - 7) = (-3 - 2, 1 - 7) \\
= (-5, -6)
$$
4. **Interpret the result:** The image of point $Z$ after the translation is at $(-5, -6)$.
5. **Check answer choices:** The correct choice is $(-5, -6)$.
**Final answer:** $(-5, -6)$