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To solve this problem, we first need to determine the translation vector that moves point \( P(-2, 3) \) to point \( P'(-6, -4) \).

1. **Calculate the translation vector:**
   \[
   \text{Translation vector} = (x', y') - (x, y) = (-6 - (-2), -4 - 3) = (-6 + 2, -4 - 3) = (-4, -7)
   \]

   So, the translation vector is \( (-4, -7) \).

2. **Apply the translation to the point \( (5, -1) \):**
   \[
   (5, -1) + (-4, -7) = (5 - 4, -1 - 7) = (1, -8)
   \]

3. **Conclusion:**
   The image of the point \( (5, -1) \) under the same translation is \( (1, -8) \).

**Answer: B. \( (1, -8) \)**

Would you like a detailed breakdown of each step or have any other questions?

### Related Questions

1. How do you determine the translation vector between two points?
2. Can you explain translations in the context of geometric transformations?
3. What is the general formula for translating a point by a vector?
4. How would you find the original point if you know the translated image and the translation vector?
5. What other transformations can be applied to points on a coordinate plane?

### Tip
When translating points, always remember to apply the vector's components to the respective \( x \) and \( y \) coordinates to obtain the translated point.

Point P'(-6, -4) is the image of point P(-2, 3) under a translation. What is the image of (5, -1) under the same translation?

Discover how to find the image of a point under a given translation in coordinate geometry. This example walks through determining the translation vector and applying it to a new point, covering key concepts suitable for Grades 7-9 students.

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