To solve this problem, we need to rotate the point $(8, -3)$ 180 degrees counterclockwise about the center $(7, -1)$.

Steps to solve:

1. Translate the points: First, we translate the center $(7, -1)$ to the origin by subtracting the coordinates of the center from both the point and the center. $\text{Translation:} \quad (x, y) \rightarrow (x - 7, y + 1)$ Applying this to the point $(8, -3)$: $(8 - 7, -3 + 1) = (1, -2)$

2. Rotate 180 degrees: Rotating 180 degrees about the origin negates both the x and y coordinates: $(x, y) \rightarrow (-x, -y)$ Applying this to $(1, -2)$: $(-1, 2)$

3. Translate back: Finally, translate the point back to the original center by adding the coordinates of the center to the rotated point: $(-1 + 7, 2 - 1) = (6, 1)$

The coordinates of the image after a 180-degree rotation are $(6, 1)$, which corresponds to option B.

Would you like more details on the process or have any questions?

Related questions:

1. How do you rotate a point 90 degrees clockwise about an arbitrary center?
2. What happens when you rotate a point 270 degrees counterclockwise about the origin?
3. Can you explain how to find the midpoint between two points?
4. How does reflection across a line differ from rotation?
5. What are the coordinates of a point reflected across the y-axis?

Tip: When rotating points around a center that isn't the origin, translating the points to and from the origin simplifies calculations.