To find the equation of the line passing through the points (-4, 7) and (6, 2), we follow these steps:

1. Calculate the slope (m):

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1) = (-4, 7)$ and $(x_2, y_2) = (6, 2)$.

Substitute the values: $m = \frac{2 - 7}{6 - (-4)} = \frac{-5}{6 + 4} = \frac{-5}{10} = -\frac{1}{2}$

2. Use point-slope form:

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the point $(-4, 7)$ and the slope $m = -\frac{1}{2}$, we get: $y - 7 = -\frac{1}{2}(x - (-4)) = -\frac{1}{2}(x + 4)$

3. Simplify the equation:

Distribute the $-\frac{1}{2}$: $y - 7 = -\frac{1}{2}x - 2$ Add 7 to both sides: $y = -\frac{1}{2}x + 5$

4. Conclusion:

The equation of the line is $y = -\frac{1}{2}x + 5$, which matches the third option in the image.

Would you like more details on this? Here are some related questions:

1. How do we derive the slope formula?
2. What is the significance of point-slope form in finding the equation of a line?
3. Can you find the equation using a different point like (6, 2)?
4. What are parallel lines, and how would their equations relate to this one?
5. What is the difference between slope-intercept and point-slope form?

Tip: Always check if the given points satisfy the equation by substituting their coordinates into the equation you find!