We are tasked with finding the equation of a line that passes through the point $(8, -1)$ and is parallel to the line that passes through the points $(-2, 2)$ and $(6, 6)$.

Step 1: Find the slope of the line passing through $(-2, 2)$ and $(6, 6)$.

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute $(-2, 2)$ as $(x_1, y_1)$ and $(6, 6)$ as $(x_2, y_2)$: $m = \frac{6 - 2}{6 - (-2)} = \frac{4}{6 + 2} = \frac{4}{8} = \frac{1}{2}$ So, the slope of the line passing through $(-2, 2)$ and $(6, 6)$ is $\frac{1}{2}$.

Step 2: Use the slope for the parallel line.

Since the line we want is parallel to this one, it will have the same slope. Therefore, the slope of the desired line is also $\frac{1}{2}$.

Step 3: Use the point-slope form to find the equation of the line.

The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ We are given the point $(8, -1)$ and the slope $m = \frac{1}{2}$. Substituting these values: $y - (-1) = \frac{1}{2}(x - 8)$ Simplify: $y + 1 = \frac{1}{2}(x - 8)$ Distribute the $\frac{1}{2}$: $y + 1 = \frac{1}{2}x - 4$ Subtract 1 from both sides: $y = \frac{1}{2}x - 5$

Step 4: Select the correct answer.

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

Final Answer:

The correct equation is: $\boxed{y = \frac{1}{2}x - 5}$

Would you like any further explanation or details?

Here are some related questions to explore:

1. How do you find the slope of a line passing through two points?
2. What does it mean for two lines to be parallel in terms of their slopes?
3. How can you derive the point-slope form of a line's equation?
4. How do you convert from point-slope form to slope-intercept form?
5. What is the difference between parallel and perpendicular lines in terms of their slopes?

Tip: When two lines are parallel, their slopes are always equal!