To find the equation of the line passing through the points $(-6, 1)$ and $(6, -9)$, we can follow these steps:

Step 1: Calculate the slope

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of the points $(-6, 1)$ and $(6, -9)$:

$m = \frac{-9 - 1}{6 - (-6)} = \frac{-9 - 1}{6 + 6} = \frac{-10}{12} = \frac{-5}{6}$

Step 2: Use the point-slope form

The point-slope form of the equation of a line is:

$y - y_1 = m(x - x_1)$

We can use either of the two points. Let's use the point $(-6, 1)$ and the slope $m = -5/6$.

$y - 1 = \frac{-5}{6}(x - (-6)) = \frac{-5}{6}(x + 6)$

Step 3: Simplify the equation

Now, distribute the slope on the right-hand side:

$y - 1 = \frac{-5}{6}(x + 6)$ $y - 1 = \frac{-5}{6}x - 5$

Finally, add 1 to both sides to isolate $y$:

$y = \frac{-5}{6}x - 4$

Final Answer:

The equation of the line in point-slope form is:

$y = \frac{-5}{6}x - 4$

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you convert this point-slope equation to slope-intercept form?
2. What would be the equation if the line were horizontal or vertical?
3. How would you find the distance between the two points $(-6, 1)$ and $(6, -9)$?
4. Can you find the midpoint of the line segment joining the two points?
5. How do you determine if two lines are parallel or perpendicular?

Tip: For any line equation, if you know two points, calculating the slope is the first key step to find the full equation.