To find the equation of the line passing through the points $(8, 10)$ and $(-4, -6)$, we follow these steps:

1. Calculate the slope ($m$) of the line using the formula:

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

Here, $(x_1, y_1) = (8, 10)$ and $(x_2, y_2) = (-4, -6)$.

$m = \frac{-6 - 10}{-4 - 8} = \frac{-16}{-12} = \frac{4}{3}$

2. Use the point-slope form of the line equation:

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

We can use either point $(8, 10)$ or $(-4, -6)$. We'll use $(8, 10)$:

$y - 10 = \frac{4}{3}(x - 8)$

3. Simplify the equation to the slope-intercept form ($y = mx + b$):

$y - 10 = \frac{4}{3}(x - 8)$ $y - 10 = \frac{4}{3}x - \frac{32}{3}$ $y = \frac{4}{3}x - \frac{32}{3} + 10$ $y = \frac{4}{3}x - \frac{32}{3} + \frac{30}{3}$ $y = \frac{4}{3}x - \frac{2}{3}$

So, the equation of the line passing through the points $(8, 10)$ and $(-4, -6)$ is:

$y = \frac{4}{3}x - \frac{2}{3}$

Would you like more details or have any questions about this? Here are some related questions to expand your understanding:

1. How do you derive the slope formula for a line?
2. What is the point-slope form of a line, and why is it useful?
3. How can you convert from point-slope form to slope-intercept form?
4. What is the significance of the slope in the line equation?
5. How can you determine if two lines are parallel using their slopes?

Tip: Always double-check your calculations when determining the slope and rearranging equations to avoid mistakes.