To find the equation of a line passing through two points, $(x_1, y_1)$ and $(x_2, y_2)$, we can use the point-slope form of the line equation. Here's the process:

Step 1: Find the slope of the line

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substituting the given points $(2, -5)$ and $(-8, 5)$:

$m = \frac{5 - (-5)}{-8 - 2} = \frac{5 + 5}{-8 - 2} = \frac{10}{-10} = -1$

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 points. Let's use the point $(2, -5)$:

$y - (-5) = -1(x - 2)$

Simplifying:

$y + 5 = -1(x - 2)$

Distribute the $-1$:

$y + 5 = -x + 2$

Subtract 5 from both sides:

$y = -x + 2 - 5$

$y = -x - 3$

Final equation:

The equation of the line is:

$y = -x - 3$

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

1. What would be the equation if the points were different?
2. How do you find the equation of a vertical line?
3. How can you write the equation in standard form?
4. How does the slope of a line affect its graph?
5. How can you verify that the equation is correct by substituting the points?

Tip: To quickly identify the slope between two points, remember that a positive slope rises, and a negative slope falls.