To find the equation of the line that passes through the points $(-5, 1)$ and $(5, -5)$, we will use the point-slope form of the equation of a line:

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

where $m$ is the slope of the line, and $(x_1, y_1)$ is one of the points the line passes through.

Step 1: Find the slope $m$

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

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

Using the points $(-5, 1)$ and $(5, -5)$:

$m = \frac{-5 - 1}{5 - (-5)} = \frac{-6}{10} = -\frac{3}{5}$

Step 2: Use point-slope form

Now that we have the slope $m = -\frac{3}{5}$, we can use one of the points to write the equation. Let’s use $(-5, 1)$:

$y - 1 = -\frac{3}{5}(x - (-5))$ $y - 1 = -\frac{3}{5}(x + 5)$

Step 3: Simplify

Now, distribute the slope and simplify:

$y - 1 = -\frac{3}{5}x - 3$

Add 1 to both sides to isolate $y$:

$y = -\frac{3}{5}x - 2$

Thus, the equation of the line is:

$y = -\frac{3}{5}x - 2$

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Would you like further details or clarification on any step? Here are some related questions you may find interesting:

1. How can we find the equation of a line given two points in 3D space?
2. What does the slope of a line represent in real-world situations?
3. How do you convert the equation of a line from point-slope form to slope-intercept form?
4. How do you calculate the distance between two points in a 2D plane?
5. What is the significance of the y-intercept in the equation of a line?

Tip: Remember that the slope of a line tells you how steep the line is—if the slope is negative, the line decreases as you move from left to right.