To write the equation of the line passing through the points $(-5, -1)$ and $(5, -7)$, we can use the point-slope form or slope-intercept form of the equation of a line.

Step 1: Calculate the slope ($m$)

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

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

Substituting the points $(-5, -1)$ and $(5, -7)$:

$m = \frac{-7 - (-1)}{5 - (-5)} = \frac{-7 + 1}{5 + 5} = \frac{-6}{10} = -\frac{3}{5}$

Step 2: Use the point-slope form

The point-slope form of a line is given by:

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

Using the point $(-5, -1)$ and the slope $m = -\frac{3}{5}$:

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

Simplifying:

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

Step 3: Simplify to slope-intercept form (if desired)

Distribute the slope:

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

Now subtract 1 from both sides to isolate $y$:

$y = -\frac{3}{5}x - 4$

Final Equation:

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

$y = -\frac{3}{5}x - 4$

Let me know if you'd like more details or have any questions about the steps!

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Here are some related questions to explore:

1. How do you find the equation of a line when you know its slope and a point on the line?
2. What is the difference between point-slope form and slope-intercept form?
3. How do you convert the point-slope form to the standard form of a linear equation?
4. What are parallel and perpendicular lines, and how are their slopes related?
5. How do you interpret the slope of a line in real-world contexts?

Tip: To make graphing lines easier, always calculate the slope and then use one point to solve for the y-intercept.