To find the equation of the line passing through two points $(-5, 1)$ and $(5, -3)$, we can use the point-slope form of the equation of a line:

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

Where:

• $(x_1, y_1)$ is a point on the line.
• $m$ is the slope of the line.

Step 1: Calculate the slope $m$

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

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

Substitute the points $(-5, 1)$ and $(5, -3)$:

$m = \frac{-3 - 1}{5 - (-5)} = \frac{-4}{10} = -\frac{2}{5}$

Step 2: Use the point-slope form

Now that we know the slope $m = -\frac{2}{5}$, we can use the point $(-5, 1)$ and plug it into the point-slope formula:

$y - 1 = -\frac{2}{5}(x + 5)$

Step 3: Simplify to slope-intercept form

Expand and simplify the equation:

$y - 1 = -\frac{2}{5}(x + 5)$ $y - 1 = -\frac{2}{5}x - 2$ $y = -\frac{2}{5}x - 1$

Thus, the equation of the line in slope-intercept form is:

$y = -\frac{2}{5}x - 1$

Summary:

The equation of the line passing through the points $(-5, 1)$ and $(5, -3)$ is:

$y = -\frac{2}{5}x - 1$

Would you like any more details or further steps?

Here are five related questions to deepen your understanding:

1. How do you derive the slope-intercept form from the point-slope form?
2. What does the slope of a line represent geometrically?
3. How do you find the equation of a line when you are given the slope and a point on the line?
4. What does it mean for two lines to be parallel or perpendicular in terms of their slopes?
5. How can you graph the equation $y = -\frac{2}{5}x - 1$?

Tip: Always ensure the correct order when substituting values for slope and points into formulas.