The task is to find the equation of a line passing through the points $(5, -1)$ and $(-1, 1)$.

To find the equation of the line, 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 one of the points on the line, and
• $m$ is the slope of the line, calculated by:

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

Step 1: Calculate the slope $m$

Using the points $(x_1, y_1) = (5, -1)$ and $(x_2, y_2) = (-1, 1)$:

$m = \frac{1 - (-1)}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}$

Step 2: Use the point-slope form

Now, using point $(5, -1)$ and the calculated slope $m = -\frac{1}{3}$, we can substitute into the point-slope form:

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

This simplifies to:

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

Now, you can simplify further to get it into slope-intercept form if needed:

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

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

$y = -\frac{1}{3}x + \frac{2}{3}$

So, the equation of the line is:

$y = -\frac{1}{3}x + \frac{2}{3}$

Would you like more details on this process or have any further questions? Here are some related questions you could explore:

1. How do you convert the equation from point-slope to slope-intercept form?
2. What is the significance of the slope in the equation of a line?
3. How do you graph the equation of a line given two points?
4. Can you find the equation of the line passing through two other points, say $(2, 4)$ and $(6, -2)$?
5. How would the equation change if the slope was positive instead of negative?

Tip: To always find the slope, remember it’s the change in $y$-values divided by the change in $x$-values. This is often called "rise over run."