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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."

Find an equation for the line that passes through the points (5, -1) and (-1, 1).

Learn how to calculate the equation of a line passing through the points (5, -1) and (-1, 1) using the point-slope form. The problem involves calculating the slope and substituting into the formula for the line equation. Suitable for students in Grades 8-10.

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