Question

how do i identify a slope? is it the amount of spaces a line goes through vertically as it slants? for example, if one intersect pint is at a top left corner of threee horizontal spaces, and the next intersect is at the bottom right, then the slope would likely be 3, right?

Answer 1

To identify the slope of a line in a coordinate system, we use the concept of "rise over run." The slope (m) is calculated by looking at the change in the vertical direction (rise) compared to the change in the horizontal direction (run).

The formula for slope is:

\[
m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where:
- \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

### Steps to Identify Slope:

1. **Identify Two Points on the Line**: For example, let's say Point A is at \((x_1, y_1)\) and Point B is at \((x_2, y_2)\).

2. **Calculate the Rise**: This is the vertical change from Point A to Point B.
\[
\text{rise} = y_2 - y_1
\]

3. **Calculate the Run**: This is the horizontal change from Point A to Point B.
\[
\text{run} = x_2 - x_1
\]

4. **Divide the Rise by the Run**: This gives you the slope.
\[
m = \frac{\text{rise}}{\text{run}}
\]

### Example:

If you have two points, say:
- Point A: (1, 3) (let's say this is the top left corner)
- Point B: (4, 1) (this is the bottom right corner)

1. **Calculate the Rise**:
\[
\text{rise} = y_2 - y_1 = 1 - 3 = -2
\]

2. **Calculate the Run**:
\[
\text{run} = x_2 - x_1 = 4 - 1 = 3
\]

3. **Calculate the Slope**:
\[
m = \frac{\text{rise}}{\text{run}} = \frac{-2}{3}
\]

### Conclusion

In your example, it seems you were trying to determine the vertical and horizontal changes, but the interpretation of the slope you provided is slightly off. It wouldn't just be "3" because you're looking at two dimensions (vertical and horizontal). By following the "rise over run" method, you can accurately determine the slope of a line.