Question

How to find the slope

Answer 1

To find the slope of a line, you can use the formula:

\[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}
\]

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

Here’s how to calculate it step by step:

1. **Identify two points** on the line. For example, let’s take points \(A(x_1, y_1)\) and \(B(x_2, y_2)\).

2. **Subtract the y-coordinates**: \(y_2 - y_1\).
- This gives you the change in the vertical direction (rise).

3. **Subtract the x-coordinates**: \(x_2 - x_1\).
- This gives you the change in the horizontal direction (run).

4. **Divide the rise by the run** to find the slope:
- \(m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\).

### Example

Let's say you have the points \(A(2, 3)\) and \(B(5, 7)\):

1. \(x_1 = 2\), \(y_1 = 3\)
2. \(x_2 = 5\), \(y_2 = 7\)

Calculating the slope:

\[
m = \frac{7 - 3}{5 - 2} = \frac{4}{3}
\]

So the slope of the line that passes through the points \(A\) and \(B\) is \(\frac{4}{3}\).

### Special Cases

- If the line is horizontal (same y-values), the slope is \(0\).
- If the line is vertical (same x-values), the slope is undefined.

### Using the Slope-Intercept Form

If you have a linear equation in the slope-intercept form, \(y = mx + b\), the slope is simply the coefficient \(m\).

Feel free to ask if you need further clarification or have a different context in mind for finding the slope!