To find the slope from two points, use the slope formula and plug in the coordinates of the points.

Slope formula (the key idea)

If you have two points on a line, (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​), the slope mmm is

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​

This is often described as “rise over run” (change in yyy divided by change in xxx).

Step‑by‑step method

  1. Label your points
    • First point → (x1,y1)(x_1,y_1)(x1​,y1​)
    • Second point → (x2,y2)(x_2,y_2)(x2​,y2​)
      It doesn’t matter which you call “1” or “2” as long as you stay consistent (top and bottom use the same order).
  1. Find the change in y (rise)
    • Compute y2−y1y_2-y_1y2​−y1​.
  1. Find the change in x (run)
    • Compute x2−x1x_2-x_1x2​−x1​.
  1. Divide rise by run
    • m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​.
  1. Simplify the fraction
    • Reduce the fraction if possible. A negative sign can go in the numerator, denominator, or in front of the whole fraction (they all mean the same slope).

Concrete example

Say the line goes through (1,−2)(1,-2)(1,−2) and (3,−6)(3,-6)(3,−6).

  1. Label:
    • (x1,y1)=(1,−2)(x_1,y_1)=(1,-2)(x1​,y1​)=(1,−2)
    • (x2,y2)=(3,−6)(x_2,y_2)=(3,-6)(x2​,y2​)=(3,−6)
  2. Compute rise:
    • y2−y1=−6−(−2)=−6+2=−4y_2-y_1=-6-(-2)=-6+2=-4y2​−y1​=−6−(−2)=−6+2=−4
  3. Compute run:
    • x2−x1=3−1=2x_2-x_1=3-1=2x2​−x1​=3−1=2
  4. Divide:
    • m=−42=−2m=\dfrac{-4}{2}=-2m=2−4​=−2

So the slope of the line through those two points is −2-2−2.

Handy HTML table of quick facts

Here’s a small HTML table summarizing the core ideas:

html

<table>
  <tr>
    <th>Concept</th>
    <th>Details</th>
  </tr>
  <tr>
    <td>Slope formula</td>
    <td>m = (y₂ - y₁) / (x₂ - x₁) [web:1][web:3]</td>
  </tr>
  <tr>
    <td>Meaning of slope</td>
    <td>“Rise over run”: change in y divided by change in x [web:3]</td>
  </tr>
  <tr>
    <td>Point labeling</td>
    <td>You can swap which point is 1 or 2, as long as you stay consistent in numerator and denominator [web:1][web:3]</td>
  </tr>
  <tr>
    <td>Zero slope</td>
    <td>If y₂ = y₁, the slope is 0 (horizontal line) [web:3]</td>
  </tr>
  <tr>
    <td>Undefined slope</td>
    <td>If x₂ = x₁, denominator is 0 and slope is undefined (vertical line) [web:3][web:10]</td>
  </tr>
</table>

Tiny story to remember it

Imagine you’re hiking from one campsite to another on a straight trail.

  • The x-coordinate is how far you’ve walked forward.
  • The y-coordinate is how high above sea level you are.

If your second campsite is higher than the first, the “rise” y2−y1y_2-y_1y2​−y1​ is positive, and the slope is positive: you’ve been climbing.
If it’s lower, the rise is negative, and the slope is negative: you’ve been going downhill.

TL;DR:
Label the points (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​), then compute m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​, simplify, and that’s your slope.

Information gathered from public forums or data available on the internet and portrayed here.