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how to find slope from two points

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.