how to find the slope of a line

To find the slope of a line, you measure how much it goes up or down compared to how much it goes left or right, written as “rise over run.”

What “slope” means

Slope tells you how steep a line is and whether it’s going up or down as you move to the right.
Positive slope: line goes up as you move right.
Negative slope: line goes down as you move right.
Zero slope: perfectly horizontal line.
Undefined slope: perfectly vertical line.

Main formula for slope

If you know two points on the line, say
(x1,y1)(x_1,y_1)(x1,y1) and (x2,y2)(x_2,y_2)(x2,y2), then the slope mmm is:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}m=x2−x1y2−y1

You can remember this as:

rise = change in y = y2−y1y_2-y_1y2−y1
run = change in x = x2−x1x_2-x_1x2−x1
slope = rise ÷ run.

Step‑by‑step: using two points

Suppose your line goes through the points (2,5)(2,5)(2,5) and (9,19)(9,19)(9,19).

Label the points
- (x1,y1)=(2,5)(x_1,y_1)=(2,5)(x1,y1)=(2,5)
- (x2,y2)=(9,19)(x_2,y_2)=(9,19)(x2,y2)=(9,19)
Find the change in y (rise)
- y2−y1=19−5=14y_2-y_1=19-5=14y2−y1=19−5=14
Find the change in x (run)
- x2−x1=9−2=7x_2-x_1=9-2=7x2−x1=9−2=7
Divide rise by run
- m=147=2m=\dfrac{14}{7}=2m=714=2

So the slope of the line through those two points is 222. That means every time x increases by 1, y increases by 2.

Finding slope from a graph (rise over run)

If you have a graph instead of coordinates:

Pick any two clear points on the line that hit grid intersections.
From the first point, count how far you go up or down to get to the second point (this is the rise).
Then count how far you go right to get to the second point (this is the run).
Slope = rise ÷ run.

Example:

From point A to point B, suppose you go up 3 units and right 2 units.
Then slope m=32m=\dfrac{3}{2}m=23.

If you go down instead of up, the rise is negative. For example, down 4 and right 1 gives slope m=−41=−4m=\dfrac{-4}{1}=-4m=1−4=−4.

Special cases to know

Horizontal line (flat):
- Same y‑value for every point, like (−3,3)(−3,3)(−3,3) and (2,3)(2,3)(2,3).
- y2−y1=3−3=0y_2-y_1=3-3=0y2−y1=3−3=0, so m=0/(x2−x1)=0m=0/(x_2-x_1)=0m=0/(x2−x1)=0.
- Slope = 0.
Vertical line (straight up and down):
- Same x‑value for every point, like (4,1)(4,1)(4,1) and (4,7)(4,7)(4,7).
- x2−x1=4−4=0x_2-x_1=4-4=0x2−x1=4−4=0, so you’d divide by 0, which is not allowed.
- Slope is “undefined.”

Quick reference table (HTML)

Here’s an HTML table summarizing slope situations:

html

<table>
  <thead>
    <tr>
      <th>Type of line</th>
      <th>Example points</th>
      <th>Calculation</th>
      <th>Slope value</th>
      <th>What it looks like</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Positive slope</td>
      <td>(2, 5) and (9, 19)</td>
      <td>(19 - 5) / (9 - 2) = 14 / 7</td>
      <td>m = 2</td>
      <td>Rises as you move right</td>
    </tr>
    <tr>
      <td>Negative slope</td>
      <td>(1, 4) and (3, 0)</td>
      <td>(0 - 4) / (3 - 1) = -4 / 2</td>
      <td>m = -2</td>
      <td>Falls as you move right</td>
    </tr>
    <tr>
      <td>Zero slope</td>
      <td>(-3, 3) and (2, 3)</td>
      <td>(3 - 3) / (2 - (-3)) = 0 / 5</td>
      <td>m = 0</td>
      <td>Perfectly horizontal</td>
    </tr>
    <tr>
      <td>Undefined slope</td>
      <td>(4, 1) and (4, 7)</td>
      <td>(7 - 1) / (4 - 4) = 6 / 0</td>
      <td>Undefined</td>
      <td>Perfectly vertical</td>
    </tr>
  </tbody>
</table>

Tiny story to remember it

Imagine walking up a hill:

Every step forward is your run.
Every step up is your rise.
A steep hiking trail has a big rise for a small run → large slope.
A flat sidewalk has almost no rise → slope near 0.
A ladder standing straight up would be like a vertical line → “infinite” steepness → undefined slope.

TL;DR

Use the formula m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2−x1y2−y1.
Think of slope as “how much y changes when x goes over by 1.”
Remember: rise over run, positive goes up, negative goes down, 0 is flat, vertical is undefined.