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Slope tells a change story.

“Rise over run” is a calculation. Slope is the rate that calculation describes: how much y changes whenever x moves one unit.

8 minute readUpdated August 22, 2026Algebra 1
The one-sentence idea: If a line has slope 3, then every move of 1 unit to the right goes with a move of 3 units up.

What slope measures

Imagine tracking the height of a ramp while you walk forward. Your horizontal movement is the run. The change in height is the rise. Slope compares those two changes:

slope = change in y ÷ change in x

or m = (y₂ − y₁) / (x₂ − x₁)

The numerator and denominator are both changes. That is why the subtraction order must match. If you calculate y₂ − y₁ on top, calculate x₂ − x₁ on the bottom.

Example: read a slope from two points

Suppose a line passes through (2, 3) and (6, 11).

  1. Change in y: 11 − 3 = 8.
  2. Change in x: 6 − 2 = 4.
  3. Slope: 8 ÷ 4 = 2.

The result means that y rises by 2 whenever x increases by 1. You could also move right 4 and up 8; the ratio simplifies to the same rate.

Positive, negative, zero, or undefined

Positive slope

The line rises as you read from left to right. Both changes have the same sign, so their quotient is positive.

Negative slope

The line falls as you read from left to right. One change is positive and the other is negative, so the quotient is negative.

Zero slope

A horizontal line has no vertical change. Zero divided by a nonzero run is zero.

Undefined slope

A vertical line has no horizontal change. Dividing by zero is undefined, so a vertical line’s slope is undefined—not “infinite.”

Where slope lives in y = mx + b

In slope-intercept form, m is slope and b is the y-intercept. The intercept tells you where the line crosses the y-axis. The slope tells you what to do from there.

For y = −2x + 5:

  • Start at (0, 5) because b = 5.
  • Use slope −2/1: move right 1 and down 2.
  • Repeat that move to create more points.

Three common slope mistakes

  1. Mixing subtraction order. Keep the same point first in the numerator and denominator.
  2. Reading left to right as mandatory. You may start at either point; consistent subtraction gives the same slope.
  3. Forgetting the units. A slope of 50 could mean 50 miles per hour, $50 per ticket, or 50 liters per minute. The context completes the meaning.

Try it

A line passes through (−1, 4) and (3, −4). What is its slope?

Show the worked answer

Change in y: −4 − 4 = −8. Change in x: 3 − (−1) = 4. Slope = −8/4 = −2. For every 1 unit right, the line moves 2 units down.

Next, sketch either point on graph paper and use “right 1, down 2” to locate the other points. The arithmetic and the visual route should agree.