Slope: Steepness as Rise Over Run

What slope measures

Picture walking from left to right along a line. The slope answers one question: for every step sideways, how much do you go up or down? It is a ratio — a comparison of vertical change to horizontal change. We capture it with the phrase rise over run: rise is the vertical change, run is the horizontal change.

A positive slope rises as you move right; a negative slope falls. A bigger absolute value means a steeper line. We almost always call the slope m.

The slope formula from two points

Give me any two points on a line, (x1, y1) and (x2, y2), and I can compute the slope. The rise is the difference in the y-values; the run is the difference in the x-values, taken in the same order:

         y2 - y1
    m = ---------
         x2 - x1

Example: line through (1, 2) and (4, 8)

         8 - 2     6
    m = ------- = ---- = 2
         4 - 1     3

Check by swapping the order of the points:

         2 - 8     -6
    m = ------- = ---- = 2     same answer
         1 - 4     -3

Zero slope and undefined slope

Two special lines push the formula to its edges. A horizontal line never rises, so the rise is 0 and the slope is 0. A vertical line never runs, so the run is 0 — and dividing by zero is undefined, so a vertical line has no slope at all (we say the slope is undefined, not zero).

Horizontal line through (1, 3) and (5, 3):
         3 - 3     0
    m = ------- = ---- = 0          slope is 0
         5 - 1     4

Vertical line through (2, 1) and (2, 7):
         7 - 1     6
    m = ------- = ----  -> divide by 0
         2 - 2     0               slope UNDEFINED