Run — Definition, Formula & Examples

Run is the horizontal distance (left-to-right change) between two points on a line. It is the bottom part of the 'rise over run' formula for slope.

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ , the run is defined as the difference $x_2 - x_1$ , representing the change in the $x$ -coordinates. A positive run indicates movement to the right; a negative run indicates movement to the left.

Key Formula

\text{run} = x_2 - x_1

Where:

$x_1$ = The x-coordinate of the first point
$x_2$ = The x-coordinate of the second point

How It Works

To find the run, subtract the

x

-coordinate of the first point from the

x

-coordinate of the second point. You pair the run with the rise (the vertical change) to calculate slope:

m = \frac{\text{rise}}{\text{run}}

. On a coordinate grid, the run is the horizontal leg of the right triangle you can draw between any two points on a line.

Worked Example

Problem: Find the run between the points (2, 3) and (8, 7).

Identify x-coordinates: The first point has

x_1 = 2

and the second has

x_2 = 8

Subtract: Subtract the first from the second.

\text{run} = 8 - 2 = 6

Interpret: The line moves 6 units to the right between these two points. Combined with a rise of

7 - 3 = 4

, the slope would be

\frac{4}{6} = \frac{2}{3}

Answer: The run is 6.

Why It Matters

Every time you calculate slope in algebra or graph a linear equation, you need the run. Understanding it separately helps you read rate-of-change problems correctly, such as finding speed (distance per hour) or cost per item.

Common Mistakes

Mistake: Confusing run with rise by subtracting the y-coordinates instead of the x-coordinates.

Correction: Run is always the change in x (horizontal). Rise is the change in y (vertical). Remember: run goes along the ground — left and right.