If you subtract the *x*-values,
then the result is called the **run**;
If you subtract the *y*-values,
then the result is called the **rise**.
Sometimes the symbols Δ*x* and Δ*y*
are used for these:

- run = Δ
*x*=*x*_{2}−*x*_{1}. - rise = Δ
*y*=*y*_{2}−*y*_{1}.

Note that the rise is positive if you move upwards on the graph while travelling from point 1 to point 2 (in addition to possibly moving horizontally as well); it's negative if you move downwards, and it's zero if you move only horizontally. Similarly, the run is positive if you move rightwards on the graph while travelling from point 1 to point 2 (in addition to possibly moving vertically as well); it's negative if you move leftwards, and it's zero if you move only vertically. In general, the rise tells you how far you move upwards, and the run tells you how for you move rightwards.

(Technically, rise and run are properties
of a *directed* line segment,
in which you travel from one endpoint to the other rather than the reverse.
However, the direction makes no difference
to any of the things that we'll use the rise and run for
―distance, midpoint, and slope―,
as long as you're consistent about it.)

The **distance** between two points
(or equivalently, the length of the line segment between them)
can be found using the Pythagorean Theorem;
the rise and run (or to be precise, their absolute values)
are the lengths of the two short sides of a right triangle,
and the line segment whose length we want is the long side.
If *d* is this distance,
then

- Δ
*x*^{2}+ Δ*y*^{2}=*d*^{2}, so *d*= √(Δ*x*^{2}+ Δ*y*^{2}), or- distance =
√(run
^{2}+ rise^{2}).

- (
*x*,*y*) = (*x*_{1}+ ½Δ*x*,*y*_{1}+ ½Δ*y*), so - (
*x*,*y*) = ((*x*_{1}+*x*_{2})/2, (*y*_{1}+*y*_{2})/2), or - midpoint =
(average
*x*-value, average*y*-value).

- slope = rise ÷ run, or
*m*= Δ*y*/Δ*x*

The slope describes the directions in which you can travel along the line.

- Lines with positive slope run up–right and down–left; lines with negative slope run down–right and up–left.
- Lines whose slope has a large absolute value are steep; lines whose slope has a small absolute value are shallow.
- Horizontal lines have a slope of exactly zero; vertical lines have a slope which is undefined (which you can think of as an infinite slope).

*y*=*m**x*+*b*.

- The slope is the number
*m*. - The
*y*-intercept is (0,*b*). - The variables
*x*and*y*stay in the equation.

If you don't know the *y*-intercept,
you can still use this equation if you know one of the points;
plug it in for *x* and *y*, and solve for *b*.
Or use this optional formula:

*y*=*m*(*x*−*x*_{1}) +*y*_{1}.

Conversely, if you have an equation for the line,
then solve it for *y*;
you now know what the slope and *y*-intercept are.
It's easy to draw a graph using any point and the slope.

If *a* is the *x*-coordinate of any point,
then the equation for a vertical line is always simply:

*x*=*a*.

*y*=*b*;

- Parallel lines:
*m*_{1}=*m*_{2}. - Perpendicular lines:
*m*_{1}*m*_{2}= −1, or*m*_{2}= −1/*m*_{1}.

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This web page was written by Toby Bartels, last edited on 2024 August 15. Toby reserves no legal rights to it.

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