Editing Slope (section)

== Definition ==

[[File:Slope of lines illustrated.jpg|thumb|400px|right|Slope illustrated for {{nowrap|1=''y'' = (3/2)''x'' − 1}}. Click on to enlarge]]
[[File:Gradient of a line in coordinates from -12x+2 to +12x+2.gif|400px|thumbnail|right|Slope of a line in coordinates system, from {{nowrap|1=''f''(''x'') = −12''x'' + 2}} to {{nowrap|1=''f''(''x'') = 12''x'' + 2}}]]
The slope of a line in the plane containing the ''x'' and ''y'' axes is generally represented by the letter ''m'',<ref>An early example of this convention can be found in {{cite book |last=Salmon |first=George |author-link=George Salmon |year=1850 |url=https://archive.org/details/treatiseonconics00salm_1/page/14/ |pages=14–15 |title=A Treatise on Conic Sections |location=Dublin |publisher=Hodges and Smith |edition=2nd }}</ref> and is defined as the change in the ''y'' coordinate divided by the corresponding change in the ''x'' coordinate, between two distinct points on the line. This is described by the following equation:

:<math>m = \frac{\Delta y}{\Delta x} = \frac{\text{vertical} \, \text{change} }{\text{horizontal} \, \text{change} }= \frac{\text{rise}}{\text{run}}.</math>
(The Greek letter ''[[delta (letter)|delta]]'', Δ, is commonly used in mathematics to mean "difference" or "change".)

Given two points <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>, the change in <math>x</math> from one to the other is <math>x_2-x_1</math> (''run''), while the change in <math>y</math> is <math>y_2-y_1</math> (''rise''). Substituting both quantities into the above equation generates the formula:
:<math>m = \frac{y_2 - y_1}{x_2 - x_1}.</math>
The formula fails for a vertical line, parallel to the <math>y</math> axis (see [[Division by zero]]), where the slope can be taken as [[infinity|infinite]], so the slope of a vertical line is considered undefined.

=== Examples ===
Suppose a line runs through two points: ''P''&nbsp;=&nbsp;(1,&nbsp;2) and ''Q''&nbsp;=&nbsp;(13,&nbsp;8). By dividing the difference in <math>y</math>-coordinates by the difference in <math>x</math>-coordinates, one can obtain the slope of the line:
:<math>m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(8 - 2)}{(13 - 1)} = \frac{6}{12} = \frac{1}{2}.</math>
:Since the slope is positive, the direction of the line is increasing. Since |''m''| < 1, the incline is not very steep (incline <&thinsp;45°).

As another example, consider a line which runs through the points (4,&nbsp;15) and (3,&nbsp;21). Then, the slope of the line is 
:<math>m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6.</math>
:Since the slope is negative, the direction of the line is decreasing. Since |''m''| > 1, this decline is fairly steep (decline >&thinsp;45°).