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Graphs of Linear Equations in Two Variables
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Slope of a Line

The slope of a line refers to the measure of the steepness of the line. If P( x 1 ,   y 1 ) and Q( x 2 ,   y 2 ) are two points on a line, then the slope m of the line is given by

m= y 2 − y 1 x 2 − x 1 ,

where x 2 ≠ x 1 .

The difference y 2 − y 1 is called the vertical change (also called rise) and x 2 − x 1 is called the horizontal change (also called run). The slope of a line gives the ratio of the vertical change to the horizontal change when going from one point to another on the line.

Example

Find the slope of the line passing through the points (1,  3) and (5,  6).



Solution

Let ( x 1 ,   y 1 )=(1,  3) and ( x 2 ,   y 2 )=(5,  6); that is, x 1 =1, y 1 =3, x 2 =5, and y 2 =6. Therefore, the slope of the line is given by

m= y 2 − y 1 x 2 − x 1 = 6−3 5−1 = 3 4 .