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In addition to its familiar meaning, the word "slope" has precise mathematical meaning. The slope of a line is the rise over the run, or the change in
*y*
divided by the change in
*x*
. To find the slope of a line, pick *any* two points on the line. Then subtract their x-coordinates and subtract their y-coordinates **in the same order**. Divide the difference of the
*y*
-coordinates by the difference of the
*x*
- coordinates:

Given two points (x_{1},y_{1}) and (x_{2},y_{2}) on a line, the slope of the line is equal to:

m= =

*Example 1.* Find the slope of the line which passes through the points
(2, 5)
and
(0, 1)
:

Slope

If a line has a positive slope (i.e.
*m* > 0
), then
*y*
always increases when
*x*
increases and
*y*
always decreases when
*x*
decreases. Thus, the graph of the line starts at the bottom left and goes towards the top right.

Often, however, the slope of a line is negative. A negative slope implies that
*y*
always decreases when
*x*
increases and
*y*
always increases when
*x*
decreases. Here is an example of a graph with negative slope:

Negative Slope

Thus, as

Sometimes, we will see equations whose graphs are horizontal lines. These are graphs in which
*y*
remains constant -- that is, in which
*y*
_{1} - *y*
_{2} = 0
for any two points on the line:

Graph of a Horizontal Line

We will also see equations whose graphs are vertical lines. These are graphs in which
*x*
remains constant -- that is, in which
*x*
_{1} - *x*
_{2} = 0
for any two points on the line:

Graph of a Vertical Line

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