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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):
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:
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:
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:
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