# Determining the Slope of a Line

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*slope.*There are different ways to determine the slope of a line given the way in which the function is expressed.

## Slope

The slope of a line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is the ratio of the vertical change $(\Delta y)$ to the horizontal change $(\Delta x)$ between the points. The variable $m$ is most commonly used to represent slope.

$m=\dfrac{\Delta y}{\Delta x}$

The words *rise* and *run* are sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to $\Delta y$ and run corresponds to $\Delta x.$

This gives the following definition for the slope of a line.

$m=\dfrac{\Delta y}{\Delta x}=\dfrac{\text{rise}}{\text{run}}$

## Determine the Slope of a Line from its Graph

The slope of a line measures the change between points on the line. When presented graphically, it's common to use the following definition of slope. $m=\dfrac{\text{rise}}{\text{run}}$

Here, rise represents the vertical distance between points and run represents the horizontal distance. Consider the line shown. The slope of the line will be found using the following method.

## Types of Slope

The slope of a line measures how $y$ changes relative to $x.$ There are different types of slope depending on the direction of the line.

*undefined*slope corresponds with a linear function that is a vertical line.

## Slope Formula

The slope of a line can be found algebraically using the following rule.

$m=\dfrac{y_2-y_1}{x_2-x_1}$

## Exercises

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