## The Slope of a Line

## Introduction

In mathematics, slope describes the “steepness” of a line. Lines can be viewed in a coordinate plane to get a better idea of their exact slope. Once in a coordinate plane, individual points on the line can be found… these points can be used to determine the slope ratio of the line.

It helps to think of slope as the "steepness" of a staircase. The red arrows to the left appear as a single step. Other steps to the left and right have the same size and shape as this step. The slope of the line can be found using any two points on the line.

## Lesson

The ** slope** of a line is the ratio of the rise compared to the run. Rise is the amount of vertical change between two points of the line, and run is the amount of horizontal change between those same two points.

The formula for slope is

Don’t be intimidated by the mathematical notation above. In (x_{1}, y_{1}), the subscript 1’s merely reflect the fact that the coordinate is picked as the first coordinate in the problem. Similarly the subscript 2’s in (x_{2}, y_{2}) represent the second coordinate of the problem.

**Example 1:** Find the slope of the line to the right

Solution:

Substitute the coordinates into the formula:

A second way to look at example #1 is to simply find the rise and run of the two points. If you can look at the graph and accurately determine the rise and run, then this method can be used to find the slope of the line.

Scroll over the graph to see the rise and the run of the line.

The rise is 4 and the run is 2. These values can be used to find the slope:

**Negative Slope**

Lines that increase in value as they move from left to right have a *positive slope*. Lines that go down as they move from left to right are said to have a *negative slope*.

**Example 2: ** Find the slope of the line to the right.

Solution:

You can find the slope of the line using either the formula or the slope ratio. The formula is shown below.

The final answer here is a negative fraction. The negative part makes sense since the line goes down as it moves from left to right. It is also sensible that the answer would less than 1 since the angle of elevation appears less than 45°. Lines that have a slope of 1 have an angle of elevation of exactly 45°.

A line with a slope of exactly 1 will have the same amount of rise and run. For each unit of rise, there will be exactly one unit of run. Slope can be estimated by looking at a line and seeing how it compares to a horizontal, vertical, and 45° diagonal line.

**Find Slope without using a Graph**

In each of the previous two examples, the line was shown on the coordinate plane. Being able to visually see the coordinate plane is a great help in determining whether a line has a positive or negative slope as well as estimating the slope itself. However, many problems do not provide a visual of the line. Be especially careful when using the formula to do problems such as example 3.

**Example 3:** Find the slope of the line that contains the points (2, 2) and (4, 18).

Solution:

The slope can be found using the slope formula:

If you are unsure of your answer, do a quick sketch of the approximate location of these points to get an idea of what the line looks like visually. Scroll over the coordinate plane to the left to reveal the line from example 3. This line appears to be positive and quite steep, so the answer “8” makes sense for the slope.

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**Zero and Undefined Slope**

The following two examples demonstrate how the slope formula can be used to find zero and undefined slopes.

**Example 4: **Find the slope of the line.

Solution:

Put the coordinates into the slope formula:

The line in this example is a horizontal line. Every horizontal line has a slope of zero.

**Example 5:** Find the slope of the line.

Solution:

Put the coordinates into the slope formula:

It is not mathematically possible to divide by zero. The symbol ∞ represents “infinity.” Another way to describe this slope is “undefined.”

**Related Links:**** **

For more information on graphing lines and related topics, try one of the links below.

**Resource Pages**

- Graphing Linear Equations (Intro) Resource Page
- Graphing Linear Equations (Advanced) Resource Page

**Related Lessons**

- Introduction to Slope
- The Slope-Intercept Form of an Equation
- Changing an Equation into Slope-Intercept Form

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