Chapter 1 Linear Equations
1.4 Slope of a Line
Learning Objectives
By the end of this section, you will be able to:
- Find the slope of a line
- Graph a line given a point and the slope
- Graph a line using its slope and intercept
- Choose the most convenient method to graph a line
- Graph and interpret applications of slope–intercept
- Use slopes to identify parallel and perpendicular lines
Find the Slope of a Line
When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter.
In mathematics, the measure of the steepness of a line is called the slope of the line.
The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing pipes, and the steepness of the stairs are all applications of slope. As you ski or jog down a hill, you are experiencing slope.
We can assign a numerical value to the slope of a line by finding the ratio of the rise to the run. The rise measures the vertical change, while the run measures the horizontal change as you move along the line from one point to another. This is shown in the illustration below. Slope is a rate of change.
Slope of a Line
The slope of a line is [latex]m=\frac{rise}{run}[/latex].
The rise measures the vertical change, and the run measures the horizontal change.
To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise, and the horizontal distance is called the run.
Find the slope of a line from its graph using
[latex]m=\frac{rise}{run}[/latex].
- Locate two points on the line whose coordinates are integers.
- Starting with one point, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope: [latex]m=\frac{rise}{run}[/latex].
Example 1
Find the slope of the line shown.
| Locate two points on the graph whose
coordinates are integers. |
[latex](0,5)[/latex] and [latex](3,3)[/latex] |
| Starting at [latex](0,5)[/latex], sketch a right triangle to
[latex](3,3)[/latex] as shown in this graph. |
|
| Count the rise—since it goes down, it is negative. | The rise is [latex]-2[/latex] |
| Count the run. | The run is [latex]3[/latex]. |
| Use the slope formula. | [latex]m=\frac{rise}{run}[/latex] |
| Substitute the values of the rise and run. | [latex]m=\frac{-2}{3}[/latex] |
| Simplify. | [latex]m=\frac{-2}{3}[/latex] |
| The slope of the line is [latex]\frac{-2}{3}[/latex]. | |
| So y decreases by 2 units as x increases by 3 units. |
Exercise 1
How do we find the slope of horizontal and vertical lines? To find th