Parallel and Perpendicular Lines
Writing the Equation of a Line Parallel to a Given Line
Parallel lines have a few important characteristics: they are included in the same plane, they do not intersect, and they have equal slopes. As you work through the problems in this interactivity, you will learn how to write the equation of a line that is parallel to a given line and that passes through a given point. Click the player button to begin.
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Writing the Equation of a Line Perpendicular to a Given Line
There are a few important attributes that make perpendicular lines unique: they intersect to form right angles, their slopes have opposite reciprocal values, and the product of their slopes is −1. In this interactivity, you will learn how to write the equation of a line that is perpendicular to a given line and that passes through a given point. Click the player button to begin.
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Alternate Strategy to Writing the Equations of Parallel and Perpendicular Lines
It is possible to use the point-slope form of a linear equation in the process of writing an equation in slope-intercept form.
Example: Write an equation in slope-intercept form of the line that passes through the point (1, 2) and is parallel to the line y = 7x + 8.
Step 1: Determine the slope m of the parallel line.
The slope of a line parallel to the line y = 7x + 8 is 7.
Step 2: Use the slope and the given point to write a linear equation.
Begin by writing an equation in point-slope form. Let m = 7 and (x1, y1 ) = (1, 2).
| y − y1 = m (x − x1) | |
y − 2 = 7 (x − 1) |
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Now, use the distributive property and inverse operations to write the equation in slope-intercept form.
| y − y1 = m (x − x1) | ||
| y − 2 = 7 (x − 1) | Distributive Property |
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| y − 2 = 7x − 7 |
| +2 | +2 | Add 2 to both sides |
| _____________________ |
| y = 7x − 5 |
The slope-intercept form of the equation of the line that passes through the point (1, 2) and is parallel to the line y = 7x + 8 is y = 7x − 5.
This process can be extended to write the equations of perpendicular lines, as well. How would Step 1 be different if you were asked to write the equation of a perpendicular line instead of a parallel line?
Horizontal and Vertical Lines
In the cases of horizontal and vertical lines, there are a few important things to remember.
- A horizontal line is parallel to a horizontal line.
- A vertical line is parallel to a vertical line.
- A horizontal line is perpendicular to a vertical line.
Example: Write the equation of a line that is parallel to the line x = 4 and passes through the point (1, 5).
The line x = 4 is vertical. So a line parallel to x = 4 will also be vertical.
Recall that the equation of a vertical line is defined by its constant x-value. Because the parallel line passes through the point (1, 5), you can conclude that the constant x-value is 1.
Therefore, the equation of the parallel line is x = 1.
Example: Write the equation of a line that is perpendicular to x = 4 and passes through the point (1, 5).
As stated earlier, the line x = 4 is vertical. So, a line perpendicular to x = 4 will be horizontal.
The equation of a horizontal line is defined by its constant y-value. Because the perpendicular line passes through the point (1, 5), you can conclude that the constant y-value is 5.
Therefore, the equation of the perpendicular line is y = 5.
Practical Problem
Example: The total cost of membership at a local gym can be modeled by the equation y = 50x + 35, where x represents the length of membership in months and y represents the total cost.
A competing gym charges members the same monthly rate, as well as a one-time fee of $25. Write an equation to model the total cost of membership at the competing gym.
At the competing gym, members are charged a constant monthly rate of $50, in addition to a one-time fee of $25. Therefore, the membership cost can be modeled by the equation y = 50x + 25.
Do the equations included in this scenario represent parallel lines, perpendicular lines, or neither?
Parallel and Perpendicular Lines Review
Now that you have learned about parallel and perpendicular lines, it is time to test your knowledge. This interactivity will help you review the information covered throughout this topic. Click the player button to get started.
Did you answer the content review questions incorrectly? Do you want more instruction or extra practice? If so, view the video Writing the Equation of Parallel and Perpendicular Lines from eMediaVASM.