<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

5.5: Families of Lines

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated7 minsto complete
%
Progress
Practice Families of Lines
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated7 minsto complete
%
Estimated7 minsto complete
%
Practice Now
MEMORY METER
This indicates how strong in your memory this concept is
Turn In

What if you were given the equation of a line like y=4x3 and you wanted to find the equation of a line that is parallel or perpendicular to it that passes through the point (2, 1). How could you find the equation of this line? After completing this Concept, you'll be able to write equations of perpendicular and parallel lines.

Watch This

CK-12 Foundation: 0505S Equations of Parallel and Perpendicular Lines (H264)

Guidance

We can use the properties of parallel and perpendicular lines to write an equation of a line parallel or perpendicular to a given line. You might be given a line and a point, and asked to find the line that goes through the given point and is parallel or perpendicular to the given line. Here’s how to do this:

  1. Find the slope of the given line from its equation. (You might need to re-write the equation in a form such as the slope-intercept form.)
  2. Find the slope of the parallel or perpendicular line—which is either the same as the slope you found in step 1 (if it’s parallel), or the negative reciprocal of the slope you found in step 1 (if it’s perpendicular).
  3. Use the slope you found in step 2, along with the point you were given, to write an equation of the new line in slope-intercept form or point-slope form.

Example A

Find an equation of the line perpendicular to the line y=3x+5 that passes through the point (2, 6).

Solution

The slope of the given line is -3, so the perpendicular line will have a slope of 13.

Now to find the equation of a line with slope 13 that passes through (2, 6):

Start with the slope-intercept form: y=mx+b.

Plug in the slope: y=13x+b.

Plug in the point (2, 6) to find b: 6=13(2)+bb=623b=163513.

The equation of the line is y=13x+513.

Example B

Find the equation of the line parallel to 6x5y=12 that passes through the point (-5, -3).

Solution

Rewrite the equation in slope-intercept form: 6x5y=125y=6x12y=65x125.

The slope of the given line is 65, so we are looking for a line with slope 65 that passes through the point (-5, -3).

Start with the slope-intercept form: y=mx+b.

Plug in the slope: y=65x+b.

Plug in the point (-5, -3): n3=65(5)+b3=6+bb=3

The equation of the line is y=65x+3.

Investigate Families of Lines

A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: one where the slope is the same and one where the yintercept is the same.

Family 1: Keep the slope unchanged and vary the yintercept.

The figure below shows the family of lines with equations of the form y=2x+b:

All the lines have a slope of –2, but the value of b is different for each line.

Notice that in such a family all the lines are parallel. All the lines look the same, except that they are shifted up and down the yaxis. As b gets larger the line rises on the yaxis, and as b gets smaller the line goes lower on the yaxis. This behavior is often called a vertical shift.

Family 2: Keep the yintercept unchanged and vary the slope.

The figure below shows the family of lines with equations of the form y=mx+2:

All the lines have a yintercept of two, but the slope is different for each line. The steeper lines have higher values of m.

Example C

Write the equation of the family of lines satisfying the given condition.

a) parallel to the xaxis

b) through the point (0, -1)

c) perpendicular to 2x+7y9=0

d) parallel to x+4y12=0

Solution

a) All lines parallel to the xaxis have a slope of zero; the yintercept can be anything. So the family of lines is y=0x+b or just y=b.

b) All lines passing through the point (0, -1) have the same yintercept, b=1. The family of lines is: y=mx1.

c) First we need to find the slope of the given line. Rewriting 2x+7y9=0 in slope-intercept form, we get y=27x+97. The slope of the line is 27, so we’re looking for the family of lines with slope 72.

The family of lines is y=72x+b.

d) Rewrite x+4y12=0 in slope-intercept form: y=14x+3. The slope is 14, so that’s also the slope of the family of lines we are looking for.

The family of lines is y=14x+b.

Watch this video for help with the Examples above.

CK-12 Foundation: Equations of Parallel and Perpendicular Lines

Vocabulary

  • A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: one where the slope is the same and one where the yintercept is the same.
  • Notice that in such a family all the lines are parallel. All the lines look the same, except that they are shifted up and down the yaxis. As b gets larger the line rises on the yaxis, and as b gets smaller the line goes lower on the yaxis. This behavior is often called a vertical shift.

Guided Practice

Find the equation of the line perpendicular to x5y=15 that passes through the point (-2, 5).

Solution

Re-write the equation in slope-intercept form: x5y=155y=x+15y=15x3.

The slope of the given line is 15, so we’re looking for a line with slope -5.

Start with the slope-intercept form: y=mx+b.

Plug in the slope: y=5x+b.

Plug in the point (-2, 5): 5=5(2)+bb=510b=5

The equation of the line is y=5x5.

Practice

  1. Find the equation of the line parallel to 5x2y=2 that passes through point (3, -2).
  2. Find the equation of the line perpendicular to y=25x3 that passes through point (2, 8).
  3. Find the equation of the line parallel to 7y+2x10=0 that passes through the point (2, 2).
  4. Find the equation of the line perpendicular to y+5=3(x2) that passes through the point (6, 2).
  5. Line S passes through the points (2, 3) and (4, 7). Line T passes through the point (2, 5). If Lines S and T are parallel, name one more point on line T. (Hint: you don’t need to find the slope of either line.)
  6. Lines P and Q both pass through (-1, 5). Line P also passes through (-3, -1). If P and Q are perpendicular, name one more point on line Q. (This time you will have to find the slopes of both lines.)
  7. Write the equation of the family of lines satisfying the given condition.
    1. All lines that pass through point (0, 4).
    2. All lines that are perpendicular to 4x+3y1=0.
    3. All lines that are parallel to y3=4x+2.
    4. All lines that pass through the point (0, -1).
  8. Name two lines that pass through the point (3, -1) and are perpendicular to each other.
  9. Name two lines that are each perpendicular to y=4x2. What is the relationship of those two lines to each other?
  10. Name two perpendicular lines that both pass through the point (3, -2). Then name a line parallel to one of them that passes through the point (-2, 5).

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Grades:
Date Created:
Aug 13, 2012
Last Modified:
Apr 11, 2016
Files can only be attached to the latest version of Modality
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.484.3.L.2
Here