# 5.5: Families of Lines

**At Grade**Created by: CK-12

**Practice**Families of Lines

What if you were given the equation of a line like

### 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:

- 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.)
- 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).
- 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

Now to find the equation of a line with slope

Start with the slope-intercept form:

Plug in the slope:

Plug in the point (2, 6) to find

**The equation of the line is**

#### Example B

Find the equation of the line parallel to

**Solution**

Rewrite the equation in slope-intercept form:

The slope of the given line is

Start with the slope-intercept form:

Plug in the slope:

Plug in the point (-5, -3):

**The equation of the line is**

**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

**Family 1:** Keep the slope unchanged and vary the

The figure below shows the family of lines with equations of the form

All the lines have a slope of –2, but the value of

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 **vertical shift.**

Family 2: Keep the

The figure below shows the family of lines with equations of the form

All the lines have a

#### Example C

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

a) parallel to the

b) through the point (0, -1)

c) perpendicular to

d) parallel to

**Solution**

a) All lines parallel to the

b) All lines passing through the point (0, -1) have the same

c) First we need to find the slope of the given line. Rewriting

**The family of lines is**

d) Rewrite

**The family of lines is**

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 they– intercept 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
y− axis. Asb gets larger the line rises on they− axis, and asb gets smaller the line goes lower on they− axis. This behavior is often called a**vertical shift.**

### Guided Practice

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

**Solution**

Re-write the equation in slope-intercept form:

The slope of the given line is

Start with the slope-intercept form:

Plug in the slope:

Plug in the point (-2, 5):

**The equation of the line is**

### Practice

- Find the equation of the line parallel to
5x−2y=2 that passes through point (3, -2). - Find the equation of the line perpendicular to
y=−25x−3 that passes through point (2, 8). - Find the equation of the line parallel to
7y+2x−10=0 that passes through the point (2, 2). - Find the equation of the line perpendicular to
y+5=3(x−2) that passes through the point (6, 2). - Line
S passes through the points (2, 3) and (4, 7). LineT passes through the point (2, 5). If LinesS andT are parallel, name one more point on lineT . (**Hint:**you don’t need to find the slope of either line.) - Lines
P andQ both pass through (-1, 5). LineP also passes through (-3, -1). IfP andQ are perpendicular, name one more point on lineQ . (This time you will have to find the slopes of both lines.) - Write the equation of the family of lines satisfying the given condition.
- All lines that pass through point (0, 4).
- All lines that are perpendicular to
4x+3y−1=0 . - All lines that are parallel to
y−3=4x+2 . - All lines that pass through the point (0, -1).

- Name two lines that pass through the point (3, -1) and are perpendicular to each other.
- Name two lines that are each perpendicular to
y=−4x−2 . What is the relationship of those two lines to each other? - 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).

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### Image Attributions

Here you'll learn how to write the equation of a line that is parallel or perpendicular to a second line given that second line's equation and one of the points it passes through. You'll also investigate families of lines.