# 4.6.4: Families of Lines

- Page ID
- 4324

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Families of Lines

Think about the members of your family. You probably all have some things in common, but you're definitely not all identical. The same is true of a family of lines. What could a family of lines have in common? What might be different?

**Families of Lines**

A straight line has two very important properties, its slope and its y−intercept. The slope tells us how steeply the line rises or falls, and the y−intercept tells us where the line intersects the y−axis. In this Concept, we will look at two 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: where the slope is the same and where the y−intercept is the same.

**Family 1: The slope is the same**

Remember that lines with the same slope are parallel. Each line on the Cartesian plane below has an identical slope with different y−intercepts. All the lines look the same but they are shifted up and down the y−axis as shown in the graph below. As b gets larger the line rises on the y−axis and as b gets smaller the line goes lower on the y−axis. This behavior is often called a **vertical shift**.

**Let's write the equation for the red line in the image above:**

We can see from the graph that the equation has a y-intercept of 1. Since all the lines have the same slope, we can look at any line to determine the slope, so the slope is −2. Therefore, the equation of the red line is:

y=−2x+1.

**Family 2: The y−intercept is the same**

The graph below shows several lines with the same y−intercept but with varying slopes.

**Let's write the equation for the brown line in the image above:**

All the lines share the same y-intercept, which is 2. Looking at the graph, the slope is -1. Thus, the equation is:

y=−x+2.

**Now, let's write a general equation for each family of lines shown in the images in this Concept.**

- For family 1, the red line has the equation y=−2x+1. Since all the lines share the same slope, we keep the slope of -2. But they all have different y-intercepts, so we will use b:

y=−2x+b.

- For family 2, the brown line has the equation y=−x+2. Since all the lines share the same y-intercept but have different slopes:

y=mx+2.

**Examples**

Example 4.6.4.1

Earlier, you were asked what a family of lines could have in common and what could be different.

**Solution**

As shown in this concept, there are two important parts of a line, the y−intercept and the slope, that a family of lines can have in common. A family of lines does not need to have both the y−intercept and slope in common, just one.

Example 4.6.4.2

Write the equation of the family of lines perpendicular to 6x+2y=24.

**Solution**

First we must find the slope of 6x+2y=24:

slope=−6/2=−3.

Now we find the slope of any line perpendicular to our original line:

−3⋅m=−1

(−3⋅m)/−3=−1/−3

m=1/3

The family of lines perpendicular to 6x+2y=24 will have a slope of m=1/3. They will all have different y-intercepts:

y=1/3x+b.

**Review**

- What is a family of lines?
- Find the equation of the line parallel to 5x−2y=2 that passes through the point (3, –2).
- Find the equation of the line perpendicular to y=−25x−3 that passes through the 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).
- Find the equation of the line through (2, –4) perpendicular to y=2/7x+3.
- Find the equation of the line through (2, 3) parallel to y=3/2x+5.

In 8–11, write the equation of the family of lines satisfying the given condition.

- All lines pass through point (0, 4).
- All lines are perpendicular to 4x+3y−1=0.
- All lines are parallel to y−3=4x+2.
- All lines pass through point (0, –1).
- Write an equation for a line parallel to the equation graphed below.
- Write an equation for a line perpendicular to the equation graphed below and passing through the point (0, –1).

**Quick Quiz**

1. Write an equation for a line with a slope of 4/3 and a y−intercept of (0, 8).

2. Write an equation for a line containing (6, 1) and (7, –3).

3. A plumber charges $75 for a 2.5-hour job and $168.75 for a 5-hour job.

Assuming the situation is linear, write an equation to represent the plumber’s charge and use it to predict the cost of a 1-hour job.

4. Rewrite in standard form: y=6/5x+11.

5. Sasha took tickets for the softball game. Student tickets were $3.00 and adult tickets were $3.75. She collected a total of $337.50 and sold 75 student tickets. How many adult tickets were sold?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.9.

### Vocabulary

Term | Definition |
---|---|

vertical shifts |
When all lines look the same but they are shifted up and down the y−axis, this behavior is called a . As b gets larger, the line rises on the y−axis, and as b gets smaller the line goes lower on the y−axis.vertical shift |

### Additional Resources

PLIX: Play, Learn, Interact, eXplore: **Families of Lines: Room in Perspective**

Video:

Activity: **Families of Lines Discussion Questions**

Practice: **Families of Lines**

Real World Application: **Parallel & Perpendicular Lines**