# 5.9: Families of Lines

**Basic**Created by: CK-12

**Practice**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? In this Concept, you'll learn about two types of families of lines and how to write general equations for each type of family.

### Guidance

A straight line has two very important properties, its **slope** and its \begin{align*}y-\end{align*}**intercept**. The slope tells us how steeply the line rises or falls, and the \begin{align*}y-\end{align*}

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 \begin{align*}y-\end{align*}

**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 \begin{align*}y-\end{align*}**vertical shift**.

#### Example A

*Write the equation for the red line in the image above.*

**Solution:**

We can see from the graph that the equation has a \begin{align*}y\end{align*}

\begin{align*}y=-2x+1.\end{align*}

**Family 2: The \begin{align*}y-\end{align*} y−intercept is the same**

The graph below shows several lines with the same \begin{align*}y-\end{align*}

#### Example B

*Write the equation for the brown line in the image above.*

**Solution:**

All the lines share the same \begin{align*}y\end{align*}

\begin{align*}y=-x+2.\end{align*}

#### Example C

*Write a general equation for each family of lines shown in the images in this Concept.*

**Solutions:**

For family 1, the red line has the equation \begin{align*}y=-2x+1.\end{align*}

\begin{align*}y=-2x+b.\end{align*}

For family 2, the brown line has the equation \begin{align*}y=-x+2.\end{align*}

\begin{align*}y=mx+2.\end{align*}

### Video Review

### Guided Practice

*Write the equation of the family of lines perpendicular to \begin{align*}6x+2y=24\end{align*} 6x+2y=24.*

**Solution:**

First we must find the slope of \begin{align*}6x+2y=24\end{align*}

\begin{align*}slope=-\frac{6}{2}=-3.\end{align*}

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

\begin{align*}-3\cdot m=-1\end{align*}

\begin{align*}\frac{-3\cdot m}{-3}=\frac{-1}{-3}\end{align*}

\begin{align*} m=\frac{1}{3}\end{align*}

The family of lines perpendicular to \begin{align*}6x+2y=24\end{align*}

\begin{align*}y=\frac{1}{3}x+b.\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Equations of Parallel and Perpendicular Lines (9:13)

- What is a
*family of lines?* - Find the equation of the line parallel to \begin{align*}5x-2y=2\end{align*}
5x−2y=2 that passes through the point (3, –2). - Find the equation of the line perpendicular to \begin{align*}y=-\frac{2}{5}x-3\end{align*}
y=−25x−3 that passes through the point (2, 8). - Find the equation of the line parallel to \begin{align*}7y+2x-10=0\end{align*}
7y+2x−10=0 that passes through the point (2, 2). - Find the equation of the line perpendicular to \begin{align*}y+5=3(x-2)\end{align*}
y+5=3(x−2) that passes through the point (6, 2). - Find the equation of the line through (2, –4) perpendicular to \begin{align*}y=\frac{2}{7} x+3\end{align*}
y=27x+3 . - Find the equation of the line through (2, 3) parallel to \begin{align*}y=\frac{3}{2} x+5\end{align*}
y=32x+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 \begin{align*}4x+3y-1=0\end{align*}
4x+3y−1=0 . - All lines are parallel to \begin{align*}y-3=4x+2\end{align*}
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 \begin{align*}\frac{4}{3}\end{align*}

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: \begin{align*}y=\frac{6}{5} x+11\end{align*}

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?

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### Image Attributions

Here you'll learn about two types of families of lines and how to write general equations for each.