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

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

#### 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 **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

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

The graph below shows several lines with the same

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

All the lines share the same

#### 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 differenty -intercepts, so we will useb :

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

### Examples

#### Example 1

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

As shown in this concept, there are two important parts of a line, the

#### Example 2

Write the equation of the family of lines perpendicular to

First we must find the slope of

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

The family of lines perpendicular to

### 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=27x+3 . - Find the equation of the line through (2, 3) parallel to
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
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

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:

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.