# 4.6.4: Families of Lines


## 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.

1. 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.

1. 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

1. What is a family of lines?
2. Find the equation of the line parallel to 5x−2y=2 that passes through the point (3, –2).
3. Find the equation of the line perpendicular to y=−25x−3 that passes through the point (2, 8).
4. Find the equation of the line parallel to 7y+2x−10=0 that passes through the point (2, 2).
5. Find the equation of the line perpendicular to y+5=3(x−2) that passes through the point (6, 2).
6. Find the equation of the line through (2, –4) perpendicular to y=2/7x+3.
7. 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.

1. All lines pass through point (0, 4).
2. All lines are perpendicular to 4x+3y−1=0.
3. All lines are parallel to y−3=4x+2.
4. All lines pass through point (0, –1).
5. Write an equation for a line parallel to the equation graphed below.
6. 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?

### 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 vertical shift. As b gets larger, the line rises on the y−axis, and as b gets smaller the line goes lower on the y−axis.

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

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