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# Families of Lines

## Lines sharing a point or slope

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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\begin{align*}y-\end{align*}intercept. The slope tells us how steeply the line rises or falls, and the y\begin{align*}y-\end{align*}intercept tells us where the line intersects the y\begin{align*}y-\end{align*}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\begin{align*}y-\end{align*}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\begin{align*}y-\end{align*}intercepts. All the lines look the same but they are shifted up and down the y\begin{align*}y-\end{align*}axis as shown in the graph below. As b\begin{align*}b\end{align*} gets larger the line rises on the y\begin{align*}y-\end{align*}axis and as b\begin{align*}b\end{align*} gets smaller the line goes lower on the y\begin{align*}y-\end{align*}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\begin{align*}y\end{align*}-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\begin{align*}-2\end{align*}. Therefore, the equation of the red line is:

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

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

The graph below shows several lines with the same y\begin{align*}y-\end{align*}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\begin{align*}y\end{align*}-intercept, which is 2. Looking at the graph, the slope is -1. Thus, the equation is:

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

#### 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.\begin{align*}y=-2x+1.\end{align*} Since all the lines share the same slope, we keep the slope of -2. But they all have different y\begin{align*}y\end{align*}-intercepts, so we will use b\begin{align*}b\end{align*}:

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

1. For family 2, the brown line has the equation y=x+2.\begin{align*}y=-x+2.\end{align*} Since all the lines share the same y\begin{align*}y\end{align*}-intercept but have different slopes:

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

### 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 y\begin{align*}y-\end{align*}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\begin{align*}y-\end{align*}intercept and slope in common, just one.

#### Example 2

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

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

slope=62=3.\begin{align*}slope=-\frac{6}{2}=-3.\end{align*}

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

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

3m3=13\begin{align*}\frac{-3\cdot m}{-3}=\frac{-1}{-3}\end{align*}

m=13\begin{align*} m=\frac{1}{3}\end{align*}

The family of lines perpendicular to 6x+2y=24\begin{align*}6x+2y=24\end{align*} will have a slope of m=13\begin{align*} m=\frac{1}{3}\end{align*}. They will all have different y\begin{align*}y\end{align*}-intercepts:

y=13x+b.\begin{align*}y=\frac{1}{3}x+b.\end{align*}

### Review

1. What is a family of lines?
2. Find the equation of the line parallel to 5x2y=2\begin{align*}5x-2y=2\end{align*} that passes through the point (3, –2).
3. Find the equation of the line perpendicular to y=25x3\begin{align*}y=-\frac{2}{5}x-3\end{align*} that passes through the point (2, 8).
4. Find the equation of the line parallel to 7y+2x10=0\begin{align*}7y+2x-10=0\end{align*} that passes through the point (2, 2).
5. Find the equation of the line perpendicular to y+5=3(x2)\begin{align*}y+5=3(x-2)\end{align*} that passes through the point (6, 2).
6. Find the equation of the line through (2, –4) perpendicular to y=27x+3\begin{align*}y=\frac{2}{7} x+3\end{align*}.
7. Find the equation of the line through (2, 3) parallel to y=32x+5\begin{align*}y=\frac{3}{2} x+5\end{align*}.

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+3y1=0\begin{align*}4x+3y-1=0\end{align*}.
3. All lines are parallel to y3=4x+2\begin{align*}y-3=4x+2\end{align*}.
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 43\begin{align*}\frac{4}{3}\end{align*} and a y\begin{align*}y-\end{align*}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=65x+11\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?

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

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### Vocabulary Language: English Spanish

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.