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Families of Lines

Lines sharing a point or slope

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Practice Families of Lines
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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? 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*}intercept tells us where the line intersects the \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 \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 \begin{align*}y-\end{align*}intercepts. All the lines look the same but they are shifted up and down the \begin{align*}y-\end{align*}axis. As \begin{align*}b\end{align*} gets larger the line rises on the \begin{align*}y-\end{align*}axis and as \begin{align*}b\end{align*} gets smaller the line goes lower on the \begin{align*}y-\end{align*}axis. This behavior is often called a 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*}-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 \begin{align*}-2\end{align*}. Therefore, the equation of the red line is:

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

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

The graph below shows several lines with the same \begin{align*}y-\end{align*}intercept but with varying slopes.

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*}-intercept, which is 2. Looking at the graph, the slope is -1. Thus, the equation is:

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

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

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Guided Practice

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

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*} will have a slope of \begin{align*} m=\frac{1}{3}\end{align*}. They will all have different \begin{align*}y\end{align*}-intercepts:

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

Explore More

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)

1. What is a family of lines?
2. Find the equation of the line parallel to \begin{align*}5x-2y=2\end{align*} that passes through the point (3, –2).
3. Find the equation of the line perpendicular to \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 \begin{align*}7y+2x-10=0\end{align*} that passes through the point (2, 2).
5. Find the equation of the line perpendicular to \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 \begin{align*}y=\frac{2}{7} x+3\end{align*}.
7. Find the equation of the line through (2, 3) parallel to \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 \begin{align*}4x+3y-1=0\end{align*}.
3. All lines are parallel to \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 \begin{align*}\frac{4}{3}\end{align*} and a \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: \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 view the Explore More answers, open this PDF file and look for section 5.9.

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.