# 5.7: Equations of Parallel Lines

**Basic**Created by: CK-12

**Practice**Equations of Parallel Lines

Suppose a coordinate plane were transposed over the map of a city, and Main Street had the equation \begin{align*}y=2x+5\end{align*}. If Broad Street ran parallel to Main Street, and it passed through the point (3, 14), what would be the equation of Broad Street? How do you know? In this Concept, given the equation of one line, you'll learn how to find the equation of a second line that is parallel to the first line, as long as you know a point that the second line passes through.

### Guidance

In a previous Concept, you learned how to identify parallel lines.

**Parallel lines** have the same slope.

Each of the graphs below have the same slope, which is 2. According to the definition, all these lines are parallel.

#### Example A

Are \begin{align*}y=\frac{1}{3} x-4\end{align*} and \begin{align*}-3x+9y=18\end{align*} parallel?

**Solution:** The slope of the first line is \begin{align*}\frac{1}{3}\end{align*}. Any line parallel to this must also have a slope of \begin{align*}\frac{1}{3}\end{align*}.

Find the slope of the second equation: \begin{align*}A=-3\end{align*} and \begin{align*}B=9\end{align*}.

\begin{align*}slope=\frac{-A}{B}=\frac{3}{9} \rightarrow \frac{1}{3}\end{align*}

These two lines have the same slope so they are parallel.

**Writing Equations of Parallel Lines**

Sometimes, you will asked to write the equation of a line parallel to a given line that goes through a given point. In the following example, you will see how to do this.

#### Example B

*Find the equation parallel to the line \begin{align*}y=6x-9\end{align*} passing through (–1, 4)*.

**Solution:**

Parallel lines have the same slope, so the slope will be 6. You have a point and the slope, so you can use point-slope form.

\begin{align*}y-y_1& =m(x-x_1)\\ y-4& =6(x+1)\end{align*}

You could rewrite it in slope-intercept form:

\begin{align*}y& =6x+6+4\\ y& =6x+10\end{align*}

#### Example C

*Find the equation of the line parallel to the line \begin{align*}y-5=2(x+3)\end{align*} passing through (1, 1)*.

**Solution:**

First, we notice that this equation is in point-slope form, so let's use point-slope form to write this equation.

\begin{align*}y-y_1=m(x-x_1)&& \ \text{Starting with point-slope form}. \\ y-1=2(x-1)&& \ \text{Substituting in the slope and point}. \\ y-1=2x-2&& \ \text{Distributing on the left}.\\ y-1+1=2x-2+1, && \ \text{Rearranging into slope-intercept form}.\\ y=2x-1 \end{align*}

### Guided Practice

*Find the equation of the line parallel to the line \begin{align*}2x-3y=24\end{align*} passing through (2, -6)*.

**Solution:**

Since this is in standard form, we must first find the slope. For \begin{align*}Ax+By=C\end{align*}, recall that the slope is \begin{align*}m=-\frac{A}{B}\end{align*}. Since \begin{align*}A=2\end{align*} and \begin{align*}B=-3\end{align*}:

\begin{align*}m=-\frac{A}{B}=-\frac{2}{-3}=\frac{2}{3}.\end{align*}

Now that we have the slope, we can plug it in:

\begin{align*}y-y_1=m(x-x_1)&& \ \text{Starting with point-slope form}. \\ y-2=\frac{2}{3}(x+6)&& \ \text{Substituting in the slope and point}. \\ y-2=4x-12&& \ \text{Distributing on the left}.\\ y-2+2=4x-12+2, && \ \text{Rearranging into slope-intercept form}.\\ y=4x-10 \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)

- Define
*parallel lines*.

Determine the slope of a line parallel to each line given.

- \begin{align*}y=-5x+7\end{align*}
- \begin{align*}2x+8y=9\end{align*}
- \begin{align*}x=8\end{align*}
- \begin{align*}y=-4.75\end{align*}
- \begin{align*}y-2= \frac{1}{5}(x+3)\end{align*}

For the following equations, find the line parallel to it through the given point.

- \begin{align*}y=-\frac{3}{5}x+2; (0,-2)\end{align*}
- \begin{align*}5x-2y=7; (2,-10)\end{align*}
- \begin{align*}x=y; (2,3)\end{align*}
- \begin{align*}x=-5; (-2,-3) \end{align*}

**Mixed Review**

- Graph the equation \begin{align*}2x-y=10\end{align*}.
- On a model boat, the stack is 8 inches high. The actual stack is 6 feet tall. How tall is the mast on the model if the actual mast is 40 feet tall?
- The amount of money charged for a classified advertisement is directly proportional to the length of the advertisement. If a 50-word advertisement costs $11.50, what is the cost of a 70-word advertisement?
- Simplify \begin{align*}\sqrt{112}\end{align*}.
- Simplify \begin{align*}\sqrt{12^2-7^2}\end{align*}.
- Is \begin{align*}\sqrt{3}-\sqrt{2}\end{align*} rational, irrational, or neither? Explain your answer.
- Solve for \begin{align*}s: \ 15s=6(s+32)\end{align*}.

### Image Attributions

Here you'll learn to tell if two lines are parallel, and given the equation of a line, you'll learn how to find the equation of a second line that is parallel to it, as long as you know a point on the second line.