5.7: 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*}
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} x4\end{align*}
Solution: The slope of the first line is \begin{align*}\frac{1}{3}\end{align*}
Find the slope of the second equation: \begin{align*}A=3\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=6x9\end{align*}
Solution:
Parallel lines have the same slope, so the slope will be 6. You have a point and the slope, so you can use pointslope form.
\begin{align*}yy_1& =m(xx_1)\\
y4& =6(x+1)\end{align*}
You could rewrite it in slopeintercept 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*}y5=2(x+3)\end{align*}
Solution:
First, we notice that this equation is in pointslope form, so let's use pointslope form to write this equation.
\begin{align*}yy_1=m(xx_1)&& \ \text{Starting with pointslope form}. \\
y1=2(x1)&& \ \text{Substituting in the slope and point}. \\
y1=2x2&& \ \text{Distributing on the left}.\\
y1+1=2x2+1, && \ \text{Rearranging into slopeintercept form}.\\
y=2x1
\end{align*}
Video Review
Guided Practice
Find the equation of the line parallel to the line \begin{align*}2x3y=24\end{align*}
Solution:
Since this is in standard form, we must first find the slope. For \begin{align*}Ax+By=C\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*}yy_1=m(xx_1)&& \ \text{Starting with pointslope form}. \\
y2=\frac{2}{3}(x+6)&& \ \text{Substituting in the slope and point}. \\
y2=4x12&& \ \text{Distributing on the left}.\\
y2+2=4x12+2, && \ \text{Rearranging into slopeintercept form}.\\
y=4x10
\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. CK12 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*}
y=−5x+7 
\begin{align*}2x+8y=9\end{align*}
2x+8y=9 
\begin{align*}x=8\end{align*}
x=8 
\begin{align*}y=4.75\end{align*}
y=−4.75 
\begin{align*}y2= \frac{1}{5}(x+3)\end{align*}
y−2=15(x+3)
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*}
y=−35x+2;(0,−2) 
\begin{align*}5x2y=7; (2,10)\end{align*}
5x−2y=7;(2,−10) 
\begin{align*}x=y; (2,3)\end{align*}
x=y;(2,3) 
\begin{align*}x=5; (2,3) \end{align*}
x=−5;(−2,−3)
Mixed Review
 Graph the equation \begin{align*}2xy=10\end{align*}
2x−y=10 .  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 50word advertisement costs $11.50, what is the cost of a 70word advertisement?
 Simplify \begin{align*}\sqrt{112}\end{align*}
112−−−√ .  Simplify \begin{align*}\sqrt{12^27^2}\end{align*}
122−72−−−−−−−√ .  Is \begin{align*}\sqrt{3}\sqrt{2}\end{align*}
3√−2√ rational, irrational, or neither? Explain your answer.  Solve for \begin{align*}s: \ 15s=6(s+32)\end{align*}
s: 15s=6(s+32) .
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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.