3.8: Parallel Lines in the Coordinate Plane
What if you wanted to figure out if two lines in a shape were truly parallel? How could you do this? After completing this Concept, you'll be able to use slope to help you to determine whether or not lines are parallel.
Watch This
CK12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneA
Watch the portion of this video that deals with Parallel Lines
Khan Academy: Equations of Parallel and Perpendicular Lines
Guidance
Recall that parallel lines are two lines that never intersect. In the coordinate plane, that would look like this:
If we take a closer look at these two lines, we see that the slopes of both are \begin{align*}\frac{2}{3}\end{align*}
This can be generalized to any pair of parallel lines in the coordinate plane. Parallel lines have the same slope.
Example A
Find the equation of the line that is parallel to \begin{align*}y=\frac{1}{3}x+4\end{align*}
Recall that the equation of a line in this form is called the slopeintercept form and is written as \begin{align*}y = mx + b\end{align*}
We know that parallel lines have the same slope, so the line we are trying to find also has \begin{align*}m=\frac{1}{3}\end{align*}
\begin{align*}5 & = \frac{1}{3}(9)+b\\
5 & = 3 + b \qquad \text{Therefore, the equation of line is} \ y=\frac{1}{3}x2.\\
2 & = b\end{align*}
Example B
Which of the following pairs of lines are parallel?

\begin{align*}y=2x+3\end{align*}
y=−2x+3 and \begin{align*}y=\frac{1}{2}x+3\end{align*}y=12x+3

\begin{align*}y=4x2\end{align*}
y=4x−2 and \begin{align*}y=4x+5\end{align*}y=4x+5

\begin{align*}y=x+5\end{align*}
y=−x+5 and \begin{align*}y=x+1\end{align*}y=x+1
Because all the equations are in \begin{align*}y=mx+b\end{align*}
Example C
Find the equation of the line that is parallel to the line through the point marked with a blue dot.
First, notice that the equation of the line is \begin{align*}y=2x+6\end{align*}
\begin{align*}y & =2x+b\\
2 & = 2(2) + b\\
2 & = 4+b\\
6 & = b\end{align*}
The equation of the parallel line is \begin{align*}y=2x+6\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneB
Vocabulary
Two lines in the coordinate plane with the same slope are parallel and never intersect. Slope measures the steepness of a line.
Guided Practice
1. Which of the following pairs of lines are parallel?

\begin{align*}y=3x+1\end{align*}
y=−3x+1 and \begin{align*}y=3x1\end{align*}y=3x−1

\begin{align*}2x3y=6\end{align*}
2x−3y=6 and \begin{align*}3x+2y=6\end{align*}3x+2y=6

\begin{align*}5x+2y=4\end{align*}
5x+2y=−4 and \begin{align*}5x+2y=8\end{align*}5x+2y=8

\begin{align*}x3y=3\end{align*}
x−3y=−3 and \begin{align*}x+3y=9\end{align*}x+3y=9

\begin{align*}x+y=6\end{align*}
x+y=6 and \begin{align*}4x+4y=16\end{align*}4x+4y=−16
2. Find the equation of the line that is parallel to \begin{align*}y=\frac{1}{4}x+3\end{align*}
3. Find the equation of the lines below and determine if they are parallel.
Answers:
1. First change all equations into \begin{align*}y=mx+b\end{align*}
2. We know that parallel lines have the same slope, so the line will have a slope of \begin{align*}\frac{1}{4}\end{align*}
\begin{align*}7 & = \frac{1}{4}(8)+b\\ 7 & = 2 + b\\ 9 & = b\end{align*}
The equation of the parallel line is \begin{align*}y = \frac{1}{4}x9\end{align*}.
3. The top line has a \begin{align*}y\end{align*}intercept of 1. From there, use “rise over run” to find the slope. From the \begin{align*}y\end{align*}intercept, if you go up 1 and over 2, you hit the line again, \begin{align*}m = \frac{1}{2}\end{align*}. The equation is \begin{align*}y=\frac{1}{2}x+1\end{align*}.
For the second line, the \begin{align*}y\end{align*}intercept is 3. The “rise” is 1 and the “run” is 2 making the slope \begin{align*}\frac{1}{2}\end{align*}. The equation of this line is \begin{align*}y=\frac{1}{2}x3\end{align*}.
The lines are parallel because they have the same slope.
Practice
Determine the equation of the line that is parallel to the given line, through the given point.
 \begin{align*}y=5x+1; \ (2, \ 3)\end{align*}
 \begin{align*}y=\frac{2}{3}x2; \ (9, 1)\end{align*}
 \begin{align*}x4y=12; \ (16, \ 2)\end{align*}
 \begin{align*}3x+2y=10; \ (8, \ 11)\end{align*}
 \begin{align*}2x  y = 15; \ (3, \ 7)\end{align*}
 \begin{align*}y = x  5; \ (9, \ 1)\end{align*}
 \begin{align*}y = 3x  4; \ (2, \ 3)\end{align*}
Then, determine if the two lines are parallel or not.
For the line and point below, find a parallel line, through the given point.
Image Attributions
Here you'll learn properties of parallel lines in the coordinate plane, and how slope can help you to determine whether or not two lines are parallel.