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
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
Recall that the equation of a line in this form is called the slopeintercept form and is written as
We know that parallel lines have the same slope, so the line we are trying to find also has
Example B
Which of the following pairs of lines are parallel?

y=−2x+3 andy=12x+3

y=4x−2 andy=4x+5

y=−x+5 andy=x+1
Because all the equations are in
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
The equation of the parallel line is
Watch this video for help with the Examples above.
CK12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneB
Guided Practice
1. Which of the following pairs of lines are parallel?

y=−3x+1 andy=3x−1

2x−3y=6 and3x+2y=6

5x+2y=−4 and5x+2y=8

x−3y=−3 andx+3y=9

x+y=6 and4x+4y=−16
2. Find the equation of the line that is parallel to
3. Find the equation of the lines below and determine if they are parallel.
Answers:
1. First change all equations into
2. We know that parallel lines have the same slope, so the line will have a slope of
The equation of the parallel line is
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.
Interactive Practice
Explore More
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.