3.9: Parallel Lines in the Coordinate Plane
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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, the slopes are both \dfrac{2}{3}.
This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different y−intercepts.
What if you were given two parallel lines in the coordinate plane? What could you say about their slopes?
Video
Example \PageIndex{1}
Find the equation of the line that is parallel to y=\dfrac{1}{4}x+3 and passes through (8, -7).
Solution
We know that parallel lines have the same slope, so the line will have a slope of dfrac{1}{4}. Now, we need to find the y−intercept. Plug in 8 for x and -7 for y to solve for the new y−intercept (b).
\begin{align*}−7=\dfrac{1}{4} (8)+b \\ −7&=2+b \\ −9 &=b \end{align*}
The equation of the parallel line is y=\dfrac{1}{4}x−9.
Example \PageIndex{2}
Are the lines 3x+4y=7 and y=\dfrac{3}{4} x+1 parallel?
Solution
First we need to rewrite the first equation in slope-intercept form.
\begin{align*}3x+4y &=7 \\ 4y &=−3x+7 \\ y &=−\dfrac{3}{4}x+\dfrac{7}{4} \end{align*}.
The slope of this line is −\dfrac{3}{4} while the slope of the other line is \dfrac{3}{4}. Because the slopes are different the lines are not parallel.
Example \PageIndex{3}
Find the equation of the line that is parallel to y=−\dfrac{1}{3} x+4 and passes through (9, -5).
Solution
Recall that the equation of a line is y=mx+b, where m is the slope and b is the y−intercept. We know that parallel lines have the same slope, so the line will have a slope of −13. Now, we need to find the y−intercept. Plug in 9 for x and -5 for y to solve for the new y−intercept (b).
\begin{align*}−5&=−\dfrac{1}{3} (9)+b \\ −5&=−3+b \\−2&=b\end{align*}
The equation of parallel line is y=−\dfrac{1}{3} x−2.
Example \PageIndex{4}
Find the equation of the lines below and determine if they are parallel.

Solution
The top line has a y−intercept of 1. From there, use “rise over run” to find the slope. From the y−intercept, if you go up 1 and over 2, you hit the line again, m=\dfrac{1}{2}. The equation is y=\dfrac{1}{2} x+1.
For the second line, the y−intercept is -3. The “rise” is 1 and the “run” is 2 making the slope \dfrac{1}{2}. The equation of this line is y=\dfrac{1}{2} x−3.
The lines are parallel because they have the same slope.
Example \PageIndex{5}
Find the equation of the line that is parallel to the line through the point marked with a blue dot.

Solution
First, notice that the equation of the line is y=2x+6 and the point is (2, -2). The parallel would have the same slope and pass through (2, -2).
\begin{align*}y &=2x+b \\ −2 &=2(2)+b \\ −2&=4+b \\−6&=b \end{align*}
The equation of the parallel line is y=2x+−6.
Review
Determine if each pair of lines are parallel. Then, graph each pair on the same set of axes.
- y=4x−2 and y=4x+5
- y=−x+5 and y=x+1
- 5x+2y=−4 and 5x+2y=8
- x+y=6 and 4x+4y=−16
Determine the equation of the line that is parallel to the given line, through the given point.
- y=−5x+1; (−2,3)
- y=\dfrac{2}{3}x−2; (9,1)
- x−4y=12; (−16,−2)
- 3x+2y=10; (8,−11)
Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.
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Figure \PageIndex{4}
For the line and point below, find a parallel line, through the given point.
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Figure \PageIndex{5} -
Figure \PageIndex{6} -
Figure \PageIndex{7} -
Figure \PageIndex{8}
Review (Answers)
To see the Review answers, open this PDF file and look for section 3.8.
Vocabulary
Term | Definition |
---|---|
Parallel | Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope. |
Additional Resource
Interactive Element
Video: Equations of Parallel and Perpendicular Lines
Activities: Parallel Lines in the Coordinate Plane Discussion Questions
Study Aids: Lines in the Coordinate Plane
Practice: Parallel Lines in the Coordinate Plane
Real World: Parallel Lines In The Coordiante Plane