3.9: Parallel Lines in the Coordinate Plane
- Page ID
- 4771
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Lines with the same slope that never intersect.
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