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3.9: Parallel Lines in the Coordinate Plane

  • Page ID
    4771
  • 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:

    f-d_3ceeb0ba9414cc883c09344f3559186ae97bd2434e1d67cdfe4483d5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    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.

    f-d_38dddf73576d1db57503d88e4e7d53ef9e5a222daa943706d3ed39c8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    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.

    f-d_f0bd89659d33ec8f1e5b8818076c5def893454eb65bf21b79a0d1ad5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    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.

    1. \(y=4x−2\) and \(y=4x+5\)
    2. \(y=−x+5\) and \(y=x+1\)
    3. \(5x+2y=−4\) and \(5x+2y=8\)
    4. \(x+y=6\) and \(4x+4y=−16\)

    Determine the equation of the line that is parallel to the given line, through the given point.

    1. \(y=−5x+1; (−2,3)\)
    2. \(y=\dfrac{2}{3}x−2; (9,1)\)
    3. \(x−4y=12; (−16,−2)\)
    4. \(3x+2y=10; (8,−11)\)

    Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.

    1. f-d_a1a4be37aa1e1410f7d2e1122e65bc5e45db8453bddb4eff9d1051c6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{4}\)

    For the line and point below, find a parallel line, through the given point.

    1. f-d_05a25757fee5b331bb5c3c6a874b9e06df0d5e28b072d3b885499076+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{5}\)
    2. f-d_6f77419f1c33f04c22afd960cb80d7aff8a8c298d62bb94b844796bc+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{6}\)
    3. f-d_60c42e2eece766976d0364099752bac90aa3e8f86b1ab9a38f73a90d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    4. f-d_aeb339745cd70930d568b3219b8f87fdac0579c4efc1828c2e047e57+IMAGE_TINY+IMAGE_TINY.png
      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

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