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3.9: Parallel Lines in the Coordinate Plane

3.9: Parallel Lines in the Coordinate Plane

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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 \(y=2x+−6\).