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# 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:

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.

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.

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

## 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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