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# 3.8: Parallel Lines in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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What if you wanted to figure out if two lines in a shape were truly parallel? How could you do this? After completing this Concept, you'll be able to use slope to help you to determine whether or not lines are parallel.

### Watch This

Watch the portion of this video that deals with Parallel Lines

### Guidance

Recall that 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, we see that the slopes of both are $\frac{2}{3}$ .

This can be generalized to any pair of parallel lines in the coordinate plane. Parallel lines have the same slope.

#### Example A

Find the equation of the line that is parallel to $y=-\frac{1}{3}x+4$ and passes through (9, -5).

Recall that the equation of a line in this form is called the slope-intercept form and is written as $y = mx + b$ where $m$ is the slope and $b$ is the $y-$ intercept. Here, $x$ and $y$ represent any coordinate pair, $(x, \ y)$ on the line.

We know that parallel lines have the same slope, so the line we are trying to find also has $m=-\frac{1}{3}$ . Now, we need to find the $y-$ intercept. 4 is the $y-$ intercept of the given line, not our new line . We need to plug in 9 for $x$ and -5 for $y$ (this is our given coordinate pair that needs to be on the line) to solve for the new $y-$ intercept $(b)$ .

$-5 & = -\frac{1}{3}(9)+b\\-5 & = -3 + b \qquad \text{Therefore, the equation of line is} \ y=-\frac{1}{3}x-2.\\-2 & = b$

#### Example B

Which of the following pairs of lines are parallel?

• $y=-2x+3$ and $y=\frac{1}{2}x+3$
• $y=4x-2$ and $y=4x+5$
• $y=-x+5$ and $y=x+1$

Because all the equations are in $y=mx+b$ form, you can easily compare the slopes by looking at the values of $m$ . To be parallel, the lines must have equal values for $m$ . The second pair of lines is the only one that is parallel.

#### Example C

Find the equation of the line that is parallel to the line through the point marked with a blue dot.

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).

$y & =2x+b\\-2 & = 2(2) + b\\-2 & = 4+b\\-6 & = b$

The equation of the parallel line is $y=2x+-6$ .

Watch this video for help with the Examples above.

### Vocabulary

Two lines in the coordinate plane with the same slope are parallel and never intersect. Slope measures the steepness of a line.

### Guided Practice

1. Which of the following pairs of lines are parallel?

• $y=-3x+1$ and $y=3x-1$
• $2x-3y=6$ and $3x+2y=6$
• $5x+2y=-4$ and $5x+2y=8$
• $x-3y=-3$ and $x+3y=9$
• $x+y=6$ and $4x+4y=-16$

2. Find the equation of the line that is parallel to $y=\frac{1}{4}x+3$ and passes through (8, -7).

3. Find the equation of the lines below and determine if they are parallel.

1. First change all equations into $y=mx+b$ form so that you can easily compare the slopes by looking at the values of $m$ . The third and fifth pair of lines are the only ones that are parallel.

2. We know that parallel lines have the same slope, so the line will have a slope of $\frac{1}{4}$ . Now, we need to find the $y-$ intercept. Plug in 8 for $x$ and -7 for $y$ to solve for the new $y-$ intercept $(b)$ .

$-7 & = \frac{1}{4}(8)+b\\-7 & = 2 + b\\-9 & = b$

The equation of the parallel line is $y = \frac{1}{4}x-9$ .

3. 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, $m = \frac{1}{2}$ . The equation is $y=\frac{1}{2}x+1$ .

For the second line, the $y-$ intercept is -3. The “rise” is 1 and the “run” is 2 making the slope $\frac{1}{2}$ . The equation of this line is $y=\frac{1}{2}x-3$ .

The lines are parallel because they have the same slope.

### Practice

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

1. $y=-5x+1; \ (-2, \ 3)$
2. $y=\frac{2}{3}x-2; \ (9, 1)$
3. $x-4y=12; \ (-16, \ -2)$
4. $3x+2y=10; \ (8, \ -11)$
5. $2x - y = 15; \ (3, \ 7)$
6. $y = x - 5; \ (9, \ -1)$
7. $y = 3x - 4; \ (2, \ -3)$

Then, determine if the two lines are parallel or not.

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

Jul 17, 2012

Jun 11, 2014