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

CK-12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneA

Watch the portion of this video that deals with Parallel Lines

Khan Academy: Equations of Parallel and Perpendicular 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

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

Recall that the equation of a line in this form is called the slope-intercept form and is written as

We know that parallel lines have the same slope, so the line we are trying to find also has *not our new line*. We need to plug in 9 for *new*

#### Example B

Which of the following pairs of lines are parallel?

y=−2x+3 andy=12x+3

y=4x−2 andy=4x+5

y=−x+5 andy=x+1

Because all the equations are in **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

The equation of the parallel line is

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneB

### Guided Practice

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

y=−3x+1 andy=3x−1

2x−3y=6 and3x+2y=6

5x+2y=−4 and5x+2y=8

x−3y=−3 andx+3y=9

x+y=6 and4x+4y=−16

2. Find the equation of the line that is parallel to

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

**Answers:**

1. First change all equations into *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 *new*

The equation of the parallel line is

3. The top line has a \begin{align*}y-\end{align*}intercept of 1. From there, use “rise over run” to find the slope. From the \begin{align*}y-\end{align*}intercept, if you go up 1 and over 2, you hit the line again, \begin{align*}m = \frac{1}{2}\end{align*}. The equation is \begin{align*}y=\frac{1}{2}x+1\end{align*}.

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

The lines are **parallel** because they have the same slope.

### Interactive Practice

### Explore More

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

- \begin{align*}y=-5x+1; \ (-2, \ 3)\end{align*}
- \begin{align*}y=\frac{2}{3}x-2; \ (9, 1)\end{align*}
- \begin{align*}x-4y=12; \ (-16, \ -2)\end{align*}
- \begin{align*}3x+2y=10; \ (8, \ -11)\end{align*}
- \begin{align*}2x - y = 15; \ (3, \ 7)\end{align*}
- \begin{align*}y = x - 5; \ (9, \ -1)\end{align*}
- \begin{align*}y = 3x - 4; \ (2, \ -3)\end{align*}

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

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.8.