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 \begin{align*}\frac{2}{3}\end{align*} .

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 @$\begin{align*}y=-\frac{1}{3}x+4\end{align*}@$ 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 @$\begin{align*}y = mx + b\end{align*}@$ where @$\begin{align*}m\end{align*}@$ is the slope and @$\begin{align*}b\end{align*}@$ is the @$\begin{align*}y-\end{align*}@$ intercept. Here, @$\begin{align*}x\end{align*}@$ and @$\begin{align*}y\end{align*}@$ represent any coordinate pair, @$\begin{align*}(x, \ y)\end{align*}@$ on the line.

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

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

#### Example B

Which of the following pairs of lines are parallel?

- @$\begin{align*}y=-2x+3\end{align*}@$ and @$\begin{align*}y=\frac{1}{2}x+3\end{align*}@$

- @$\begin{align*}y=4x-2\end{align*}@$ and @$\begin{align*}y=4x+5\end{align*}@$

- @$\begin{align*}y=-x+5\end{align*}@$ and @$\begin{align*}y=x+1\end{align*}@$

Because all the equations are in
@$\begin{align*}y=mx+b\end{align*}@$
form, you can easily compare the slopes by looking at the values of
@$\begin{align*}m\end{align*}@$
. To be parallel, the lines must have equal values for
@$\begin{align*}m\end{align*}@$
. 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 @$\begin{align*}y=2x+6\end{align*}@$ 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 @$\begin{align*}y=2x+-6\end{align*}@$ .

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter3ParallelLinesintheCoordinatePlaneB

### Vocabulary

Two lines in the coordinate plane with the same slope are
**
parallel
**
and never intersect.

**measures the steepness of a line.**

*Slope*### Guided Practice

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

- @$\begin{align*}y=-3x+1\end{align*}@$ and @$\begin{align*}y=3x-1\end{align*}@$

- @$\begin{align*}2x-3y=6\end{align*}@$ and @$\begin{align*}3x+2y=6\end{align*}@$

- @$\begin{align*}5x+2y=-4\end{align*}@$ and @$\begin{align*}5x+2y=8\end{align*}@$

- @$\begin{align*}x-3y=-3\end{align*}@$ and @$\begin{align*}x+3y=9\end{align*}@$

- @$\begin{align*}x+y=6\end{align*}@$ and @$\begin{align*}4x+4y=-16\end{align*}@$

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

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

**
Answers:
**

1. First change all equations into
@$\begin{align*}y=mx+b\end{align*}@$
form so that you can easily compare the slopes by looking at the values of
@$\begin{align*}m\end{align*}@$
. 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
@$\begin{align*}\frac{1}{4}\end{align*}@$
. Now, we need to find the
@$\begin{align*}y-\end{align*}@$
intercept. Plug in 8 for
@$\begin{align*}x\end{align*}@$
and -7 for
@$\begin{align*}y\end{align*}@$
to solve for the
*
new
*
@$\begin{align*}y-\end{align*}@$
intercept
@$\begin{align*}(b)\end{align*}@$
.

@$$\begin{align*}-7 & = \frac{1}{4}(8)+b\\ -7 & = 2 + b\\ -9 & = b\end{align*}@$$

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

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

### Practice

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.