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 \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.

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?

- @$y=-3x+1@$ 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.

**
Answers:
**

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.

### Interactive Practice

### Practice

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

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

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

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