### Parallel Lines in the Coordinate Plane

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.

#### Finding the Equation of a Parallel Line

Find the equation of the line that is parallel to 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 where is the slope and is the intercept. Here, and represent any coordinate pair, on the line.

We know that parallel lines have the same slope, so the line we are trying to find also has . Now, we need to find the intercept. 4 is the intercept of the given line, *not our new line*. We need to plug in 9 for and -5 for (this is our given coordinate pair that needs to be on the line) to solve for the *new* intercept .

#### Identifying Parallel Lines

Which of the following pairs of lines are parallel?

- and

- and

- and

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

#### Finding the Equation of a Parallel Line given a Graph

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 and the point is (2, -2). The parallel would have the same slope and pass through (2, -2).

The equation of the parallel line is .

### Examples

#### Example 1

Which of the following pairs of lines are parallel?

- and

- and

- and

- and

- and

First change all equations into form so that you can easily compare the slopes by looking at the values of . The *third* and *fifth* pair of lines are the only ones that are parallel.

#### Example 2

Find the equation of the line that is parallel to and passes through (8, -7).

We know that parallel lines have the same slope, so the line will have a slope of . Now, we need to find the intercept. Plug in 8 for and -7 for to solve for the *new* intercept .

The equation of the parallel line is .

#### Example 3

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

The top line has a intercept of 1. From there, use “rise over run” to find the slope. From the intercept, if you go up 1 and over 2, you hit the line again, . The equation is .

For the second line, the intercept is -3. The “rise” is 1 and the “run” is 2 making the slope . The equation of this line is .

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

### Review

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

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

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

### Review (Answers)

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