### 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

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*

#### Identifying Parallel Lines

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.

#### 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

The equation of the parallel line is

### Examples

#### Example 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

First change all equations into *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

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

#### Example 3

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

The top line has a \begin{align*}y-\end{align*}

For the second line, the \begin{align*}y-\end{align*}

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

### Interactive Practice

### Review

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*}
y=−5x+1; (−2, 3) - \begin{align*}y=\frac{2}{3}x-2; \ (9, 1)\end{align*}
y=23x−2; (9,1) - \begin{align*}x-4y=12; \ (-16, \ -2)\end{align*}
x−4y=12; (−16, −2) - \begin{align*}3x+2y=10; \ (8, \ -11)\end{align*}
3x+2y=10; (8, −11) - \begin{align*}2x - y = 15; \ (3, \ 7)\end{align*}
2x−y=15; (3, 7) - \begin{align*}y = x - 5; \ (9, \ -1)\end{align*}
y=x−5; (9, −1) - \begin{align*}y = 3x - 4; \ (2, \ -3)\end{align*}
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.

### Review (Answers)

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