### 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 \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.

#### Finding the Equation of a Parallel Line

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*}

#### Identifying Parallel Lines

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.

#### 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 \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*}.

### Examples

#### Example 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*}

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.

#### Example 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).

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*}.

#### 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*}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.

### 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*}
- \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.

### Review (Answers)

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