### Parallel Lines in the Coordinate Plane

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, the slopes are both \begin{align*}\frac{2}{3}\end{align*}.

This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different \begin{align*}y-\end{align*}intercepts.

What if you were given two parallel lines in the coordinate plane? What could you say about their slopes?

### Examples

#### Example 1

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 2

Are the lines \begin{align*}3x+4y=7\end{align*} and \begin{align*}y=\frac{3}{4}x+1\end{align*} parallel?

First we need to rewrite the first equation in slope-intercept form.

\begin{align*}3x+4y&=7\\4y&=-3x+7\\y&=-\frac{3}{4}x+\frac{7}{4}\end{align*}.

The slope of this line is \begin{align*}-\frac{3}{4}\end{align*} while the slope of the other line is \begin{align*}\frac{3}{4}\end{align*}. Because the slopes are different the lines are not parallel.

#### Example 3

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 is \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. We know that parallel lines have the same slope, so the line will have a slope of \begin{align*}-\frac{1}{3}\end{align*}. Now, we need to find the \begin{align*}y-\end{align*}intercept. Plug in 9 for \begin{align*}x\end{align*} and -5 for \begin{align*}y\end{align*} 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\\ -2 & = b\end{align*}

The equation of parallel line is \begin{align*}y = -\frac{1}{3}x-2\end{align*}.

#### Example 4

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.

#### Example 5

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*}.

### Review

Determine if each pair of lines are parallel. Then, graph each pair on the same set of axes.

- \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*}
- \begin{align*}5x+2y=-4\end{align*} and \begin{align*}5x+2y=8\end{align*}
- \begin{align*}x+y=6\end{align*} and \begin{align*}4x+4y=-16\end{align*}

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*}

Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.

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

### Review (Answers)

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