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# Parallel Lines in the Coordinate Plane

## Lines with the same slope that never intersect.

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Parallel Lines in the Coordinate Plane

### 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 23\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 y\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 y=14x+3\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 14\begin{align*}\frac{1}{4}\end{align*}. Now, we need to find the y\begin{align*}y-\end{align*}intercept. Plug in 8 for x\begin{align*}x\end{align*} and -7 for y\begin{align*}y\end{align*} to solve for the new y\begin{align*}y-\end{align*}intercept (b)\begin{align*}(b)\end{align*}.

779=14(8)+b=2+b=b\begin{align*}-7 & = \frac{1}{4}(8)+b\\ -7 & = 2 + b\\ -9 & = b\end{align*}

The equation of the parallel line is y=14x9\begin{align*}y = \frac{1}{4}x-9\end{align*}.

#### Example 2

Are the lines 3x+4y=7\begin{align*}3x+4y=7\end{align*} and y=34x+1\begin{align*}y=\frac{3}{4}x+1\end{align*} parallel?

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

3x+4y4yy=7=3x+7=34x+74\begin{align*}3x+4y&=7\\4y&=-3x+7\\y&=-\frac{3}{4}x+\frac{7}{4}\end{align*}.

The slope of this line is 34\begin{align*}-\frac{3}{4}\end{align*} while the slope of the other line is 34\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 y=13x+4\begin{align*}y=-\frac{1}{3}x+4\end{align*} and passes through (9, -5).

Recall that the equation of a line is y=mx+b\begin{align*}y = mx + b\end{align*}, where m\begin{align*}m\end{align*} is the slope and b\begin{align*}b\end{align*} is the y\begin{align*}y-\end{align*}intercept. We know that parallel lines have the same slope, so the line will have a slope of 13\begin{align*}-\frac{1}{3}\end{align*}. Now, we need to find the y\begin{align*}y-\end{align*}intercept. Plug in 9 for x\begin{align*}x\end{align*} and -5 for y\begin{align*}y\end{align*} to solve for the new y\begin{align*}y-\end{align*}intercept (b)\begin{align*}(b)\end{align*}.

552=13(9)+b=3+b=b\begin{align*}-5 & = -\frac{1}{3}(9)+b\\ -5 & = -3 + b\\ -2 & = b\end{align*}

The equation of parallel line is y=13x2\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 y\begin{align*}y-\end{align*}intercept of 1. From there, use “rise over run” to find the slope. From the y\begin{align*}y-\end{align*}intercept, if you go up 1 and over 2, you hit the line again, m=12\begin{align*}m = \frac{1}{2}\end{align*}. The equation is y=12x+1\begin{align*}y=\frac{1}{2}x+1\end{align*}.

For the second line, the y\begin{align*}y-\end{align*}intercept is -3. The “rise” is 1 and the “run” is 2 making the slope 12\begin{align*}\frac{1}{2}\end{align*}. The equation of this line is y=12x3\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 y=2x+6\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).

y226=2x+b=2(2)+b=4+b=b\begin{align*}y & =2x+b\\ -2 & = 2(2) + b\\ -2 & = 4+b\\ -6 & = b\end{align*}

The equation of the parallel line is y=2x+6\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.

1. y=4x2\begin{align*}y=4x-2\end{align*} and y=4x+5\begin{align*}y=4x+5\end{align*}
2. y=x+5\begin{align*}y=-x+5\end{align*} and y=x+1\begin{align*}y=x+1\end{align*}
3. 5x+2y=4\begin{align*}5x+2y=-4\end{align*} and 5x+2y=8\begin{align*}5x+2y=8\end{align*}
4. x+y=6\begin{align*}x+y=6\end{align*} and 4x+4y=16\begin{align*}4x+4y=-16\end{align*}

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

1. y=5x+1; (2,3)\begin{align*}y=-5x+1; \ (-2, 3)\end{align*}
2. y=23x2; (9,1)\begin{align*}y=\frac{2}{3}x-2; \ (9, 1)\end{align*}
3. x4y=12; (16,2)\begin{align*}x-4y=12; \ (-16, -2)\end{align*}
4. \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.

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

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### Vocabulary Language: English

Parallel

Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.