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# 3.8: Parallel Lines in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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What if you wanted to figure out if two lines in a shape were truly parallel? How could you do this? After completing this Concept, you'll be able to use slope to help you to determine whether or not lines are parallel.

### Watch This

Watch the portion of this video that deals with Parallel Lines

### Guidance

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 23\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.

#### Example A

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 in this form is called the slope-intercept form and is written as 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. Here, x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} represent any coordinate pair, (x, y)\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 m=13\begin{align*}m=-\frac{1}{3}\end{align*}. Now, we need to find the y\begin{align*}y-\end{align*}intercept. 4 is the y\begin{align*}y-\end{align*}intercept of the given line, not our new line. We need to plug in 9 for x\begin{align*}x\end{align*} and -5 for y\begin{align*}y\end{align*} (this is our given coordinate pair that needs to be on the line) to solve for the new y\begin{align*}y-\end{align*}intercept (b)\begin{align*}(b)\end{align*}.

552=13(9)+b=3+bTherefore, the equation of line is y=13x2.=b

#### Example B

Which of the following pairs of lines are parallel?

• y=2x+3\begin{align*}y=-2x+3\end{align*} and y=12x+3\begin{align*}y=\frac{1}{2}x+3\end{align*}
• y=4x2\begin{align*}y=4x-2\end{align*} and y=4x+5\begin{align*}y=4x+5\end{align*}
• y=x+5\begin{align*}y=-x+5\end{align*} and y=x+1\begin{align*}y=x+1\end{align*}

Because all the equations are in y=mx+b\begin{align*}y=mx+b\end{align*} form, you can easily compare the slopes by looking at the values of m\begin{align*}m\end{align*}. To be parallel, the lines must have equal values for m\begin{align*}m\end{align*}. The second pair of lines is the only one that is parallel.

#### Example C

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

The equation of the parallel line is y=2x+6\begin{align*}y=2x+-6\end{align*}.

Watch this video for help with the Examples above.

### Vocabulary

Two lines in the coordinate plane with the same slope are parallel and never intersect. Slope measures the steepness of a line.

### Guided Practice

1. Which of the following pairs of lines are parallel?

• y=3x+1\begin{align*}y=-3x+1\end{align*} and y=3x1\begin{align*}y=3x-1\end{align*}
• 2x3y=6\begin{align*}2x-3y=6\end{align*} and 3x+2y=6\begin{align*}3x+2y=6\end{align*}
• 5x+2y=4\begin{align*}5x+2y=-4\end{align*} and 5x+2y=8\begin{align*}5x+2y=8\end{align*}
• x3y=3\begin{align*}x-3y=-3\end{align*} and x+3y=9\begin{align*}x+3y=9\end{align*}
• x+y=6\begin{align*}x+y=6\end{align*} and 4x+4y=16\begin{align*}4x+4y=-16\end{align*}

2. 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).

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

1. First change all equations into y=mx+b\begin{align*}y=mx+b\end{align*} form so that you can easily compare the slopes by looking at the values of m\begin{align*}m\end{align*}. The third and fifth pair of lines are the only ones that are parallel.

2. 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

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

3. 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.

### Practice

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

### Vocabulary Language: English

Parallel

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.

Jul 17, 2012

Jun 07, 2015