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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

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.

#### Finding the Equation of a Parallel Line

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\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?

• 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.

#### 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 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*}.

### Examples

#### Example 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*}

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.

#### Example 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).

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 3

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.

### Review

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. 3x+2y=10; (8, 11)\begin{align*}3x+2y=10; \ (8, \ -11)\end{align*}
5. 2xy=15; (3, 7)\begin{align*}2x - y = 15; \ (3, \ 7)\end{align*}
6. y=x5; (9, 1)\begin{align*}y = x - 5; \ (9, \ -1)\end{align*}
7. y=3x4; (2, 3)\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.

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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.