Equations of Parallel Lines - Definition & Examples

(Video) Writing Equations of Parallel Lines

by Friendly Math 101

Here's a video by Friendly Math 101 explaining parallel line equations.

$\large{\text{Summary}}$

Parallel lines are lines that will never intersect. Both lines can go onto infinity, and they will never meet.

If we have an equation in the form $y=mx+b$, parallel lines will have the same slope but different $y$-intercepts.

Let's look at an example. If we have a line given by the equation $y=5x-1$, what would be the equation of a line parallel to it?

A line parallel to this will have the same slope and a different intercept. Since, the slope in the equation is $m=5$ the slope for the parallel line will be $m=5$.

Finally, we need to choose any $y$-intercept that isn't the same as $b=-1$. $$y=5x+2$$ Here's one possible answer. There are essentially an infinite number of options.

Changing only the $y$-intercept keeps lines parallel

Parallel lines are lines that will never intersect. One special property shared by parallel lines is that the slopes are equivalent. Intersecting lines will never have equal slope. Parallel lines' slopes are equivalent, so the only thing setting them apart from equivalent lines is the difference in $y$-intercept.

Sometimes you might have to solve for $y$ in terms of $x$ in order to see if the slopes are the same. It's helpful to convert both equations to the slope-intercept form, so they're both in the same form, making it easier to compare different slopes.

Try this problem:

Let's convert each of the equations to slope intercept form. Any of them that end with a slope of $3$ are parallel to $y=3x+7$.

$$\begin{align}2y+6 &= 6x+2\\ 2y&=6x-4\\ y&=3x-2\\ \\ 5y+2&=3x-7\\ 5y&=3x-9\\ y&=\frac35x - \frac95\\ \\ 10y+24&=30x+12\\ 10y&=30x-12\\ y&=3x-\frac65 \end{align}$$

The first and third equations have a slope of $3$, so they are parallel to $y=3x+7$.

Keeping with this pattern when solving other equations will help you in the future!

Parallel Lines

If you keep the slope of a line the same but change the $y$-intercept, you're basically moving the line uniformly (all points by the same amount) in a vertical direction. The two lines won't intersect, so we call them parallel. Lines that meet at a $90^o$ angle are perpendicular. The perpendicular slope are opposite reciprocals of the parallel slope.

Example

In the example below, the blue line, $y = 0.5x - 2$, has been moved upwards by $7$ units, to get the green line $y = 0.5x + 5$.

The two lines are parallel: they have the same slope, $0.5$, but the $y$-intercept has changed by $+7$ units, from $-2$ to $+5$.

There are infinitely many parallel lines because you can have any line with the same slope and infinitely many different $y$-intercepts. Here are some examples of parallel lines in graphs.

The graph above shows $8$ parallel lines. Notice how more lines could be graphed in between those lines. There are infinite possibilities because the real numbers are the numbers on a number line or Cartesian plane. The real numbers are infinite, so the number of parallel lines are also infinite.