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Parallel Lines

Parallel lines are lines that do not meet, ever. It's important that we include "ever" in the definition because when we look at some graphs, our lines don't cross. Here's an example:

Notice how the red and blue lines don't actually meet anywhere on our graph. However, in order to be parallel, our lines can't meet EVER.

Imagine extending both the red and blue lines into infinity. We can see then that they'd eventually cross on their lower halves. Thus, we can say that these two lines are * not parallel.*

Another way to think about it: in order for two lines to be parallel, the distance between them must be constant.

We can see in the graphs above that the lines are much closer together near the bottom and much farther apart at the top. This means that the distance between them is not constant and that the lines are not parallel.

You can also tell if two lines are parallel if they have the same slope and different $y$-intercepts. Let's look.

(Video) Parallel and Perpendicular Lines

by Mahalo Math

Mahalo Math has a good video explaining parallel lines.

$\large{\text{Summary}}$

To tell if two lines are parallel you can look at their slopes.

Parallel lines are lines that never intersect each other. They have the same slope, but different $y$-intercepts. Above are two parallel lines:

• $y=\color{Red}{2}x+2$
• $y=\color{Red}{2}x-1$

Notice that both of these lines have a slope of $2$.

Parallel and Perpendicular Lines

When two lines are graphed together, the lines will be parallel if they are increasing at the same rate—if the rates of change are the same. In this case, the graphs will never cross (unless they're the same line).

Parallel Lines

Two lines are parallel if the slopes are equal (or, if both lines are vertical). In other words, given two linear equations $f(x)=b+m_1x$ and $g(x)=b+m_2x,$ the lines will be parallel if $m_1=m_2.$ Example 8

Find a line parallel to $f(x)=6+3x$ that passes through the point $(3,0)$.

We know the line we're looking for will have the same slope as the given line, $m=3$. Using this and the given point, we can solve for the new line's vertical intercept: $$g(x)=b+3$$then at $(3,0)$, $$0=b+3(3)\\ b=-9$$ The line we're looking for is $g(x)=-9+3x$.