Parallel and Perpendicular Lines
Remember when we learned about the equation of a line back in algebra? You know, all that y = mx + b business? Well, if we think parallel lines in terms of linear functions, we know there's got to be something weird going on. After all, two lines that never intersect—ever? That's some fishy business.
The truth is that it's not weird at all. Looking at two parallel lines, we can see that they'll never crash into one another because if one of the lines moves one unit up and one unit to the right, the other does the same. If it wants to go up three units and two to the left, a line parallel to it will do the exact same. What we're getting at is that two parallel lines will always have the exact same slope. In fact, that's what makes them parallel.
Two lines have equations of y = 6x + 3 and y = 3x + 3. Are these lines parallel?
For two lines to be parallel, their slopes must be the same. In the case of linear equations, the slope of the line is the coefficient before x, otherwise known as m in y = mx + b. Here, the slope of the first line is 6 and the slope of the second line is 3. Since they're not equal, the lines aren't parallel. (In fact, they intersect at the y-intercept. Plug in x = 0 for both equations and you'll see what we mean.)
If we want to find a line parallel to another line, all we need is the slope of the original line. That takes care of the m part in y = mx + b. After that, all we need to do is plug in a point on that line (either given or chosen) and solve for b.
Be careful though! We don't want to plug in a point that's on the original line because parallel lines don't intersect. If we do that, we'll end up with the equation for the original line, not a new one that's parallel to it.
It's like when songwriters try to rhyme a word with the same exact word. That's not rhyming; that's repetition. We're looking at you, Carly Rae Jepsen.
What is a line that's parallel to y = ¼x – 3 and passes through the point (4, -3)?
Our first step is to start out with our line equation, y = mx + b. We know that parallel lines have the same slope, so we can substitute ¼ for m to get y = ¼x + b. To find b, we can plug in the point (4, -3) for x and y.
y = ¼x + b
-3 = ¼(4) + b
-3 = 1 + b
b = -4
Our final equation is y = ¼x – 4, which is parallel to y = ¼x – 3.
We won't harp too much more on the algebra of it all, but it's important to understand that it all ties together. Whether you like it or not, math is all related. And one way or another, it's going to get you.