There are special rules to help us find the equation of a line given an equation of a line that is parallel or perpendicular to it. Parallel and perpendicular lines have related slopes. Parallel lines have equivalent slopes and perpendicular lines have negative inverses of their slopes. To fully understand and apply this concept, we should be familiar with the slope and the slope-intercept form of an equation.
Working with parallel and perpendicular lines is really important not only in algebra but also in geometry. So this is a concept you are going to want to review at the end of your algebra course before you move on to geometry.
First thing, two lines are parallel if they never cross. Their slopes are the same. A lot of people think of railroad tracks when they think about parallel lines. If in your brain you picture how railroad ties are -- is that what they are called? Ties? I think so. They're always parallel. They never, ever cross and they go on forever and ever across the land. I guess not forever and ever. That's the idea of parallel lines.
If you are looking at the equations, you will know two lines are parallel without having graphed them if their slopes are the same. Let me draw a quick picture of that. Slopes, remember, means how steep the lines are. So slopes are the same, means those two lines are equally steep. That's something that might help you remember it.
This is a different concept. Two lines are called perpendicular if they intersect at a right angle, their slopes are opposite sign reciprocals.
So let's look at a picture. This is, again, not perfect, but picture of two perpendicular lines. Perpendicular lines means they cross at a right angle. So if I draw this line right here, a perpendicular line would look kind of like that. That's not great. You could tell it was perpendicular if you took your paper and you stuck it right in the corner there, it would fit perfectly, all four corners. Mine isn't great. You get the idea.
Perpendicular lines mean they cross a right angle which is 90 degrees and you will work with that a lot in geometry.
Let's talk more about opposite sign reciprocal slopes. Like, if I gave you the number -2, the reciprocal of 2 is 1/2. Notice, also, how instead of -2 I wrote positive 1/2. These are opposite signs, meaning one's positive, one's negative and they're reciprocals. Another example would be, like, 3/4 and the opposite sign reciprocal would be -4/3.
These are the kinds of slopes that we're looking for when we're talking about perpendicular lines. These are things you are just going to have to memorize: Parallel never cross, same slope; perpendicular means they cross at a right angle, opposite sign reciprocals. There's no real shortcuts to it. But these are really, really important definitions that you'll need in algebra and also in geometry.