Hey, everyone. So in previous videos, we've seen the three different ways of writing line equations in three different forms. Well, some problems, like the one we're going to work out down below, will give us an equation in any one of these forms and ask us to write lines that are parallel or perpendicular to those lines. That's what I want to show you in this video. We're going to see that parallel and perpendicular lines are really just related by the values of their slopes. Alright? So let's go ahead and take a look here.

I've got these two graphs, and I've got these two lines in this diagram over here, -3x + 2 and -3x - 4. You can see that they're graphed here. Well, if you notice that these lines are sort of, like, perfectly almost identical, they're just sort of shifted, you know, to the side. And that's the key characteristic of parallel lines. The key characteristic is that because they're sort of exactly the same and just shifted over, they actually never intersect. And that really just has to do with their slopes. Notice how the slopes here are both negative 3. So for parallel lines, the slopes are always equal to each other, and the y-intercepts are different. So in other words, they both have a slope of negative 3, but their y-intercepts are different, like 2 and negative 4. If both of these things were the same, it would actually just be describing the exact same line. So the key characteristic for parallel lines is the slopes are equal and the b's or the y-intercepts are different. Alright?

If these two things have the same slope, it's basically like the rise over the run of each of these points will always be the same, and they'll never cross each other. Alright? Now let's take a look at perpendicular lines. Here, what I've got is I've got -3x + 2. I've got the exact same lines I had over here, but now the other line, this orange one over here, has \frac{1}{3}x - 4. So, clearly, these two things have very different slopes and y-intercepts, and we can see that they do actually intersect. Now what's special about perpendicular lines is that the point where they intersect is they intersect at right angles. So in other words, this sort of this intersection point over here has an angle of 90 degrees. That's what perpendicular actually means. And, basically, what you need to know about these lines is that the slopes here are different. So the negative three is the slope of the one line, and then we have \frac{1}{3} for the slope of the other. These things are related, because notice how one is positive or sorry. One's negative and the other one's positive, and they're also reciprocals of each other. That's actually always what's going to happen between perpendicular lines. The slopes will always have opposite signs. So in other words, one was positive and one was negative, and they're also going to be reciprocals of each other. So you're basically just going to flip the fractions or whole numbers or something like that. Alright? So that's basically the difference between parallel and perpendicular lines. You may see some symbols regarding these things. You may see some, like, sort of kind of like to remember way I kind of like to remember it is that parallel lines have the symbol, and it kinda looks like the double l that's inside the word. Alright? So just in case you see that. That's basically the difference. Let's go ahead and take a look at some, example problems. Alright?

So here, what we want to do is we want to write an equation of a line that passes through a point and is parallel to some other reference line that it's giving us, y = 2x - 6. So in other words, I want to write an equation of a line that is parallel to this line over here. So parallel means that I want to have an equal slope and a different y-intercept. So in other words, I'm looking for a line in which the slope is equal to 2 because that's the slope of this line, m = 2. So it has to be the same. Alright? Now notice how this problem actually hasn't told us what form of an equation to write, and that's what we're going to have to figure out. But notice how this problem tells us that we also are going to be passing through a point. That's the equation of our line. So we're going to be passing through the point (-1, 4). And if you remember, what happens is if you're ever given the slope and a point, we always write the equation using point-slope form. So we're going to write the equation in point-slope. Alright? So I'm just going to write that out for a second. Basically, what we're going to do here is we're going to write y - y_1 = m (x - x_1). Alright. So this is the sort of now we just have to fill in the so the sort of variables. And remember the y_1 and x_1 just have to do with the point that we picked, which is (-1, 4), and the m is just the 2. So just gonna fill this out really quick. I'm gonna do the y_1, which is 4, so y - 4 = m (x - (-1)). Remember these are variables, and we don't replace them with numbers. Alright? So all we have to do is just clean this up a little bit, but this is gonna be y - 4 = 2 (x + 1). Alright? So this is the equation of a line that passes through the point (-1, 4) and is parallel to this equation over here because they have the same ms. And, in fact, if you graph these two equations out, what you're going to see is we're going to have 2x - 6, which kind of looks like, this over here passes through negative 6 and has a slope of 2. And then if you graph this point over here, we know that it's going to pass through the point (-1, 4), and it's also going to have a slope of 2. So if you draw some points out and connect them, we're going to see sort of as a rough sketch that these two lines are parallel to each other. They will never intersect. Alright.

Let's take a look at our next problem here. Here, we have to write the equation of a line that is now perpendicular. So we're looking for a line that is perpendicular to this equation over here that's written in standard form, and it has a y-intercept of 3. So what are we told? We're told that it has a y-intercept, which remember b, so b = 3. Alright? So remember which so first, we have to figure out what form we're going to write this equation in, And remember that whenever we're given a b term or asked for b, we're always going to write this in slope-intercept form. Right? So if we're given or asked for b, we write it in slope-intercept form. Okay. So if I want to write slope intercept form, I need the b, which I have, but I also need the m. So that m that I need has to be perpendicular to this line over here. In fact, what I'm actually going to do is I'm going to write this over here for seconds. So I have to have an m that is perpendicular to this line, which is in standard form. So to figure that out, first, I'm going to have to sort of isolate and solve for this y over here. What I'm going to do is I'm going to take this x, move it over, and the negative 8 and move it over. I'm going to have to subtract x, and I'm going to have to add eights from both sides. Alright? So this ends up going away, and I end up with 4y = -x + 8. Now what I have to do is I have to divide by 4 of each one through each one of the terms over here, and I get rid of the 4. So this is going to be y = -\frac{1}{4}x + 2. So remember, this is the equation not that I'm trying to find. This is the equation of this line over here just written in standard form. And I needed to write in standard form because I needed to figure out what the m is. Alright? So this m is -\frac{1}{4}. To get an m of my line that I'm trying to find, I have to do the opposite sign and the reciprocal of this number. The opposite sign is going to be a positive, and the reciprocal of this number is going to be an m that's equal to 4. So now that I have m and b, now I can go ahead and write the equation of my line that is perpendicular. So this is going to be y = mx + b. So in other words, this is going to be y = 4x + 3. So let's graph these lines and see if we've actually done this correctly. So this is an equation y = -\frac{1}{4}x + 2. So it starts here, and we go down 1 and over 4, up 1 and to the left 4. So this line is going to kind of look like this. Now our line is going to look like 4x + 3. So that's an equation that starts here, and it has a slope of 4. So we go up 4 over 1 or down 4 and then over 1, down 4 over 1. So this line is going to look kind of like this. And if you look at these two lines over here, they intersect at this point, and the intersection is definitely 90 degrees. Alright?

So that's it for this one, folks. Hopefully, this made sense. Thanks for watching.