 Now we're going to look at finding equations for parallel and perpendicular lines. First let's look at some parallel lines. These are parallel lines and what do you notice? You notice about their steepness that they're both going the same direction and they're both going at the same rate. So that would tell us that parallel lines have the same slope. Alright? So then if we go and look at the perpendicular lines, in this case we can see that they're going in opposite directions. So we know that they are at least going to have opposite signs, but their steepness are opposite the reciprocal. When we look at these then, let's see if we can use the same ideas that we found equations with. We have to know if something about the slope, we need to have a y-intercept and plug and chug what we know. So we have parallel line here that is parallel to, the slope is negative three, so parallel has the same slope, and your line contains one, two. Remember this is your x and y, so x is equal to one, y is equal to two, and we're going to find b. So rewriting it, y, which is two, is equal to m, negative three times x, which is one plus b. So if this is two is equal to negative three plus b, if we continue and just add three to both sides, we find out that five is equal to b. So our equation would be y is equal to slope, negative three times our x, because we need the variable holder, plus our b, which is five. Let's try another one. Now we have an equation and a point, but it's parallel to this line, and the slope of this line, remember that's the coefficient on x, so the slope that here is seven, so our slope is also seven, same slope for parallel. And again, we have our x and our y, so x is four, y is five. Up here it asks you, is b equal to three? It shouldn't be equal to three, because if it were equal to three, then we'd have exactly the same line, and we're just trying to find one that's parallel to it, a different line. So we don't know what b is. No, it is not equal to three. So let's find the equation. I'm going to start up here above this b for plugging in my equation, y, which is five, is equal to m, seven times x, which is four, plus the b I don't know. So five is equal to 28 plus b. If I subtract 28 from both sides, I should end up with negative 23. So that would give me y is equal to the slope, which is seven times x, plus my y, yes, y-intercept b, which is negative 23. One more for parallel. What if I have an equation that's not said equal to y, but I still have to know the slope of that line. So I'm going to take this equation, and over here I'm going to say negative x plus 2y is equal to five, and I'm going to solve for y so that I can find the slope. I'm going to put it in y-intercept form. So I'm going to add the x to both sides. So 2y is equal to x plus five, the x in front, because it's going to look like y equal to x plus b that way. Divide everything by two and everything by two. So that's going to be x over two, or one-half x, plus five over two. Now this is my slope, one-half x. So the slope is one-half. And then they gave me the point negative four and negative nine, and I have to find my b. So I had to solve for y to find out what the slope was, and then I can follow like we've normally done. Y is negative nine equal to m one-half times my x, which is negative four plus my b. Negative nine, one-half times negative four would be half of negative four is negative two plus b. Add two to both sides. Negative nine plus two would be negative seven. So y should be equal to my slope, which we said was one-half x minus seven. All right, let's look at perpendicular lines then. Perpendicular, remember, means opposite reciprocal. So if m one is equal to three, then the perpendicular slope is going to be equal to negative one-third. So we want to find the perpendicular line to y is equal to one-half x plus one-hundred. So the slope here is one-half, and that makes my slope the opposite reciprocal. So it's going to be a negative, and the reciprocal of one-half would be two. So it's negative two. X is one, y is two from my point. We don't know what b is. So then we're going to have y, which is two, is equal to m negative two times x, which is one plus b. So two is equal to negative two plus b. And adding two to both sides, b is going to be the fourth. So y is equal to my new slope, negative two x plus four, using my n and my b. Let's try another one. What if it's not again in the y-intercept form? We still can find the slope. We need to find the slope so we can then find the perpendicular slope. We already know that x and y are six and five, so let's put those in there since we know that already. And if I take six x minus two y and solve it for y, that's a negative nine. Then I want to subtract six x from both sides. So negative two y is equal to negative six x minus nine. And divide everything by negative two. So negative six over negative two x minus nine over negative two. Negative six divided by negative two is positive three x minus, or actually it's a negative divided by negative, so it's plus nine over two. But remember, this is the only thing we care about. This slope is three, so the perpendicular one is going to be negative. This is positive, so we need a negative and then the reciprocal of three is one-third. So plugging into our equation, y is five, is equal to n, which is negative one-third, times six plus b. Five is equal to negative one-third, times six, with two plus b. Adding two to both sides, seven will equal b. So now I know that y is equal to my new slope. And here's my b, so y is equal to negative one-third x plus seven. Finally, let's do a tricky one. A perpendicular to y equal eight and contains the point negative 21, negative 25. So you're looking at this and saying y equal eight. How do I know a slope? Well, think about it. What kind of line is y equal eight? All the y's are eight. So you've got these x's and you've got these y's and these are all eight. One, two, three. If all the y's are eight, it's going to be a horizontal line. Horizontal lines have zero slope. So the perpendicular line is going to be a no slope, which is a x equal equation. So we look at this one and we say no slope. So we really can't do any algebra to it. But we know that x is negative 21. y is negative 25. Plug in what we know. But this is all we really need to know. It's going to be a vertical line and that's an x equal equation and what is x equal to through our point? Negative 21.