 This video will talk about different ways to find equations of a line given information about that line. So we're going to be doing these in the form of y equal mx plus b, which is the slope intercept form. And you'll notice that there are four unknowns, x, y, m, and b. We need, in order to be able to find these equations, at least a slope and a point. If we have at least that, then we should be able to write the equation. This gives us an x and a y and an m to plug and chug, which would mean that we just have b as an unknown. As long as we have three of the unknowns, then we can solve for the fourth one, because we know how to solve one variable equations. So our first example, write the equation of a line with a slope of negative three and contains the point zero, nine. Well, if you look at this, we know that slope is going to be negative three. And there's two ways to do this problem. The first way we're going to do it is to think that x is equal to zero and y is equal to negative nine. So when we do that, we now know that y equal mx plus b is the form that we're going into. So we plug and chug, y we said was negative nine, and that's equal to slope, which we said was negative three, plus our x, which we said over there was zero, and then plus b. Well, anything times zero is just zero, so we really have the b is equal to negative nine. So now we want to write our equation as y equals some slope times x plus and then some b. And we know that our slope is negative three, and our y-intercept we know to be negative nine, or our b to be negative nine, so y is equal to negative three x minus nine would be the equation for that line. So if you look at this equation again, we still know that slope is negative three, but if we look a little closer, when x is equal to zero, then we know that we have the y-intercept. So if we would have recognized that this point was zero negative nine, that would have told us that b was negative nine, and now we're just ready to rewrite the equation, because remember you want to plug in for m and you want to plug in for b. So we rewrite our equation as y is equal to negative three times x plus our b, which in this case is minus nine. Exact same equation, just a second way to do it, but that only works if our point starts with a zero in the x. If it starts with something different, then we have to go through that same process we did earlier. Let's try again. Now we know slope is three-fourths, and it contains the point x equal negative eight, and y is equal to negative one, and we know m is three-fourths. So again, it might be helpful for us to just write y is equal to mx plus b, and then plug and chug what we know. Y is negative one, slope is three-fourths, and x is negative eight this time, and we'll say plus b. Our negative one is our y, and that's not changing, but now we're going to take negative eight times three-fourths. And if you remember, we can do that four goes into negative eight, negative two times, and then we just have to multiply three times negative two and get negative six, and then plus our b. And if we add six to both sides, then we will find out what b is, and that'll be negative one plus six is five equal to b. So our b is five. These are the two pieces of information that we need to plug into the equation. Okay? Write it as y equal, and then something x, and then plus, in this case, our y intercept. Slope is three-fourths, so three-fourths, x, and plus five, which is our y intercept. What happens though? Now that all they tell us are two points. They didn't give us a slope, but if you remember, from the very beginning, we said you need a slope and a point. It doesn't matter which point we're going to use eventually, but we do have to find the slope with both points. So let's start there. Y, which is nine, minus my other y, which is negative one, which is the same as plus one. And x, starting with the same x from the point I started my y in, so negative three minus two. And when I do that, nine plus one is ten. Negative three minus two would be minus five. So we know that our slope is actually negative two. We have a choice here of which one of these points we want to use. And it really doesn't matter. I'm going to choose this point. And the only reason why I'm choosing this point is because the numbers are littler. That's really the only reason I'm going to do that. So I know that x is going to be two, and y is going to be negative one, and I don't know what b is. So plug and chug into y equal mx plus b. Y, that looks like a seven, but that's actually just a negative one. And that's equal to my slope, which we said is negative two, times my x, which we said was positive two over here, x is two. And then plus b. One negative one is equal to negative four plus b. And if we add four to both sides to get rid of the four on the side that has the b, then we find out that b is equal to negative one plus four would be three. So again, here's what we need to write our equation. Y is equal to slope, which is negative two times x. I like to put my y in red. And then plus our b, which is a positive three. The last thing we can do is talk about if a line is parallel or perpendicular to another line. Remember we talked about parallel lines having the same slope in a previous video. So if this slope is two, then that tells me that the slope I have to use is two. And that also tells me from here, I know that x is equal to negative three. And y is equal to negative five. Plug and chug into y equal mx plus b. Y is equal to negative five, put that in red. M is two. And x is negative three. Plus b, negative five, it's going to be equal to negative six when we multiply plus our b. So we add six to both sides and we find out that b is equal to one. So if I write my equation, then y is equal to two times x plus one, because it's a positive one. And if I can do parallel, then I can also do perpendicular. And perpendicular slopes, remember, are negative reciprocals. So line one has a slope that is two. And my line then, which is perpendicular to it, would need to be a reciprocal, which would be one-half. And the opposite are negative reciprocals, so I need to put a negative there. I think I prefer to put opposite up here instead of negative, because if this had been a negative, you might not think what to do. It's the opposite of whatever sign we had. So it's a negative one-half slope that I'm going to use. So my slope is negative one-half. And again, plug and chug, and we know that x is equal to four, y is equal to seven. We found everything that we know. So y is seven, and slope is negative one-half. It's a negative one times my x, which is four, plus my b. Seven is equal to, and negative one-half times four, one-half of four would be two, and the opposite of it would be negative two, plus b. So we're going to add two to both sides, which will give us b equal to nine. And if I take all of this information into my equation, I will have y is equal to negative one-half times x plus b, which is a positive nine.