 So let's take a look at another form of the equation of a line. So if a line is vertical or horizontal, we've seen that we can use either point to write an equation for the line and get the same equation. But what if the line isn't horizontal or vertical? So if we consider the line through 2, 5, and 1, 3, we found the slope and then wrote down the equation of the line using the point 2, 5. But what if we'd use the other point 1, 3? Well, we could still write down the equation of the line, but it looks different, even though they are the same line. And so the question we have to ask is, which one of these is correct? To answer that question, let's do a little bit of algebra and solve both of these equations for y. The first equation, when we solve for y, gives us, what if we solve the other equation for y? And we see that after we've solved both equations for y, the two equations are in fact the same. And so it doesn't matter which point we use. And what this suggests is that there's another useful form of the equation of the line, which is known as the slope-intercept form. So we say that the linear equation is in slope-intercept form when it is in the form y equals mx plus b. And it's worth comparing this to our equation in the point-slope form. The advantage to the point-slope form is that as soon as you have a point on the line and a slope, then you can write down the equation of the line. So, for example, as soon as we found that the slope of the line between 2, 5, and 1, 3 was 2, we could use the point 2, 5 to write down the equation of the line immediately. What this means is that you never want to write the equation of a line in slope-intercept form. Slope-intercept form is only something you do after you get the equation of a line. So, for example, let's put the equation in slope-intercept form. And the thing to recognize here is that the equation is in slope-intercept form when it's in the form y equals something. So we've essentially solved our equation for y. So let's solve this equation for y. And now our equation is in slope-intercept form. How about writing the equation of a line in slope-intercept form? So let's try to write the equation of the line through 3, 1 with slope negative 2 in slope-intercept form. So the important thing to remember is that the easiest way to write the equation of a line is point-slope form. And once we have a point and a slope, we can write down the equation of the line immediately. Now, because the problem asks us to do this, we should rewrite this in the slope-intercept form. And we can do that by expanding and solving for y. Now, in the grand scheme of things, there's actually no good reason to write the equation in slope-intercept form. Point-slope form is the easiest and fastest way to write the equation of a line, and we really don't need any other form. On the other hand, if an equation in slope-intercept form should fall out of the sky and hit you on the head, then you can immediately identify certain properties of the corresponding line. I'd also watch where I was walking. For example, suppose you're given the equation y equals 3x minus 7, and you want to find the y-intercept. The y-intercept occurs when x is equal to 0, so we'll substitute that in and find our y-value. And remember, the intercept is a point, so the intercept is going to be 0, negative 7. And this suggests the following theorem. The line with equation y equals mx plus b has y-intercept, well, you know, memorizing theorems in formulas is actually the worst way to learn mathematics. It's more important to understand concepts, so you don't actually need this theorem, as long as you keep in mind what the y-intercept is. What if I want to find the slope of a line? Well, if I have the equation of the line in point-slope form, I can read off the slope immediately. But what if the line is in slope-intercept form? To answer this question, we need to remember the definition of slope, and we see that we can calculate the slope of a line between any two points. Which means that if we want to calculate the slope of this line, we need to find two points on the line. Since the equation of our line is in the form y equals a formula involving x, we can choose the value of x and compute y directly. So let's let x equals 0 and find y, which gives us one of the points on the line, 0, 4. Again, the useful thing to remember is that if we multiply a fraction by its denominator, the fraction disappears. So in this case, if we let x equals 5, our fraction will disappear. Oh, but that's too easy. Since we want to work with fractions, we'll let x equal 3 7s. And maybe not. Let's try to do things the easy way. So we'll let x equals 5, substituting that in, and we get y equals 2, and so a second point of the line is 5 2. So now I have two points, and that's all I need to calculate the slope. So the slope between the two points is, and that slope works out to be negative two-fifths. How about the slope of the line with equation y equals 2x plus 5? So again, all we need are two points on the line. So we'll let x equals 0 and find our y-coordinate, which gives us the point 0, 5. We need another point, so we'll let x equal, I don't know, 1. We like 1. If we let x equals 1, our y-coordinate is 7, and so the point 1, 7 is on the line. We have two points on the line, so we can find the slope between these two points, which works out to be 2. Or maybe we have the line y equals minus two-thirds x plus 7. So if we want to find the slope, we'll find two points. If x equals 0, then y equals 7. Since we have a fraction with a denominator of 3, we might want to let x equals 3, and so y equals 5, and so we have two points. We can calculate the slope between those two points, and it's negative two-thirds. So remember, we calculated the slope of the line with equation y equals negative two-fifths x plus 4 to be negative two-fifths. And we calculated the slope of the line with equation y equals 2x plus 5, and that slope was 2. And we calculated the slope of the line with equation y equals negative two-thirds x plus 7, and that slope was negative two-thirds. And so as the saying goes, once is an accident, twice eh, could be coincidence. But three times is a conspiracy. So many of the things that have happened on the internet can only be explained because it's controlled by a c- There is no secret cabal that controls the internet. Totally irrefutable evidence. Mathematically, we might look at this as follows. If you see a pattern, there may be a pattern. So we see the coefficient of x and the slope are the same. And so we get the following useful result. The graph of y equals mx plus b will be a straight line with slope m. For example, suppose we want to find the slope of the line y equals minus three-sevenths x plus five-thirteenths. Now, we do have our theorem, but if we forget the theorem, it's not a big deal because we can find the slope by finding two points on the line. All the theorem really does is save us a little bit of time. So let's assume we remember the theorem. We should read the fine print. This only works if we have the equation in form y equals mx plus b. So we'll check it out. Our equation is y equals minus three-sevenths x plus five-thirteenths, which is what we need. And so our theorem says that the slope will be the coefficient of x negative three-sevenths.