All right. Let's talk about the equation of a line. You're going to be using the equation on the line in a lot of your classes if you're taking an MBA program. But it's also very important just to understand what are the mathematics of describing a linear equation when you have different points. So the linear equation and the mathematical description of a data in a linear form, we're going to be revisiting again in the regression portion of this tape. So it's good to understand the basics of what's going on. Here, I have the three forms, the three ways that a line can be illustrated mathematically and can be solved mathematically. These are three expressions. There are three forms of linear equation that are generally used in mathematics. One of this is called the standard form, ax plus by equals c, where a, b and c are integers, a and b are not zero. Okay, we also have what's called the slope intercept form of a line. Here, y equals mx plus b. M is the slope of the line and b is the y-intercept of the line. Then lastly, we have what's called the point slope form, where m is the slope of the line and x_1 and y_1 is a point on that line. Most likely, you will be using what we call the slope intercept form of a line. So, I'm going to be teaching you how to use that and what it means. So, to calculate the slope intercept form of a line, first is we have to have a set of data. Right here we have what's called a demand schedule. We've got the price of a good and the corresponding quantity demanded of that good. So, this could be the price of a cup of coffee and the quantity demanded for the number of cups of coffee that you might demand, let's say in a day or a week or something like this. Now this demand schedule is very specific. What I have is I've identified something that is the exact same slope over the entirety of this schedule. In reality, this is not what you're likely to find, but it's a good place to start with our understanding of how to calculate and how to express data in a mathematical form. So, in order to calculate the slope intercept form of a line, y equals mx plus b, we have these data. We have price and quantity schedule. Now, the first step is to identify the slope. I have written here m is equal to change in y over change in x, where the change in y is y_2 minus y_1 and the change in x is x_2 minus x_1. This may be illustrated as y new minus y old or y new minus y original, and so we're always talking about like the second point minus the first point for y, the second point minus the first point for x. The y and x are representative of their axes. So, let's draw a x and y diagram of these data, and show you how you actually calculate the slope intercept form of this line. Suppose that we drew our axes here and we labeled this axes price, and we label this axes quantity. Now prizes the y-variable and quantity is the x variable. In order to calculate my algebraic expression of a line, I need y_1 and y_2 and its corresponding x_1 and x_2. Let's suppose that we used these two points right here. Points nine and three and points eight,six. So, now I'm going to identify this first one right here. This is going to be y_1. So then, this corresponding point over here is going to be my x_1. This second point right here, this eight, this is going to be my y_2. This second point right here, that's going to be my x_2. There's my y_2 and my x_2. So, now what I have is I can calculate my slope down here. So, here's my slope is going to be y_2 minus y_1. We have to be consistent. So, if my y_2 is eight, my y_1 is nine, and my x_2 is six and my x_1 is three. So, now I've got negative 1 over 3. Now, some people they get a little anxious about the fact that you're subtracting nine from eight, and they say, "Wow, this just makes me a little bit uncomfortable. Can I just switch that nine and the eight?" You can, but it would be wrong. What you want to do is you want to tell you calculate the slope. So you have to keep everything consistent. In this case, my slope is negative one-third. So then my second part of my slope intercept form is the y equals mx plus b. The b is the y-intercept. Now, we can calculate the b in two different ways. First, if we have the complete schedule, we can calculate the b by saying, at what point of quantity equals zero? What is the value of b? Because b is the intercept. B is where on this axis is my line going to intersect the axis? So, at what quantity zero, what is the value up here of my y? Well, here's where quantity is zero and the corresponding value is 10. So, I'd say, b equals 10. So, my equation would become y equals mx plus b, y equals negative 1/3x plus 10. Now of course, my y and my x, they've got slightly different letters. There's a price and there's a quantity. So, it would be most appropriate for me to express this algebra as this, price equals negative one-third quantity plus 10, and that is the slope intercept form of a line that is consistent with these data right here. So, if I was graphing them, I would have price and quantity on this axis, and I would have a line that looks like this, and this would be 10 over here, down here would be 30, and the equation of this line would be price is equal to negative one-third quantity plus 10. That's the slope intercept form of this line.
