1151 Ff - Equations of lines

Today we're going to be talking about equations of lines. We'll find that linear functions are the most important functions in the world because they're the only ones in which the rate of change of the Y coordinate with respect to the X coordinate is always constant. That is, we're basically on cruise control if you're thinking from a speed situation. These equations of lines, you'll be seeing throughout the semester because we take advantage of the constant rate in calculus to find out what's going on when the rate is not constant. We work with that over, and over, and over again. You're going to be seeing the problem, find the equation of the line, over and over again this semester and even beyond when you deal with something called the fundamental theorem of calculus. Something we talk about here will be the most basic situation in which that fundamental theorem turns out to work. To begin, what we're going to do is first talk geometrically a little bit. We're going to ask, what pieces of information do we need to know about a line to be able to determine the entire line. From a geometric standpoint, we're given two points on the line. It's fairly obvious and some might consider this to be an axiom of geometry in fact, that if you have 2 points here, there is exactly one line that fits, that contains those 2 points. No other line contains both those points at the same time. It turns out the two points are enough to give us a unique line. Also if we're given a point and the slope of the line, which is basically the direction we can look at geometrically in which the line is going in. If we we're given the point, if we were just given a point alone here, we would have all sorts of lines that would go through, that would contain, that point. Over and over and over again. As you can see, each of those lines has one specific direction. If we gave that slope in addition to the information here, that would nail down exactly which of these lines that we wanted to talk about. It turns out that that point of view, is that the point and the slope together will determine one unique line. On the other hand, if we were just given the slope of the line, and say that slope was 3, or 5, or some positive number, that would, just like given the point here, not give us enough information to determine one unique line. In fact we would have a family here of parallel lines all going in the same direction. What if we were given one point, one singular point on the line, as well as the slope? We would know our line was one of these lines, but we want to know which one it was. It might be some other parallel line that I haven't drawn. If we were given a point, on the line, say this point here, we would know since all parallel lines basically are selfish, they share no points with each other, we would know that only one of these lines could contain that point. Pretty lousy line drawn there but we'll live with it. It turns out geometrically speaking, given both a point and the slope of a line, we are able to determine one unique line. Now let's look at the problem algebraically, which is the way that you will be dealing with it in the future. By the way, it's somewhat amazing that just these small amounts of pieces of information, the 2 points or the point and the slope is enough to determine an infinite number of points. You get the whole family by just knowing those couple of pieces of information. Now let's go on and talk algebraically. The first form of an equation of a line, which also determines all the points of a line, now numerically or algebraically rather than geometrically, we're going to go with probably the more famous Y equals M X plus B. The old slope intercept form of the line. We're going to see where that comes from. We're going to do that, as well as finding other forms of the line of an equation of a line, by meeting a friend named free-lance Freddy. Let's read the problem here. Free-lance Freddy works a new job for a new hourly wage every day. He brings money for lunch and, as he works, keeps track of how much money he has, including his lunch money, with a meter. Maybe to keep his employers nervous. I don't know. The first problem about Freddy: today Freddy flips burgers at Mickey B's Burger Emporium. He forgot to bring lunch money. He has no lunch money today. He looks at his meter 5.3 hours into the job and sees that he has 47 dollars. We want to know, how much money will the meter register at 7.8 hours. Then in general, how much will the meter register at X hours. Basically what we're doing here in all of this when asked for example, how much money will a meter register at 7.8 hours, we're finding out can we determine any point that we would like along the ling that would be representing the relationship between the amount of hours Freddy works and how much money he has including his lunch money. Let's solve this problem. First of all, to answer how much money he'll have at 7.8 hours, what we would like to do, is to find out what rate, it's quite natural in fact to ask at what rate is Freddy earning money each hour. To do that, we know that he's worked 5.3 hours and he has 47 dollars. Now none of that 47 dollars has anything to do with his lunch money because he forgot to bring his lunch money. His lunch money today is 0. That entire 47 dollars came from his earning money over time. What we would like to do, this is a division problem, we have 47 dollars, what we would like to do is find out how many dollars are devoted to each one of those hours. We take 47 and divide by the number of hours, 5.3 hours, or if you will, a fraction. That turns out to be around, we're going to be very approximate in this lecture, 8.87 dollars, 8 dollars and 87 cents, in each one of those hours. 8 dollars and 87 cents per hour. Now, to answer the question then, how much money will the meter register at 7.8 hours? Well that becomes quite simple because we each hour, we devote 8.87 dollars and 8 dollars and 87 cents. All we need to do here, is we want 7.8 copies of that 8 dollars and 87 cents. We would solve that as the amount that he has at 7.8 hours, is going to be 8 dollars and 87 cents per hour times 7.8 copies of that. That turns out to be around 69 dollars and 19 cents. We don't really care too much about Free-lance Freddy. He's a nice guy but we're not too concerned about finding his amount of money hour per hour per hour. What we would like to do, at least for this lecture's purpose, is find the amount at X hours. In fact, let's call that another variable called Y. What we would do is the very same thing in the numerical example. We would have not X copies of the 8 dollars and 87 cents for each hour. That would be 8.87 times X. We have a linear equation of the form Y is equal to 8 dollars and 87 cents per hour times X hours to give us the amount of dollars he has at that time, after working for that amount of time I should say. Notice that this is the most basic of linear equations. In fact is a very pure ratio. The most important thing, just like geometrically, is we were given a certain amount of information in this 8 dollars and 87 cents per hour, is essentially what we called, in school, our slope or our rate of change of dollars per hour, per each hour. Slope is a rate. No matter how you cut it, it will always be some sort of rate. Some sort of per quantity of one variable, one quantity per some other quantity. We see, that given that slope, technically we were given a point on the line as well, that case being 0, 0. When he worked 0 hours, he had 0 dollars because he had no lunch money. We were able to find an equation that describes all point on the line X Y and a formula to be able to find those points. Now let's try to look at a different scenario with Free-lance Freddy. We have now, Freddy is teaching math at OSU. He brought 8 dollars and 75 cents in lunch money today. We now know he has some lunch money. At 0 hours we know he has 8 dollars and 75 cents, even before he begins his work. Now he looks at his meter 5.3 hours into the job and sees that he has 47 dollars. Remember that includes his lunch money. The question is the same question we asked earlier. How much money will the meter register now at 7.8 hours into his job? Let's assume for all these problems, he never really uses his lunch money. He's so involved in his work. Again, we also want to know how much will the meter register in general at X hours, which is our major goal for this lecture. Now this is an arithmetic question. Once again what we would like to know is what is the rate that he is earning his money? What is the slope of this situation? Since he's earning at a constant rate, what is that rate? To answer that, we have that 47 dollars, part of that is his lunch money. It really has nothing to do with how much money he's earning per hour. What we would like to do first is let's just work with the money he's earned over that time and knock out that lunch money out of the 47 dollars. We subtract 47 minus 8.75. What we would like to do with that now is take that amount of money, whatever it might be, and divide that by how many hours he works. This is the amount of money he's earned over that period of time, and this is the period of time which he's earned it. If you take this and crank it on your calculator, what you'll get is that he earns 7 dollars and 22 cents per hour on this job. How are we going to use that? Well, what we know is that his money comes in two parts. One is his lunch money. One is from his from his earning 7 dollars and 22 cents per hour. The question asks, is how much will the meter register at 7.8 hours? In order to solve that, basically, what we can do is take part of that, use part of the meter, the amount on the meter at 7.8 hours, once again we can call that Y if we like. How do we calculate that? We can take, some of that money is his lunch money, and the rest of that money comes from his earnings. This is like the problem from before where we had, ok every hour he gets 7.22 dollars, or 7 dollars and 22 cents, and he is going to be working for 7.8 hours goes by. We just need 7.8 copies of that. That's a multiplication problem. We wind up with, if you crank this on your calculator, he has 8 dollars and 75 cents plus 56 dollars and 32 cents he's earned. All totaled on his meter, he's going to have 65 dollars and 7 cents. Now what we would like to do here is find out what about Y in the amount on the meter at X hours. In other words, for any number or hours of working that we want, can we predict how much money Free-lance Freddy will have on his meter? Well, we do the arithmetic to kind of give us an idea of what goes on in the algebra. It's a good algebraic thinking process to do an arithmetic situation first and then mimic that when doing the general case. In this case well, we still the 8 dollars and 75 cents that he has to start with. Then we're going to have X copies of 7 dollars and 22 cents for each one of those hours that goes by. That's going to be 7.22 times X, or if you will, the Y would be, using the commutative property, switching things around of addition that is, 7 dollars and 22 cents X plus 8 dollars and 75 cents. Now you see, here is your slope, times X plus the initial value. We know this is how much he has at 0 hours. He had 8 dollars and 75 cents at that point. This is basically your Y intercept if you were graphing the line. We have slope and essentially the Y coordinate of the Y intercept. Put together helps make the equation of the line to find out exactly any amount of money he wants at any time. In other words, it determines the entire line, given the slope and the Y intercept. However, if you recall geometrically, we didn't necessarily need the Y intercept to be out point to go along with this slope to determine amounts, to determine the entire line. We can find any old point as long as we know that and the slope, we can find the line geometrically. Let's look at that algebraically. Today, Freddy is now starting pitcher for the Cleveland Indians. He brought some lunch money. We don't know how much lunch money he has. That's going to be a little bit mysterious for a while. I'm kind of hiding the Y intercept from you. Now he looks at his meter 5.3 hours into the job and sees that he has 4 hundred and 78 dollars. He also then looks at his meter 7.8 hours into the job and sees he has 700 dollars and 50 cents. Well we know the major league pays a lot of money. A lot more than teaching at OSU or working at Mickey B's burger emporium. Without figuring out how much lunch money he has, this is going to be very important here, we want to resist the temptation for finding out how much lunch money he has, how much money will the meter register at 9.34 hours? Again we want to find out how much money will he have at some other time other than what's given here. Then later how about at X hours? Then when we're finished, we'll do the same problem but then we'll go back and find out his lunch money first. For now we're going to do it without figuring out how much lunch money he has. What are we essentially given? We're given two instances of the relationship between the number of hours and how much money he has. Essentially geometrically, we're given two points on the line that he's earning with a constant rate. Now just like the other problems, it would be great and very helpful to find out just how much money he's earning per hour. This is not the same as his lunch money obviously. This is asking, what is the slope? Well even without knowing any formula, it's pretty, well hopefully a little bit easy to see, how we're going to figure that out. We know we only want deal with how much money he's earning per hour. Obviously that doesn't have anything to do with his lunch money. We know that at 7.8 hours, he has 700 dollars and 50 cents, and at another time, 5.3 hours, he has 478 dollars. Can we figure out much money he's earned over that period of time? Well sure. The amount of money he's earned, the rate of earning or his hourly wage, is how much money he earned over that period of time, that's going to be 700 dollars and 50 cents, and knock out 478 dollars. Notice here by the way that by subtracting, this is the amount earned in that amount of time. This difference, whatever that number might be, is not going to have anything to do with his lunch money because we know that his lunch money is somewhere contained within that 700 dollars and 50 cents and we also know his lunch money is contained within that 478 dollars. When we subtract here, whatever that lunch money was in here and in here, gets subtracted out. All that's left is what he's earned over that period of time. How do we figure out how much time he's taken to earn that much money? Well, this was at 7.8 hours and we knock out the 5.3 hours from before and we see that he's earned that amount of money over this difference in time. Calculating this, we find that he's earned; this will be 222 dollars and 50 cents, over a period of 2.5 hours, which winds up being 89 dollars per hour. Notice we've just found the slope. Without even memorizing, or knowing some formula, it makes sense that to find the slope given two points, we do the change of Y over the change of X, which is something you've learned in school. Hopefully it makes sense. Hopefully it made sense in school, but it not, hopefully it makes sense now. We still haven't solved the problem yet though. We want to know, how much money will the meter register at 9.34 hours? Was our original question. All we have now is our rate of change. Our constant rate, our slope, of his earning. 89 bucks per hour. We're not allowed to find his lunch money, so all we can work with is maybe those two instances that we already have. For example, let's look at, take advantage that, well we don't know how much he has at 0 hours, but we sure know how much he has at 7.8 hours into his job. We know that he has 700 dollars and 50 cents. We would like to know, what about his situation just a little bit later from 7.8 hours. What about at 9.34 hours? What we can do is the amount, again we can call this Y like before, the amount of money at 9.34 hours, what we can do is say eh, we already know that he has 700 dollars and 50 cents at some point in time. We could do this, we could take this amount of money here and just add what additional amount of money that he earned between 7.8 hours and 9.34 hours. Well, we know what the slope is. We know the slope is 89. We want to know how many 89's will take place between the 7.8 hours and the 9.34 hours. Well, that's the amount of time between those two times. How do we find that amount of time is by subtracting 9.34 hours minus 7.8 hours. Essentially we've solved the problem. This is going to give us our starting amount and then we add how much it changed over the intervening hours. Between 7.8 and 9.34 hours, with 89 dollars per hour given to us. We crank that out just with arithmetic on our calculator and we get 700 dollars and 50 cents for our start. Our change turns out to me 137 dollars and 6 cents roughly. Putting the start plus the change together, us our new amount of 837 dollars and 56 cents. That solves the problem without finding his lunch money. Notice we could have solved that same problem with the starting amount being what he had at 5.3 hours, which was 478 dollars. Let's figure out, is this the same? We're going to start with the 478 dollars and add in, what about, we want 89 dollars per hour, but now at sense 5.3 hours, not 7.8 hours. We figure out how much time took place between then. If you crank this on your calculator, I won't do it, but u will find it will be 478 plus the amount of change, again this start plus the amount of change idea, you will wind up with same amount. That makes sense because there is only one line that's going to contain those points in these slopes. We just happen to be working with a different point that time. Well, how much will the meter register at X hours? Well, we will just mimic the arithmetic. We can take one of the initial points that we want, take your pick. We saw both will wind up giving us the same point because we're on the same line. That will be our starting value plus your slope, times your difference in time between X hours and the time that we have. That's going to be X minus 7.8. We have a start plus change equation. If you want you can simplify that to get it to Y equals M X plus B form if you would like. We're not going to do that though right now because we're going to emphasize this form of the line which is called by the way, the point slope form of the line because it gives you the point, 7.8, 700.50, and it also involves the slope. From that, you can find his amount of money at any time that you wish as long as he keeps working. Now what happens in general? If we're given two points on the line, he's starting pitcher for the Indians again. He brought some lunch money again which we never did find. He looks at his meter X sub 1 hours into the job and see's he has Y sub 1 dollars. He looks at his meter X sub 2 hours into the job and sees he has Y sub 2 dollars. Once again, how much money will he have, without figuring out his lunch money, at X hours into his job? Well, we would just mimic the arithmetic we did before. The first thing we did was we found out at what rate is he earning money. That was our slope, often called M which by the way comes from the word multiplier because you're multiplying that by how many hours he works in varies to these problems. How do we find that? Well, we found out how much money he earned, over the specified period of time that we're given, and divide by the number of hours that he worked. We know that from school, but now we know if from making sense. We're just going to call that M for now rather than having to re-write that quotient over and over again. Then, what did we do to find out how much money he has at X hours? Well, we took one of these initial values we had, say Y sub 1, and we added in how much in changed here. This is like your starting amount you might recall, and then we add how much it changed. Well, how much did it change? Well, he earns M dollars per hour and we want to find over how much time since that starting value did it go. Well, that was however many hours you want, the X, minus the known number of hours, which was X sub 1. This is called the point slope form of the line. You may have remembered this from school as looking like this, an equivalent form, as Y minus Y sub 1 is equal to M times X minus X sub 1. Basically you're looking at is as the change of Y from the known value of Y, is equal to the change that we've been talking about. The amount of time that went by, times how much money he's earning every hour. You may also have learned it looking at it this way: Y minus Y 1 over X minus X sub 1 is equal to M. Basically what that tells you is whatever point X Y you are, when you calculate the slope from the known point, the change in Y over the change in X, you will always wind up with that same unique slope; basically says any line only has one slope. Now you'll be using, probably more often than the slope intercept form, you'll be using this point slope form of the line quite often in math 1151. Several different places. One will be finding, what we're going to call, the equation of the tangent line, and the corner stone of that will be finding that slope which is now going to be changing with functions. You're going to be wanting to find what is the unique slope at any particular time. Somewhat like taking a picture of your speedometer as you drive along. Although that speed might be changing, you have a unique speed at any particular time. You're going to want to find the line that seems to be going along just like the function is at that time. You'll be using the point slope form of the line to find that equation. You will also be doing that within linear approximation which basically is the point slope form of a line. Also there you will be using it within the fundamental theorem of calculus. Again where the slope is always changing, but we'll catch that when it comes up later in the semester. That would take a while to get through right now to talk about what that would mean. You will still be using this basic idea of , let's start where we know and add on how does that change, and that's basically the theme of the point slope form of a line. By the way, we did not solve one of the problems that I gave you earlier about Freddy. Let's look at that. It says, when you are finished, do the same problem, now by first finding out how much is lunch money was. We're going to kind of cheat and instead of finding the point slope form of the line, we're going to work towards finding the slope intercept form because the Y intercept, if you recall, was the amount of money he had at 0 hours which we call his lunch money. How are we going to do that from this information? Let's find his lunch money and just using basic arithmetic ideas and without delving into any formulas we've memorized from school. In fact we're going to find that you do the exact same thing you might have done in school. Algebraically we're going to do it with arithmetic. We know that he looks at his meter 5.3 hours into the job and he sees that he has 478 dollars. He also looks at his meter 7.8 hours into the job and sees he has 700 dollars and 50 cents. Obviously we did figure out what his rate of change was of earning money which was 89 dollars per hour. We know he's earning his money at 89 dollars per hour. We also know from this amount of money that are given, say with the 478 dollars, let's just work with that for now, we know that some of that 478 dollars was his lunch money. Some of it was the amount of money he earned over that period of time, the 5.3 hours. Let's figure out his lunch money from just knowing that. We know that 89 dollars times the 5.3 hours is going to give us how much money he earned over that period of time. That turns out to be 89 times 5.3, is 471 dollars and 70 cents. In that 5.3 hours, of that 478 dollars, 471 and 70 cents was what he earned with the Indians. That means the left over part, whatever was left out of the 478 dollars, was his lunch money. How are we going to figure that out? That's just subtraction. 478 dollars minus the 471.70 turns out to be 6 dollars and 30 cents. Maybe I should have rigged the numbers a little bit better. That's' not going to buy him too much up at progressive field with the Indian's. But he's playing anyway so he probably gets that in the club house. We know that he has 6 dollars and 30 cents when he comes on to the job and earns money at 89 dollars per hour. We know then, if we wanted to write this in the point slope form of the line, we can rephrase the whole problem as, he earns 89 dollars per hour and he starts with 6 dollars and 30 cents. His slope is 89. Kind of your Y intercept value is 6 dollars and 30 cents. Therefor the slope intercept form of the line would be 89 dollars X plus 6 dollars and 30 cents. Notice this process we just went through to find his lunch money, and then by the way you can always solve the rest of the problems then by using the slope intercept form now. For example you could just plug in the 9.34 into this and figure out what he has. He's worked a total of 9.34 hours at 89 dollars an hour and then you tack on the 6.30 he brought in for lunch; you're going to get the same answer we got before for his amount of money at 9.34 hours. If you notice this arithmetic process you've probably done before when you said in school, I don't care about the point slope form of the line I want the slope intercept form. I know because he's earning 89 dollars per hour, I know the equation of the line must be Y equals 89 X plus B. I also happen to know a point on the line. What did you do? Well, you often just plugged in the known value of X, 5.3, in search of B, and you then plugged in Y equal to 478. Then what did u do to solve B? You just subtracted this amount from both sides. 478 minus 89.5.3 which happens to be the exact same calculations you did before. We just calculated that product then we subtracted 478 minus that product and that gave us the 6 dollars and 30 cents. Therefor we know that's our equation of the line and we can find out how much money he has at any point. Just by reasoning, with arithmetic, we can do exactly what you did way back when with algebra and find the slope intercept form of a line. Which equation of the line is better to use? It doesn't matter. They're both representing the same line because we're given the same amount of information about it. We know geometrically, given 2 points, we can find the unique line. Given a point and the slope we can a unique line. We've just now done it algebraically and we've done it in two different ways. When it comes time to find the equation of the line in calculus class, now you have a couple of different ways to approach it.