Topic D: Systems of linear equations and their solutions
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You are traveling in some type of a strange fantasy land. And you're trying to get to the castle up here to save the princess or the prince or whomever you're trying to save. But to get there, you have to cross this river. You can't swim across it. It's a very rough river. So you have to cross this bridge. And so as you approach the bridge, this troll shows up. That's the troll. And he says, well, I'm a reasonable troll. You just have to pay $5. And when you look a little bit more carefully, you see that there actually was a sign there that says $5 toll to cross the bridge. Now, unfortunately for you, you do not have any money in your pocket. And so the troll says, well, you can't cross. But you say, I need to really, really get to that castle. And so the troll says, well, I'll take some pity on you. Instead of paying the $5, I will give you a riddle. And the riddle is this. And now, I'm speaking as the troll. I am a rich troll because I get to charge from everyone who crosses the bridge. And actually, I only accept $5 or $10 bills. It's a bit of a riddle why they accept American currency in this fantasy land. But let's just take that as a given for now. So I only take $5 or $10 bills. I'm being the troll. Obviously, if you give me a $10, I'll give you $5 back. And I know, because I count my money on a daily basis. I like to save my money as the troll. I know that I have a total of 900 bills. So let me write that down. I have a total of 900 bills, a total of 900 $5 and $10 bills. And he says, because I'm very sympathetic, I'll give you another piece of information. He says, if you were to add up the value of all of my money, which is all in $5 and $10 bills that I have, I, speaking as the troll about dollars bills, is $5,500. This is a rich troll. And so the riddle is-- exactly, exactly. And if you give the wrong answer and if you're not able to solve it in 10 minutes, he's just going to push you into the river or do something horrible to you. He says, exactly how many 5's and 10's do I, the troll, have? So the first thing I'm going to have you think about is, is this even a solvable problem? Because if it's not a solvable problem, you should probably run as fast as you can in the other direction. So now, I will tell you, yes, it is a solvable problem. And let's start thinking about it a little bit algebraically. And to do that, let's just set some variables. And I will set the variables to be what we're really trying to solve for. We're trying to solve for the number of $5 bills we have and the number of $10 bills that we have. So let's just define some variables. I'll say f for 5. Let's let f equal the number of $5 bills that we have. And I'll use the same idea. Let's let t is equal to the number $10 bills that we have. Now, given this information, and now I'm not sure if I'm speaking as-- well, let's say I'm still speaking as the troll. I'm a very sympathetic troll, and I'm going to give you hints. Given this information and setting these variables in this way, can I represent the clues in the riddle mathematically? So let's focus on the first clue. Can I represent this clue that the total of 900 $5 and $10 bills, or can I represent that mathematically, that I have a total of 900 $5 and $10 bills? Well, what's going to be our total of bills? It's going to be the number of 5's that we have, which is f. The number of 5's that we have is f. And then the number of 10s that we have is t. The total number of 5's plus the total number of 10s, that's our total number of bills. So that's going to be equal to 900. So this statement, this first clue in our riddle, can be written mathematically like this if we defined the variables like that. And I just said f for 5, because f for 5 in t for 10. Now, let's look at the second clue. Can we represent this one mathematically given these variable definitions that we created? Well, let's think separately about the value of the $5 bills and the value of the $10 bills. What's the value of all of the $5 bills? Well, each $5 bill is worth $5. So it's going to be 5 times the number of $5 bills that we have. So if I have one $5 bill, it will be $5. If I have 100 $5 bills, then it's going to be $500. How ever many $5 bills, I just multiply it by 5. That's the value of the $5 bills. Let me write that down. Value of the $5 bills. Now, same logic. What's the value of the $10 bills? Well, the value of the $10 bills is just going to be 10 times however many bills I have-- value of the $10 bills. So what's going to be the total value of my bills? Well, it's going to be the value of the $5 bills plus the value of the $10 bills? And he tells me what that total value is. It's $5,500. So if I add these two things, they're going to add up to be $5,500. So this second statement we can represent mathematically with this second equation right over here. And what we essentially have right over here, we have two equations. Each of them have two unknowns. And just using one of these equations, we can't really figure out what f and t are. You can pick a bunch of different combinations that add up to 900 here. You could pick a bunch of different combinations, where if you work out all the math, you get $5,500. So independently, these equations, you don't know what f and t are. But what we will see over the next several videos is that if you use both of this information, if you say that there's an f and a t that has to satisfy both of these equations, then you can find a solution. And this is called a system of equations. Let me write that down-- system of equations.