As I mentioned before, equations behave a lot like sentences, as they are statements that give you information. In this video, you will learn what a linear equation is, and what a system of linear equations is. As a matter of fact you will be solving your first system of linear equations. Which is extracting all the possible information from that system. Just like with systems of sentences, systems of linear equations can also be singular or non-singular based on how much information they carry. And as you already learned these concepts of real life sentences, you are more than ready to tackle them with equations. In the previous video, you saw sentences such as between the dog and the cat, one is black. For the rest of the course, you'll focus on sentences that carry numerical information such as this one. The price of an apple and a banana is $10. This sentence can easily be turned into equations as follows. If a is the price of an apple, and b is the price of a banana, then the equation stemming from the sentence is a + b = 10. Now, here's the first quiz in which you will be solving the first system of linear equations in this class. The problem is the following, you are going to a grocery store, but this is a very peculiar grocery store. In this store, the individual items don't have information about their prices. You only get the information about the total price when you pay in the register. Naturally, being a math person as you are, you're interested in figuring out the price of each item. So you keep track of the total prices of different combinations of items in order to deduce the individual prices. So the first day that you go to the store you bought an apple and a banana and they cost $10. The second day you bought an apple and two bananas and they cost $12. And the question is, how much does each fruit cost? So several things may happen, you may be able to figure out the price of the apple and banana. Or you may conclude that you don't have enough information to figure this out. Or even more, you may conclude that there was a mistake with the prices giving this information, all of these are options in the quiz. And the solution is that apple's cost $8 and bananas cost $2 each, why? Well, from day 1 you can see that an apple plus a banana is $10, from day 2, you can see that an apple plus two bananas is $12. So what was the difference between day 1 and day 2? Well in day 2 you bought one more banana than day 1. Also in day 2 you pay $2 more than in day 1. Thus you can safely conclude that that extra banana you bought on day 2 cost $2. The extra $2 you paid on day 2 were because of that extra banana you bought on day 2. And now that you know that bananas cost 2, well how much do apples cost? Well from day 1, you can see that an apple and a banana cost $10. So if a banana cost $2 then the remaining $8 must correspond to the apple. Thus, each apple costs $8 and each banana costs $2. Now here's quiz 2, the scenario is the same except the prices in the store are a little different, and you also bought different quantities of fruits. On day 1, you bought an apple and a banana and they cost $10. On day 2 you bought two apples and two bananas and they cost $20. The question is how much does each fruit cost? Remember that the options of not having enough information or having a mistake in the information given are both valid as well. For this problem, the solution is that there is not enough information to tell the actual prices, and why is this? Well you can use a similar reasoning than before. From day 1, you can deduce that an apple and a banana cost $10 from day 2, you can deduce that two apples and two bananas cost $20. But these two equations are the same thing. They may not look the same, but in disguise, they're the exact same thing. Because you see if one apple and one banana cost $10, then twice of one apple and one banana cost twice of $10, which is $20. So two apples and two bananas cost $20. Therefore the system is redundant because it basically has the same equation twice. It's like that system of sentences where both sentences stated that the dog was black, the system didn't carry enough information. Now what are the solutions to the system? Well, because the system doesn't carry enough information, the system has infinitely many solutions. Any two numbers that had to add to ten are a solution to the system. So for example if the apple's 8 and the banana's 2, then that works because apple plus bananas, 10 and two apples plus two bananas is 20. But if they're 5 and 5 that also works. If they're 8.3 and 1.7 that also works. And even then saying that the apples are free and the bananas are 10, works too. So this system has infinitely many solutions because you simply don't have enough information. You don't have the two equations to narrow it down to one single solution like you had with the complete system. And now, you're ready for a final quiz. Similar scenario except the first day you bought an apple and banana and they cost $10 and the second day you bought two apples and two bananas and they cost $24. Can you figure out how much each fruit costs? And remember there's still the options of not enough information or a mistake in the information. And the answer here is that there's no solution, why? Well in the same fashion as before, if one apple and one banana cost $10, then two apples and two bananas must cost $20. But the store charged you $24 for two apples and two bananas where are those four extra dollars? If you assume that there are no extra fees for buying more than one fruit or discounts or anything of that sort. Then you must conclude that that extra money must be due to a mistake with the register when you checked out in at least one of the two days. This means that these two equations contradict each other, just like the two sentences, the dog is black and the dog is white contradicted themselves. And this concludes that the system has no solutions. So here's our recap, in the previous three quizzes, you solved three systems of equations. The first one has the equations a + b = 10 and a + 2b =12. Because the price of an apple and banana was 10, and the price of apple and two bananas were 12. The second one has the equations a + b = 10 and 2a + 2b = 20. And the third one has the equations a + b = 10 and 2a + 2b = 24. The first one had a unique solution which was a = 8 and b = 2. For a is the price of an apple and b the price of banana. The reason this system has a unique solution is because both equations give you one different piece of information. Thus you're able to narrow down the solution to one unique solution. For this reason the system is complete and non-singular. The second system, has infinitely many solutions, which are any two numbers that add to 10. In this system, the two equations are the exact same one. So you never had a second equation to help you narrow down the solution to a unique one. This means the system is redundant and singular. And finally, the third system has no solution because the two equations contradict each other. Therefore, the system of equations is contradictory and singular. So as you see, we're using the same terminology as with systems of sentences and everything works in the exact same way. And finally, some clarification, you may have noticed the words linear equation several times, what does that mean? Well linear equation can be anything like a + b = 10, 2a + 3b = 15, 3.4a- 48.99b +2c = 122.5, anything like that. And notice that it can have as many variables as we want. But there's a special rule that must be followed, in a linear equation, variables a, b, c etc were only allowed to have numbers or scalar is attached to them. And there's also an extra number all by itself like the 122.5 here allowed to be in the equation. So in short, you can multiply the variable by scalars and then add them or subtract them and then add a constant and that's it. So what's an equation that's non-linear? Well, non-linear equations can be much more complicated. They can have squares like a squared b squared. They can have things like sine, cosine, tangent arctan, anything like that powers like b to the 5. They can have powers like 2 to the a or 3 to the b. And furthermore, you can actually multiply the a's and b's. In linear questions you can only add them, but in a non-linear equation you can have ab squared, b divided by a, 3 divided by b, things like logarithms, anything along those lines. So linear algebra is the study of linear equations like the ones in the left. And since they're much simpler then there are many things you can do with them, such as manipulating them and extracting information out of them. So we're only going to worry about the linear equations in the left. And the reason it's called linear algebra its because it's the study of linear equations.
