WEBVTT mathematics/pre-calculus/selhorst-jones 00:00:00.000 --> 00:00:01.900 Hi--welcome back to Educator.com. 00:00:01.900 --> 00:00:05.800 Today we are going to talk about variables, equations, and algebra. 00:00:05.800 --> 00:00:11.400 What is a variable? We talk about them all the time: we want to think of a variable as just being a placeholder. 00:00:11.400 --> 00:00:19.400 It is a placeholder for a number; it is a symbol that stands in for something that can come in later; it is standing in for a number. 00:00:19.400 --> 00:00:24.700 Sometimes the variable will be able to vary; it is going to be able to change, depending on what we want to do. 00:00:24.700 --> 00:00:28.000 And as the value of the variable changes, it will affect something else. 00:00:28.000 --> 00:00:33.300 It might affect the output of the function; it might affect some other dependent variable, 00:00:33.300 --> 00:00:38.900 if we see something like y = 3x, where you change x--we make it the independent variable. 00:00:38.900 --> 00:00:46.500 So, we put in different things for x, and it causes our dependent variable, y, to change, varying on what we put in for x, or something else. 00:00:46.500 --> 00:00:52.700 That is one way of looking at a variable: it is something that is allowed to vary, and it causes other things to shift around as it changes. 00:00:52.700 --> 00:00:57.200 Other times, we are just using a variable as a fixed value we don't know yet. 00:00:57.200 --> 00:01:02.500 Sometimes, it might even be multiple possible fixed values; it could be fixed values or a fixed value. 00:01:02.500 --> 00:01:10.200 But the point is that it is something that we just don't know yet; it is a placeholder for something that we want to find out more about. 00:01:10.200 --> 00:01:14.900 So, normally, we are going to be able to figure out what it is, based on the information given to us in the problem. 00:01:14.900 --> 00:01:19.800 Otherwise, it is probably not going to be a very good problem, if we can't actually solve for what the variable is. 00:01:19.800 --> 00:01:24.700 So, we will almost always have enough information to figure out what this variable is. 00:01:24.700 --> 00:01:29.000 That is the other possibility: a variable is something that we just don't know yet. 00:01:29.000 --> 00:01:33.600 It is a number that has been given a name, because we are trying to figure out more information about it. 00:01:33.600 --> 00:01:40.600 It is like if a detective is trying to find out who committed a crime; they might talk about the perpetrator, and they might find facts out 00:01:40.600 --> 00:01:45.700 about the perpetrator, until they have enough information to be able to figure out who the perpetrator actually is. 00:01:45.700 --> 00:01:51.100 "Perpetrator" is just a placeholder for some other person, until they figure out who that person is. 00:01:51.100 --> 00:01:55.100 Great; we can name variables any symbol that we want. 00:01:55.100 --> 00:02:01.100 Normally, we are going to use lowercase letters to denote variables; but occasionally, we will use Greek letters or other symbols. 00:02:01.100 --> 00:02:05.100 When we are working on word problems, we are going to choose our symbols...we want to choose the letter, 00:02:05.100 --> 00:02:11.800 or maybe other symbol that we use, based on something that helps us remember what it is representing in this word problem. 00:02:11.800 --> 00:02:15.200 What am I using this variable to get across? 00:02:15.200 --> 00:02:21.800 There are a lot of them that we regularly use; and so, we will get an idea of what they are; here we go! 00:02:21.800 --> 00:02:25.500 Any symbol could potentially be used for any meaning at all. 00:02:25.500 --> 00:02:30.000 We could make a smiley face; and I sometimes do use a smiley face to represent a number. 00:02:30.000 --> 00:02:36.100 But smiley face is a little bit harder to draw than x, so we tend to use letters that we are used to drawing. 00:02:36.100 --> 00:02:40.800 Anything could potentially be used for anything else; but here is a list of common symbols used, 00:02:40.800 --> 00:02:43.200 and what the meaning we normally associate with them is. 00:02:43.200 --> 00:02:47.000 Occasionally, we will have different meanings associated with them, depending on the problem. 00:02:47.000 --> 00:02:51.000 We might use y to talk about the number of yaks that there are at a farm; 00:02:51.000 --> 00:02:56.300 but generally, we are going to use them as we see right here--all of this right here. 00:02:56.300 --> 00:03:00.600 So, x is our most common one, probably our favorite variable of all. 00:03:00.600 --> 00:03:04.800 We use it for general use, when we are talking about horizontal location or distance. 00:03:04.800 --> 00:03:12.400 y is vertical location or distance; t normally stands in for time; n stands in for a quantity of some stuff. 00:03:12.400 --> 00:03:19.400 θ--this is a Greek letter, θ; when we encounter Greek letters, I will talk about them a little bit more, but mainly, it is just going to be θ. 00:03:19.400 --> 00:03:25.300 We draw θ by hand; you just make something sort of like you are drawing a zero or an O, 00:03:25.300 --> 00:03:29.200 and then you just draw a line straight across the middle of it; that is θ. 00:03:29.200 --> 00:03:40.800 r is radius; A is area; V is volume; and we often use a, b, c, and k to represent fixed, unchanging values-- 00:03:40.800 --> 00:03:44.400 values that are not going to vary and change into other things--things where we know 00:03:44.400 --> 00:03:48.400 that they are going to just stay the same, but we don't know what they are yet. 00:03:48.400 --> 00:03:51.300 Or we might decide on what they are later on. 00:03:51.300 --> 00:03:57.800 Anyway, this gives us a general idea of what the normal stable variables we constantly encounter are. 00:03:57.800 --> 00:04:01.100 Now, you might use x for something totally different than what we have here. 00:04:01.100 --> 00:04:04.600 You might use r for something totally different; you are not stuck to just using this. 00:04:04.600 --> 00:04:09.100 But we are going to see them in a lot of problems, and we want what we do to make sense to other people. 00:04:09.100 --> 00:04:12.900 So, it is good to go along with these conventions, usually. 00:04:12.900 --> 00:04:18.100 All right, what is a constant? A constant is a fixed, unchanging number. 00:04:18.100 --> 00:04:21.100 It is a value that does not turn into another value. 00:04:21.100 --> 00:04:29.000 So, we can have variables become different values; we might plug in x = 3, and then plug in x = 5, and then plug in x = 7. 00:04:29.000 --> 00:04:32.200 But a constant only has one thing; it just stays the same. 00:04:32.200 --> 00:04:40.900 So, any time we see a number, like 3 or 5.7 or -82 or anything that is just a number, it is a constant, because numbers do not change. 00:04:40.900 --> 00:04:44.200 After all, we don't have to worry about 3 suddenly turning into 4. 00:04:44.200 --> 00:04:48.600 It is just 3; it is going to be 3 today; it is going to be 3 tomorrow; it is going to be 3 forever. 00:04:48.600 --> 00:04:51.700 3 doesn't suddenly jump around and become a new number. 00:04:51.700 --> 00:04:55.200 Occasionally, we might refer to a symbol that is representing a number as a constant. 00:04:55.200 --> 00:04:58.400 We might say a is a constant in this problem. 00:04:58.400 --> 00:05:04.800 We might not know what value that symbol represents; but we know it cannot change--a constant is something that cannot change. 00:05:04.800 --> 00:05:11.300 And other times, we might even refer to a symbol as a variable, and just know that that variable is fixed, that it is a constant variable. 00:05:11.300 --> 00:05:16.300 It seems kind of like a contradiction in terms, but remember: we are using "variable" more for the idea of placeholder. 00:05:16.300 --> 00:05:21.300 And while sometimes it varies, sometimes it can also just be a placeholder in general. 00:05:21.300 --> 00:05:26.100 A constant is something that isn't going to move around; it is one number, and one number only. 00:05:26.100 --> 00:05:33.300 It doesn't matter if it is a symbol, or if it is actually a number; but the idea is that it is something that is not going to change. 00:05:33.300 --> 00:05:36.600 A coefficient is a multiplicative factor on a variable. 00:05:36.600 --> 00:05:46.000 So, anything that has some number multiplying in front of it, and it is a variable, like 3 times x...its coefficient is 3. 00:05:46.000 --> 00:05:51.700 Normally, it is just going to end up being a number; but occasionally, it is also going to involve other variables. 00:05:51.700 --> 00:05:55.900 So, not all coefficients are constants, and not all constants are coefficients. 00:05:55.900 --> 00:06:04.000 For example, if we have n times x, plus 7, we have n as the coefficient of x, because it is multiplied against x. 00:06:04.000 --> 00:06:09.600 But 7 is not a coefficient, because it is not multiplying against any variable. 00:06:09.600 --> 00:06:12.400 7 is a constant, though, because it is just a fixed number. 00:06:12.400 --> 00:06:17.000 So, n is a coefficient, but it might not be a constant--it might be allowed to vary. 00:06:17.000 --> 00:06:24.300 But it isn't going to be...n is probably not a constant, but it is definitely a coefficient. 00:06:24.300 --> 00:06:27.400 And 7 is not a coefficient, but it is definitely a constant. 00:06:27.400 --> 00:06:36.800 And we could even look at x as being a coefficient on n: we can look at it from n's point of view, or look at it from x's point of view. 00:06:36.800 --> 00:06:42.800 So, a coefficient is a multiplicative factor; a constant is just something that doesn't change. 00:06:42.800 --> 00:06:47.300 An expression is a string of mathematical symbols that makes sense. 00:06:47.300 --> 00:06:53.700 What do I mean by "makes sense"? Well, you can put together a string of words in English that makes no sense, 00:06:53.700 --> 00:07:03.000 like tree sound running carpet; that didn't make any sense, right?--tree sound running carpet--that was meaningless. 00:07:03.000 --> 00:07:08.100 But it was a bunch of words; to be an expression in math means that you have to make sense. 00:07:08.100 --> 00:07:15.200 So, to be an expression in English (passing this idea along as a metaphor) would mean that it has to make sense as a sentence. 00:07:15.200 --> 00:07:24.100 A string of mathematical symbols that make sense together: 2 times 3 minus 5 could be an expression. 00:07:24.100 --> 00:07:44.200 But (((((( times divide minus 4 times plus (...that doesn't make any sense; that is just a bunch of things that have been put down on paper. 00:07:44.200 --> 00:07:47.700 They have just been written down, but they don't actually mean anything. 00:07:47.700 --> 00:07:52.100 So, an expression has to make sense; that is one of the basic ideas behind it. 00:07:52.100 --> 00:07:55.600 Often, we will need to simplify an expression by converting it into something 00:07:55.600 --> 00:08:00.400 that has the exact same value, but is easier to understand, and often is just shorter. 00:08:00.400 --> 00:08:12.400 For example, we might simplify 7 + 1 + 2 into the equivalent 10: 7 + 1 is 8, and 8 + 2 is 10; so 7 + 1 + 2 has the same value as 10. 00:08:12.400 --> 00:08:17.600 They are both different expressions; they are different expressions, but they have the same value, so we can convert one to the other. 00:08:17.600 --> 00:08:20.400 We can simplify it if we want to. 00:08:20.400 --> 00:08:23.900 An equation is a statement that two expressions have the same value. 00:08:23.900 --> 00:08:31.600 We show it with an equals sign: what is on the left side of the equals sign and what is on the right side of the equals sign-- 00:08:31.600 --> 00:08:35.500 we know that those two things are the same--they have the same value. 00:08:35.500 --> 00:08:42.200 Each side of the equation might look very different: 3x + 82 looks very different that 110/2. 00:08:42.200 --> 00:08:46.600 But that equals sign is telling us that what is on the left is the same as what is on the right. 00:08:46.600 --> 00:08:51.900 It guarantees equality between the two sides. 00:08:51.900 --> 00:08:58.900 Algebra...for being able to do algebra, we need to have some sort of relationship between two or more expressions. 00:08:58.900 --> 00:09:03.100 In this course, our relationship is almost always going to end up being equality. 00:09:03.100 --> 00:09:06.700 It is going to be based on having an equals sign between two expressions. 00:09:06.700 --> 00:09:12.600 The two expressions will be equal to each other, and that gives us an equation to work with and allows us to do algebra and do some things. 00:09:12.600 --> 00:09:15.500 We could potentially have a relationship that is not based on equality. 00:09:15.500 --> 00:09:21.300 We could have an inequality, where one side is less than another side, or one side is greater that another side. 00:09:21.300 --> 00:09:24.900 Or we could have another relationship that is different than either of those. 00:09:24.900 --> 00:09:32.300 But for this course, we are going to almost entirely see equality; and that is going to make up pretty much all of the relationships we ever see in math. 00:09:32.300 --> 00:09:36.400 They will be based around knowing equality between two things. 00:09:36.400 --> 00:09:41.700 So, the two expressions will be equal to each other; and this gives us a starting point to work from. 00:09:41.700 --> 00:09:50.200 The key idea behind math...not behind all math, but behind algebra...is simple; it is intuitive; and it is incredibly important. 00:09:50.200 --> 00:10:00.600 If two things are the same, equal, then we can do the exact same operation to both things; and the results will have to be the same. 00:10:00.600 --> 00:10:09.700 Let's look at it like this: if we have a carrot here, and then we have another carrot that is exactly like that first carrot, 00:10:09.700 --> 00:10:16.700 so they are a perfect copy of one another...we have two carrots, and then we come along and pick up a knife, 00:10:16.700 --> 00:10:21.300 and we cut up this carrot with the knife, and we cut up this carrot with the same knife-- 00:10:21.300 --> 00:10:27.400 the exact same knife being used on both of them--and we cut up both of the carrots in the exact same way; 00:10:27.400 --> 00:10:34.600 we cut 1-inch sections, exactly the same, on both of them; we are going to end up having chopped carrots 00:10:34.600 --> 00:10:45.000 from the first carrot, and from the second carrot; but we know, because we did the exact same way of cutting them up, 00:10:45.000 --> 00:10:52.200 and we started with the exact same carrot--our chopped carrot piles will be exactly equal to one another. 00:10:52.200 --> 00:10:58.800 Since we start with the same carrot, and then we do the exact same kind of thing to both of the carrots, 00:10:58.800 --> 00:11:02.900 we will end up having the exact same pile of chopped carrot at the end. 00:11:02.900 --> 00:11:09.200 Now, compare that to if we had a third carrot that was exactly the same as its other two carrot "brothers," 00:11:09.200 --> 00:11:17.200 but instead of using a knife on it, we decide to shove it into a blender. 00:11:17.200 --> 00:11:19.900 We put it in a blender, and we run it for a minute. 00:11:19.900 --> 00:11:27.300 Out of that blender, we are going to get a pile of carrot mush; we are going to have a carrot mush pile. 00:11:27.300 --> 00:11:31.900 And that carrot mush pile is going to be nothing like those chopped carrots. 00:11:31.900 --> 00:11:37.000 It doesn't matter that we just started with equality; we also have to do the same thing. 00:11:37.000 --> 00:11:41.400 Starting with equality is important; but if we don't do the same thing to both objects-- 00:11:41.400 --> 00:11:46.500 we don't do the same thing to both sides of our equation--we end up with totally different things. 00:11:46.500 --> 00:11:50.900 We no longer have that relationship of equality that we really want to be operating on. 00:11:50.900 --> 00:11:55.400 If you shove the carrot into a blender, you are going to have something totally different than if you had chopped it up. 00:11:55.400 --> 00:12:02.600 If we do the exact same chopping to the two carrots, we end up getting the same pile of carrots. 00:12:02.600 --> 00:12:07.600 But if we do a totally different thing, like shove it into a blender, we have something totally different at the end. 00:12:07.600 --> 00:12:12.300 We have this pile of carrot mush; that is nothing like what we have from the other two. 00:12:12.300 --> 00:12:17.000 The idea here is that we have to have the same operation be applied to both. 00:12:17.000 --> 00:12:22.200 Doing algebra is based around this idea of doing the same thing to both sides. 00:12:22.200 --> 00:12:27.400 Now, of course, you have seen this idea before: but it is absolutely critical to remember. 00:12:27.400 --> 00:12:34.800 You have to remember this fact: always do the exact same thing to both sides. 00:12:34.800 --> 00:12:41.100 If you don't do the exact same thing to both sides, you are not doing algebra anymore; you are just making fantasies up. 00:12:41.100 --> 00:12:45.700 You have to do the same thing: if you add 7 to one side, you have to add 7 to the right side. 00:12:45.700 --> 00:12:48.600 If you square the left side, you have to square the right side. 00:12:48.600 --> 00:12:52.600 If you say higgledy-piggledy to the left side, you have to say higgledy-piggledy to the right side. 00:12:52.600 --> 00:12:59.200 The huge quantity of mistakes that students make are because they forgot to do the operation on both sides. 00:12:59.200 --> 00:13:06.600 They used it only on one side, or they used slightly different operations on the two sides. 00:13:06.600 --> 00:13:11.500 If you end up doing this, you are going to end up making mistakes; don't let this happen to you. 00:13:11.500 --> 00:13:16.600 Pay close attention when you are doing algebra--make sure you are doing the exact same thing to both sides. 00:13:16.600 --> 00:13:24.100 You have to follow all of the rules on both sides; otherwise, we are just making things up--we are no longer following algebra. 00:13:24.100 --> 00:13:29.500 When you are asked to solve an equation, you are being asked to solve for something. 00:13:29.500 --> 00:13:33.200 This usually means solving the equation for whatever variable is in it. 00:13:33.200 --> 00:13:37.400 If more than one variable is present, you will be told which variable to solve for. 00:13:37.400 --> 00:13:44.400 What does solving an equation mean? It means you are looking for the things that make the equation true. 00:13:44.400 --> 00:13:51.000 You are told that this side equals this side; the stuff on the left equals the stuff on the right. 00:13:51.000 --> 00:13:56.000 But they both have variables in them, or one side has variables in it, or one side has just one variable in it. 00:13:56.000 --> 00:14:03.600 But the point is that, depending on what that variable is, or depending on what those variables are, that equation might no longer be true. 00:14:03.600 --> 00:14:07.900 So, what you need to do is make sure that this is true. 00:14:07.900 --> 00:14:13.900 You were told that it equals one another; so you have to figure out what variable, what value for my variable, 00:14:13.900 --> 00:14:18.800 or what values for my variables, will make this equation continue to be true. 00:14:18.800 --> 00:14:23.700 I was told it was true from the beginning; so I have to make sure that it stays true. 00:14:23.700 --> 00:14:30.900 Most often, you will be able to figure out what the values are that make something true by isolating a variable (or variables) on one side. 00:14:30.900 --> 00:14:37.600 You will isolate the variable on one side, and then whatever is on the other side must be the value of that variable. 00:14:37.600 --> 00:14:38.900 How are you going to do that in general? 00:14:38.900 --> 00:14:41.400 Normally, you are going to isolate the variable by doing algebra. 00:14:41.400 --> 00:14:45.900 You will ask yourself what operation would help get this variable alone. 00:14:45.900 --> 00:14:49.800 What would I have to do to this side to be able to get this variable on its own? 00:14:49.800 --> 00:14:55.400 Then, you do that operation to both sides; you continue to apply these operations, asking yourself, 00:14:55.400 --> 00:14:58.900 one time after another, "What could I do to get this variable alone?" 00:14:58.900 --> 00:15:04.300 You keep asking yourself, "What could I do?"; you keep doing operations to both sides. 00:15:04.300 --> 00:15:08.300 And then, you keep doing this until, eventually, the variable is alone on one side, and you have solved it. 00:15:08.300 --> 00:15:17.600 You will get something in the form like x = ... of numbers--so you will know that x is equal to this stuff right here; you will have solved it. 00:15:17.600 --> 00:15:21.600 Now, keep in mind: sometimes you will not solve something by directly doing algebra. 00:15:21.600 --> 00:15:26.100 Algebra will probably be involved, but you might actually be doing something a little bit more creative. 00:15:26.100 --> 00:15:29.000 For example, we will see stuff like this when we work on polynomials. 00:15:29.000 --> 00:15:35.900 We will see cases where we are not just doing algebra; we are also trying to figure out some other stuff and think on a slightly higher level. 00:15:35.900 --> 00:15:41.100 But the key idea is that we are figuring out what makes this equation true. 00:15:41.100 --> 00:15:45.500 What are all the possible ways to make this equation be true? 00:15:45.500 --> 00:15:48.900 That is the real heart behind solving an equation. 00:15:48.900 --> 00:15:54.500 It just so happens that it is very often a good way to solve it by doing algebra and getting the variable alone, 00:15:54.500 --> 00:16:03.200 because once you get the variable alone and on one side, that tells you what value would make that original equation true. 00:16:03.200 --> 00:16:06.000 Order of operations: it is critical to remember the order of operations. 00:16:06.000 --> 00:16:11.300 We have known about this for a long time, but it still matters today; and it is going to matter for as long as you are doing math. 00:16:11.300 --> 00:16:14.000 Certain operations take precedence over others. 00:16:14.000 --> 00:16:22.300 In order, it goes: parentheses (things in parentheses go first), then exponents and roots, multiplication and division, addition and subtraction. 00:16:22.300 --> 00:16:24.700 Always pay attention to the order of operations. 00:16:24.700 --> 00:16:29.200 If you forget to do the order of operations, and you do it in a different order, disaster will befall your arithmetic. 00:16:29.200 --> 00:16:33.500 So, always make sure you are working based on this idea of the order of operations. 00:16:33.500 --> 00:16:38.000 Also, I just want to point out something: exponents and roots are two sides of the same thing. 00:16:38.000 --> 00:16:43.500 x² reverses square root: x², √x...if you take something, 00:16:43.500 --> 00:16:46.200 and you square it, and then you take its square root, they reverse one another. 00:16:46.200 --> 00:16:52.800 Multiplication and division reverse one another: if we multiply by 3, and then divide by 3, it reverses. 00:16:52.800 --> 00:16:59.500 Addition and subtraction reverse one another: if we add 5, and then we subtract 5, they reverse one another. 00:16:59.500 --> 00:17:04.500 So, exponents and roots--the reason why they go at the same time is because they are really two sides of the same thing. 00:17:04.500 --> 00:17:06.900 They have some similar idea going on behind them. 00:17:06.900 --> 00:17:09.800 We will talk about that more when we get into exponents more, later in the course. 00:17:09.800 --> 00:17:15.000 Multiplication and division: they go together at the same time, because they are two sides of the same thing; they can reverse one another. 00:17:15.000 --> 00:17:20.800 Addition and subtraction go together at the same time, because they are working together; they are, once again, things that can reverse one another. 00:17:20.800 --> 00:17:23.100 So, that is why we have these things paired together. 00:17:23.100 --> 00:17:27.400 Parentheses, exponents/roots, multiplication/division, addition/subtraction: always make sure 00:17:27.400 --> 00:17:31.800 that you are working in that order, or at least that whatever you are doing goes along with that order. 00:17:31.800 --> 00:17:36.600 Sometimes, you might be able to do things where you don't have to follow this order absolutely precisely, 00:17:36.600 --> 00:17:43.000 because you might see something like 3 times 2, plus (7 - 5). 00:17:43.000 --> 00:17:48.300 Well, because there is this plus sign in the middle, we know that we can actually do what is on the left side 00:17:48.300 --> 00:17:51.700 and what is on the right side simultaneously, because they will never talk to each other 00:17:51.700 --> 00:17:56.200 until both orders of operations have completely gone through on their two sides. 00:17:56.200 --> 00:18:02.000 So, we can just skip right to 6 + 2 = 8; we don't have to do everything there. 00:18:02.000 --> 00:18:05.600 But if you are not quite sure--if you are not really capable with the order of operations, 00:18:05.600 --> 00:18:11.600 so that you can see this sort of thing right away, always go with the order of operations very carefully, very explicitly. 00:18:11.600 --> 00:18:16.300 In the worst case, it will just take a little bit longer, but at least you will not make a mistake. 00:18:16.300 --> 00:18:19.000 Distributive property: we do not want to forget about the distributive property. 00:18:19.000 --> 00:18:22.800 It allows multiplication to act over addition when it is inside of parentheses. 00:18:22.800 --> 00:18:35.100 So, if we have 3 times (5 + k + 7), then that is equal to 3 times the first one, plus 3 times the second one, plus 3 times the third one (7)... 00:18:35.100 --> 00:18:40.000 so 3 times 5, plus 3 times k, plus 3 times 7; that is the distributive property. 00:18:40.000 --> 00:18:47.200 Always make sure you distribute to all of the terms that are inside of the parentheses; we have to distribute to everything inside of the parentheses. 00:18:47.200 --> 00:18:51.900 I see lots of students see something like this, and they say, "Oh, 3 times 5, plus k, plus 7!" 00:18:51.900 --> 00:18:57.800 No, no, no, no, no! You have to do everything inside of the parentheses; otherwise, you are not distributing. 00:18:57.800 --> 00:19:04.400 So, make sure that you are always distributing to everything in there--everything, when you are multiplying in there. 00:19:04.400 --> 00:19:10.500 All right, we can also use the distributive property in reverse, so to speak; we can go backwards, in a way. 00:19:10.500 --> 00:19:12.900 This idea is what allows us to combine like terms. 00:19:12.900 --> 00:19:17.300 For example, if we have 3x² + 7x² - 5x², well, we have x² here, 00:19:17.300 --> 00:19:21.800 x² here, and x² here; so we can just pick them all up, 00:19:21.800 --> 00:19:25.000 and we can shove them in, because they are all multiplying. 00:19:25.000 --> 00:19:30.300 We pick them all up; and it is times x²; so we have (3 + 7 - 5) times x², 00:19:30.300 --> 00:19:34.600 because if we did the distributive property again, we would get what we started with; so it must be the same thing. 00:19:34.600 --> 00:19:42.300 Now, 3 + 7 - 5--well, that just comes out to be 5: 3 + 7 is 10, minus 5 is 5; we get 5x². 00:19:42.300 --> 00:19:45.300 And that is what we are using to allow us to combine like terms. 00:19:45.300 --> 00:19:49.900 We are sort of pulling out the like term, doing the things, and then putting it back in. 00:19:49.900 --> 00:19:53.100 At this point, we have gotten so used to doing it that we don't have to explicitly do this. 00:19:53.100 --> 00:19:56.700 But for some problems, it will end up being a really useful thing to notice. 00:19:56.700 --> 00:20:03.900 So, it is important to see that we can occasionally use the distributive property in reverse; sometimes it will help us see what is going on. 00:20:03.900 --> 00:20:07.200 Substitution: this is a really important idea in math. 00:20:07.200 --> 00:20:11.800 We can use information from one equation in another equation through substitution. 00:20:11.800 --> 00:20:15.400 If we know that two things are equal to each other, we can substitute one for the other. 00:20:15.400 --> 00:20:22.800 For example, if we know that x is equal to 2z + 3, and we also have this equation that 5y = x - 2, 00:20:22.800 --> 00:20:31.300 well, we can say, "Oh, look, right here I have an x, and I also know that x is the same thing as saying 2z + 3." 00:20:31.300 --> 00:20:34.800 So, we take this information, and we plug it in for x. 00:20:34.800 --> 00:20:44.400 That is what gets us (2z + 3); we will replace that x; so we have 5y is also equal to (2z + 3) - 2. 00:20:44.400 --> 00:20:54.000 When we substitute, we need to treat the replacement the exact same way we treated what was initially there. 00:20:54.000 --> 00:20:57.200 The best way to do this is by putting your substitution in parentheses. 00:20:57.200 --> 00:21:03.400 Notice how I took 2z + 3, and I put it in parentheses up here, even though right here, it didn't start in parentheses. 00:21:03.400 --> 00:21:09.600 That is because I was substituting in for x; so I want to make sure 2z + 3 is treated the exact same way that x was treated. 00:21:09.600 --> 00:21:15.400 So, I have to put it in parentheses to make sure that it gets treated the exact same way that x got done. 00:21:15.400 --> 00:21:19.400 The best way to do this is always to just put your substitution in parentheses. 00:21:19.400 --> 00:21:25.600 It won't always be necessary: for example, on that 5y = 2z + 3, we didn't actually have to put it in parentheses there. 00:21:25.600 --> 00:21:30.000 But it will never cause us to make a mistake; it is never going to hurt us. 00:21:30.000 --> 00:21:34.600 (2z + 3) is just the same thing as 2z + 3, in this case right up here. 00:21:34.600 --> 00:21:41.200 And in other cases (like this one that we are about to talk about), it is absolutely necessary; otherwise we will make bad mistakes. 00:21:41.200 --> 00:21:47.700 Consider this really common mistake: if we know that a is equal b + 2, and we know that c is equal to a², 00:21:47.700 --> 00:21:54.900 then we can say, "Oh, a is right here; a is right here; I will take b + 2, and I will substitute it in for a." 00:21:54.900 --> 00:21:58.900 Lots of students will say, "Oh, well, it is a², so it must be b² + 2." 00:21:58.900 --> 00:22:02.900 No, that is not the same thing: we need a to be all of what it is. 00:22:02.900 --> 00:22:10.700 a is all of (b + 2), not just the b part; and c is similarly not going to be equal to b + 2². 00:22:10.700 --> 00:22:18.000 This right here is not working, because it has to be over this and this; everything needs to be put together. 00:22:18.000 --> 00:22:20.400 b² is not going to work here, as well. 00:22:20.400 --> 00:22:23.000 The thing that we have to do is: we have to have it in parentheses. 00:22:23.000 --> 00:22:29.300 The parentheses cause us to treat that a the same way that we are going to treat (b + 2). 00:22:29.300 --> 00:22:36.200 a²...since a is equal to b + 2, all of a has to be squared; all of that (b + 2) has to be squared. 00:22:36.200 --> 00:22:40.100 And the way that we get all of it is by putting it in parentheses. 00:22:40.100 --> 00:22:49.300 So, whenever you are substituting something in, make sure that whatever is getting substituted in gets plugged inside of parentheses. 00:22:49.300 --> 00:22:51.900 Otherwise, lots of bad mistakes can happen. 00:22:51.900 --> 00:22:56.400 Sometimes, when you see the problem, you will be able to say, "Oh, I don't actually have to plug it in in parentheses," 00:22:56.400 --> 00:22:59.900 at which point, yes, you might be right; sometimes it will make it a little bit faster. 00:22:59.900 --> 00:23:05.900 But really, it is a possible risk that you are taking for just putting down (). 00:23:05.900 --> 00:23:10.200 It is not that much effort to put down parentheses, and it is going to save you so many times. 00:23:10.200 --> 00:23:17.200 So, I really recommend that you put all of your substitutions, any time you are substituting something in, in parentheses. 00:23:17.200 --> 00:23:26.900 Let's do some examples: we want to simplify the following: 2 times 3² + 4((5 + 7)2 - 27). 00:23:26.900 --> 00:23:32.900 Well, we have parentheses inside of parentheses; so first, let's work on the thing inside of the parentheses. 00:23:32.900 --> 00:23:35.100 And then, inside of that, we have even more parentheses. 00:23:35.100 --> 00:23:45.100 So, first we do 5 + 7; we bring everything down--each new horizontal line is a copy of what was above it, but just put in new ways of talking about it. 00:23:45.100 --> 00:23:54.600 4 times the quantity...well, what does 5 + 7 become? 5 + 7 becomes 12, so 12 times 2 minus 27. 00:23:54.600 --> 00:24:05.100 Now, we keep doing this inside of these parentheses: first 2 times 3², plus 4...12 times 2 becomes 24, minus 27. 00:24:05.100 --> 00:24:13.800 2 times 3²...still working inside of these parentheses...24 - 27 becomes -3. 00:24:13.800 --> 00:24:18.400 Now, we have 4 times -3, so now there is no longer anything happening inside of the parentheses. 00:24:18.400 --> 00:24:22.200 So, what is next on the order of operations? Parentheses, then exponents and roots. 00:24:22.200 --> 00:24:28.700 So, 2 times 3²...3² is 9, plus 4(-3). 00:24:28.700 --> 00:24:41.900 Next, we have multiplication: 18 + 4(-3)...-12; finally, we are down to addition and subtraction: 18 + -12 becomes 6; our answer is 6. 00:24:41.900 --> 00:24:45.600 One thing I would like to point out is: if we are really good at math, we might have been able to say, 00:24:45.600 --> 00:24:51.100 "Oh, look, there is a plus sign between these two sides, so these two sides aren't going to be able to talk to each other 00:24:51.100 --> 00:24:54.100 until they have done everything they have to do on their own two sides." 00:24:54.100 --> 00:25:02.500 So, we could have gone right down to saying 2 times 3²...that is the same thing as 2 times 9, which is the same thing as 18. 00:25:02.500 --> 00:25:08.300 And then, we would have kept doing our stuff on the right side, but we could have been simultaneously doing everything on the left side, 00:25:08.300 --> 00:25:13.800 because they are not able to talk to each other, because they have plus signs between them and everything else. 00:25:13.800 --> 00:25:17.700 That is a more advanced trick, and you are probably at the point where you can start seeing this sort of thing. 00:25:17.700 --> 00:25:21.900 But if you have difficulty with the order of operations, you end up making mistakes like this sometimes. 00:25:21.900 --> 00:25:26.100 Be careful and go through it really carefully, and make sure you have that stuff completely understood. 00:25:26.100 --> 00:25:28.600 You need that foundation before math is going to be able to work. 00:25:28.600 --> 00:25:31.800 It is the grammar of math; it is like knowing the grammar of English. 00:25:31.800 --> 00:25:34.800 If you don't put words in the right order, it is just nonsense. 00:25:34.800 --> 00:25:39.500 If you don't follow the operations in the right order, it is just nonsense; we are not able to speak the same language 00:25:39.500 --> 00:25:43.800 as everyone else is speaking in math, and what everyone else is expecting us to be able to do 00:25:43.800 --> 00:25:48.900 when we are working on problems or solving things...or engineering bridges...whatever we are going to do with math. 00:25:48.900 --> 00:26:00.500 All right, Example 2: Use the distributive property to simplify 5(x + x²) + 3(x + y) - 7(x² + x + y). 00:26:00.500 --> 00:26:07.000 So, 5 times (x + x²) becomes 5x + 5x². 00:26:07.000 --> 00:26:16.100 Plus 3(x + y) becomes + 3x + 3y; minus 7...oh, here is something we have to be careful about. 00:26:16.100 --> 00:26:21.500 It is not just going to be minus 7x², but minus 7 is the entire thing. 00:26:21.500 --> 00:26:27.500 So, it is that -7 that gets distributed; it is easier to see this as +, and then a -7. 00:26:27.500 --> 00:26:40.100 + -7: -7x² + -7x + -7y--we have to make sure we distribute that negative, as well. 00:26:40.100 --> 00:26:45.400 We see a minus, but it means that the "negative-ness" has to be distributed to everything inside of there. 00:26:45.400 --> 00:26:49.800 Now, at this point, so we can see things a little bit more easily, let's move things together. 00:26:49.800 --> 00:26:56.100 5x²...and here is a little trick: if you are not sure...we have 1, 2, 3, 4, 5, 6, 7 00:26:56.100 --> 00:26:59.000 different terms here--lots of different terms here to have to work with. 00:26:59.000 --> 00:27:05.900 We can say, "Let's mark off each one; we will make a little tick mark after we write it on the next line, so we don't get confused, 00:27:05.900 --> 00:27:09.300 accidentally use the same thing twice, or not even use it once." 00:27:09.300 --> 00:27:16.800 So, 5x² +...what is another thing involving x²? -7x². 00:27:16.800 --> 00:27:31.800 Plus...what comes next? It looks like we can work on the x's next: 5x (tick there) + 3x (tick there) - 7x (plus -7x); 00:27:31.800 --> 00:27:38.000 and then finally, we have the y's: + 3y + -7y. 00:27:38.000 --> 00:27:40.600 Those tick marks just help us keep track of what we are doing. 00:27:40.600 --> 00:27:44.900 They are not necessary, but it makes it easier to follow, so we don't accidentally make any mistakes. 00:27:44.900 --> 00:27:51.400 5x² + -7x²...those will combine to become -2x². 00:27:51.400 --> 00:28:01.000 5x + 3x + -7x...we have 8x - 7x; we have 1x, which we just write as x. 00:28:01.000 --> 00:28:11.700 And 3y + -7y becomes -4y; -4y we can also just write as minus 4y; and there is our answer. 00:28:11.700 --> 00:28:18.900 Third example: we want to solve for x, so the first thing we do is ask ourselves, "How can I get x by itself?" 00:28:18.900 --> 00:28:25.200 How can I get it isolated on one side, where it is just the variable, and only one of the variable, and nothing else there? 00:28:25.200 --> 00:28:32.900 So, we say, "Well, it is inside of a fraction; we want it to be on top, and we want it to be the only thing there." 00:28:32.900 --> 00:28:36.300 So, we are going to have to somehow change this fraction; how do we change a fraction? 00:28:36.300 --> 00:28:40.900 Well, multiply by x + 3, and that will destroy the denominator. 00:28:40.900 --> 00:28:47.900 Great--but if we multiply by x + 3, then this 2 is going to get hit, and this 3 is going to get hit, by the x + 3. 00:28:47.900 --> 00:28:54.700 We have to hit everything on both sides, so the 3 will get hit by x + 3; the fraction will get hit by x + 3; and the 2 will get hit by x + 3. 00:28:54.700 --> 00:28:59.300 So, the first thing we want to do is have some way of being able to have it operate on fewer things. 00:28:59.300 --> 00:29:02.800 Let's try to get it to operate just on the fraction, at least on one side. 00:29:02.800 --> 00:29:10.500 What we will do is start by subtracting 2 from both sides; that will make it easier to have a simple time with that x + 3. 00:29:10.500 --> 00:29:12.700 We won't have anything else getting in the way. 00:29:12.700 --> 00:29:18.800 That gets us 1 = 5/(x + 3). 00:29:18.800 --> 00:29:25.400 Now, we can multiply by (x + 3); and while we will still have to multiply the 1 (we have to multiply both sides), 00:29:25.400 --> 00:29:28.400 we will have at least a little less stuff in the way. 00:29:28.400 --> 00:29:40.200 We multiply by (x + 3) over here, and we multiply by (x + 3) over here; so (x + 3) times (x + 3) on the bottom...they cancel each other out. 00:29:40.200 --> 00:29:45.200 (x + 3) times 1...that is just going to become (x + 3). 00:29:45.200 --> 00:29:48.500 Since we canceled out the (x + 3) on the bottom, we have 5 here. 00:29:48.500 --> 00:29:50.700 Now, we ask ourselves, "How can I get that x alone?" 00:29:50.700 --> 00:30:02.400 Oh, it is not too hard from here: we just subtract 3 from both sides: minus 3, minus 3; we get x = 2, and there is our answer; great. 00:30:02.400 --> 00:30:08.200 Example 4--the final example: this one is a little bit tough, but we can totally understand what is going on. 00:30:08.200 --> 00:30:14.200 x = 2z; y = z + 4; we want to solve for a in terms of z. 00:30:14.200 --> 00:30:21.100 So, we have a in this equation down here; and we have x², and we have y, and we have x. 00:30:21.100 --> 00:30:25.200 So, z doesn't currently show up in this equation; we want to solve for a in terms of z. 00:30:25.200 --> 00:30:34.600 What that means is that we want to get a = .... with z's...z's are going to be inside of that stuff on the right side. 00:30:34.600 --> 00:30:39.400 And we are going to have a by itself: that is what "solve for a in terms of z" means. 00:30:39.400 --> 00:30:48.300 a equals stuff involving z; it may be multiple z's; it may be just one z; but it is going to be a = [stuff involving z]. 00:30:48.300 --> 00:30:51.600 But notice: it is not going to involve x; it is not going to involve y. 00:30:51.600 --> 00:30:58.900 We are told to solve in terms of z, so it is going to be only in terms of z and other actual numbers--that is, constant numbers. 00:30:58.900 --> 00:31:06.100 So, if we know that x = 2z and y = z + 4, we need to get that z stuff to show up here, and we need to get rid of the y and get rid of the x. 00:31:06.100 --> 00:31:17.300 So, we will use substitution: x = 2z and y = z + 4. 00:31:17.300 --> 00:31:23.400 So, right here, we have a y; here we have an x; here we have an x. 00:31:23.400 --> 00:31:29.600 So, let's do substitution: we have the left side--it will still be the same: 2a - 26 =. 00:31:29.600 --> 00:31:47.700 What comes in for x? 2z comes in; so (2z)² + 4 - 2(3...what goes inside for y? (z + 4)), minus... 00:31:47.700 --> 00:31:52.800 what goes inside for x? (2z); close up that. 00:31:52.800 --> 00:32:00.600 What we have is our original equation, but we have now gotten rid of x and gotten rid of y, and we only have z's and a's in here. 00:32:00.600 --> 00:32:03.200 Now, we are able to solve for a in terms of z. 00:32:03.200 --> 00:32:06.000 So, let's simplify what we have on the right side. 00:32:06.000 --> 00:32:14.700 2a - 26 =...we have this plus sign in the middle, so we can work out what is on the left and what is on the right simultaneously. 00:32:14.700 --> 00:32:19.400 We don't have to worry about them interfering with each other, even though they don't show up at the same time in the order of operations. 00:32:19.400 --> 00:32:22.900 The only time they will be able to talk to each other is when we get all the way down to +. 00:32:22.900 --> 00:32:28.400 So, we can have stuff on the left and stuff on the right work simultaneously to make it a little bit faster. 00:32:28.400 --> 00:32:36.700 (2z)²...we square the 2; we square the z; so 2² and z², 00:32:36.700 --> 00:32:50.700 plus 4 minus 2...we go inside...3 times z, plus 3 times 4, minus 2z. 00:32:50.700 --> 00:33:00.000 2a - 26 =...square 2; we get 4; square z; we don't know what z is, so it just stays as z²; 00:33:00.000 --> 00:33:24.100 plus 4 minus 2 times (3z + 12 now - 2z)...keep simplifying...2a - 26 = 4z² + 4 - 2(3z - 2z becomes just 1z + 12). 00:33:24.100 --> 00:33:43.000 We can now distribute this -2: 2a - 26 = 4z² + 4...we distribute the -2, so remember, it is plus a -2; so we get + -2z + -24. 00:33:43.000 --> 00:33:48.700 Now, we are in a position to be able to keep simplifying the right side to its most fundamental level. 00:33:48.700 --> 00:33:52.800 It equals 4z²...we don't have any other z²s, so it is just 4z². 00:33:52.800 --> 00:34:02.700 Plus 4...well, let's put our constants in there again; so we will go to + -2z, so - 2z, and 4 + -24 becomes - 20. 00:34:02.700 --> 00:34:05.000 At this point, we can now do our algebra. 00:34:05.000 --> 00:34:20.200 We will add 26 to both sides, and we will get 2a = 4z² - 2z + 26, so plus...oops, I accidentally wrote what I was saying... 00:34:20.200 --> 00:34:36.900 minus 2z still; add 26; -20 plus 26 becomes just 6; divide both sides by 2: a = (4z² - 2z + 6), all over 2. 00:34:36.900 --> 00:34:46.500 And we can simplify that: 4z² becomes 2z², minus 2z; that becomes -z; plus 6--that becomes + 3. 00:34:46.500 --> 00:34:54.800 So, it is a = 2z² - z + 3; a equals stuff-just-involving-z. 00:34:54.800 --> 00:34:57.500 Solve for a in terms of z; great. 00:34:57.500 --> 00:35:05.200 I really want to point out that the reason we were able to get that right is because we put parentheses when we substituted in. 00:35:05.200 --> 00:35:10.000 If we hadn't done that, we would not have had our square go onto both the 2 and the z. 00:35:10.000 --> 00:35:15.500 We would not have had our 3 distribute to both the z and the + 4; we wouldn't have had our subtraction... 00:35:15.500 --> 00:35:18.100 well, our subtraction actually still would have subtracted 2z. 00:35:18.100 --> 00:35:21.500 But if we didn't put in those parentheses, we would have definitely made some mistakes. 00:35:21.500 --> 00:35:25.500 It is absolutely critical to put in parentheses when we are substituting. 00:35:25.500 --> 00:35:28.000 Otherwise, mistakes will just start popping up everywhere. 00:35:28.000 --> 00:35:31.000 All right, I hope all that made sense; we will see you at Educator.com later--goodbye!