WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today we are going to talk about variables, equations, and algebra.
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What is a variable? We talk about them all the time: we want to think of a **variable** as just being a placeholder.
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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.
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Sometimes the variable will be able to vary; it is going to be able to change, depending on what we want to do.
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And as the value of the variable changes, it will affect something else.
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It might affect the output of the function; it might affect some other dependent variable,
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if we see something like y = 3x, where you change x--we make it the independent variable.
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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.
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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.
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Other times, we are just using a variable as a fixed value we don't know yet.
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Sometimes, it might even be multiple possible fixed values; it could be fixed values or a fixed value.
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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.
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So, normally, we are going to be able to figure out what it is, based on the information given to us in the problem.
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Otherwise, it is probably not going to be a very good problem, if we can't actually solve for what the variable is.
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So, we will almost always have enough information to figure out what this variable is.
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That is the other possibility: a variable is something that we just don't know yet.
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It is a number that has been given a name, because we are trying to figure out more information about it.
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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
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about the perpetrator, until they have enough information to be able to figure out who the perpetrator actually is.
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"Perpetrator" is just a placeholder for some other person, until they figure out who that person is.
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Great; we can name variables any symbol that we want.
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Normally, we are going to use lowercase letters to denote variables; but occasionally, we will use Greek letters or other symbols.
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When we are working on word problems, we are going to choose our symbols...we want to choose the letter,
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or maybe other symbol that we use, based on something that helps us remember what it is representing in this word problem.
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What am I using this variable to get across?
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There are a lot of them that we regularly use; and so, we will get an idea of what they are; here we go!
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Any symbol could potentially be used for any meaning at all.
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We could make a smiley face; and I sometimes do use a smiley face to represent a number.
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But smiley face is a little bit harder to draw than x, so we tend to use letters that we are used to drawing.
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Anything could potentially be used for anything else; but here is a list of common symbols used,
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and what the meaning we normally associate with them is.
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Occasionally, we will have different meanings associated with them, depending on the problem.
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We might use y to talk about the number of yaks that there are at a farm;
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but generally, we are going to use them as we see right here--all of this right here.
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So, x is our most common one, probably our favorite variable of all.
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We use it for general use, when we are talking about horizontal location or distance.
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y is vertical location or distance; t normally stands in for time; n stands in for a quantity of some stuff.
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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.
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We draw θ by hand; you just make something sort of like you are drawing a zero or an O,
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and then you just draw a line straight across the middle of it; that is θ.
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r is radius; A is area; V is volume; and we often use a, b, c, and k to represent fixed, unchanging values--
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values that are not going to vary and change into other things--things where we know
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that they are going to just stay the same, but we don't know what they are yet.
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Or we might decide on what they are later on.
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Anyway, this gives us a general idea of what the normal stable variables we constantly encounter are.
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Now, you might use x for something totally different than what we have here.
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You might use r for something totally different; you are not stuck to just using this.
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But we are going to see them in a lot of problems, and we want what we do to make sense to other people.
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So, it is good to go along with these conventions, usually.
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All right, what is a constant? A **constant** is a fixed, unchanging number.
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It is a value that does not turn into another value.
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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.
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But a constant only has one thing; it just stays the same.
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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.
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After all, we don't have to worry about 3 suddenly turning into 4.
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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.
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3 doesn't suddenly jump around and become a new number.
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Occasionally, we might refer to a symbol that is representing a number as a constant.
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We might say a is a constant in this problem.
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We might not know what value that symbol represents; but we know it cannot change--a constant is something that cannot change.
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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.
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It seems kind of like a contradiction in terms, but remember: we are using "variable" more for the idea of placeholder.
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And while sometimes it varies, sometimes it can also just be a placeholder in general.
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A constant is something that isn't going to move around; it is one number, and one number only.
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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.
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A **coefficient** is a multiplicative factor on a variable.
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So, anything that has some number multiplying in front of it, and it is a variable, like 3 times x...its coefficient is 3.
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Normally, it is just going to end up being a number; but occasionally, it is also going to involve other variables.
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So, not all coefficients are constants, and not all constants are coefficients.
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For example, if we have n times x, plus 7, we have n as the coefficient of x, because it is multiplied against x.
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But 7 is not a coefficient, because it is not multiplying against any variable.
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7 is a constant, though, because it is just a fixed number.
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So, n is a coefficient, but it might not be a constant--it might be allowed to vary.
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But it isn't going to be...n is probably not a constant, but it is definitely a coefficient.
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And 7 is not a coefficient, but it is definitely a constant.
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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.
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So, a coefficient is a multiplicative factor; a constant is just something that doesn't change.
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An **expression** is a string of mathematical symbols that makes sense.
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What do I mean by "makes sense"? Well, you can put together a string of words in English that makes no sense,
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like tree sound running carpet; that didn't make any sense, right?--tree sound running carpet--that was meaningless.
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But it was a bunch of words; to be an expression in math means that you have to make sense.
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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.
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A string of mathematical symbols that make sense together: 2 times 3 minus 5 could be an expression.
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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.
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They have just been written down, but they don't actually mean anything.
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So, an expression has to make sense; that is one of the basic ideas behind it.
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Often, we will need to simplify an expression by converting it into something
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that has the exact same value, but is easier to understand, and often is just shorter.
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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.
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They are both different expressions; they are different expressions, but they have the same value, so we can convert one to the other.
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We can simplify it if we want to.
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An **equation** is a statement that two expressions have the same value.
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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--
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we know that those two things are the same--they have the same value.
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Each side of the equation might look very different: 3x + 82 looks very different that 110/2.
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But that equals sign is telling us that what is on the left is the same as what is on the right.
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It guarantees equality between the two sides.
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Algebra...for being able to do algebra, we need to have some sort of relationship between two or more expressions.
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In this course, our relationship is almost always going to end up being equality.
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It is going to be based on having an equals sign between two expressions.
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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.
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We could potentially have a relationship that is not based on equality.
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We could have an inequality, where one side is less than another side, or one side is greater that another side.
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Or we could have another relationship that is different than either of those.
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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.
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They will be based around knowing equality between two things.
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So, the two expressions will be equal to each other; and this gives us a starting point to work from.
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The key idea behind math...not behind all math, but behind algebra...is simple; it is intuitive; and it is incredibly important.
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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.
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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,
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so they are a perfect copy of one another...we have two carrots, and then we come along and pick up a knife,
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and we cut up this carrot with the knife, and we cut up this carrot with the same knife--
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the exact same knife being used on both of them--and we cut up both of the carrots in the exact same way;
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we cut 1-inch sections, exactly the same, on both of them; we are going to end up having chopped carrots
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from the first carrot, and from the second carrot; but we know, because we did the exact same way of cutting them up,
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and we started with the exact same carrot--our chopped carrot piles will be exactly equal to one another.
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Since we start with the same carrot, and then we do the exact same kind of thing to both of the carrots,
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we will end up having the exact same pile of chopped carrot at the end.
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Now, compare that to if we had a third carrot that was exactly the same as its other two carrot "brothers,"
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but instead of using a knife on it, we decide to shove it into a blender.
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We put it in a blender, and we run it for a minute.
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Out of that blender, we are going to get a pile of carrot mush; we are going to have a carrot mush pile.
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And that carrot mush pile is going to be nothing like those chopped carrots.
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It doesn't matter that we just started with equality; we also have to do the same thing.
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Starting with equality is important; but if we don't do the same thing to both objects--
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we don't do the same thing to both sides of our equation--we end up with totally different things.
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We no longer have that relationship of equality that we really want to be operating on.
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If you shove the carrot into a blender, you are going to have something totally different than if you had chopped it up.
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If we do the exact same chopping to the two carrots, we end up getting the same pile of carrots.
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But if we do a totally different thing, like shove it into a blender, we have something totally different at the end.
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We have this pile of carrot mush; that is nothing like what we have from the other two.
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The idea here is that we have to have the same operation be applied to both.
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Doing algebra is based around this idea of doing the same thing to both sides.
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Now, of course, you have seen this idea before: but it is absolutely critical to remember.
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You have to remember this fact: always do the exact same thing to both sides.
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If you don't do the exact same thing to both sides, you are not doing algebra anymore; you are just making fantasies up.
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You have to do the same thing: if you add 7 to one side, you have to add 7 to the right side.
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If you square the left side, you have to square the right side.
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If you say higgledy-piggledy to the left side, you have to say higgledy-piggledy to the right side.
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The huge quantity of mistakes that students make are because they forgot to do the operation on both sides.
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They used it only on one side, or they used slightly different operations on the two sides.
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If you end up doing this, you are going to end up making mistakes; don't let this happen to you.
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Pay close attention when you are doing algebra--make sure you are doing the exact same thing to both sides.
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You have to follow all of the rules on both sides; otherwise, we are just making things up--we are no longer following algebra.
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When you are asked to solve an equation, you are being asked to solve for something.
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This usually means solving the equation for whatever variable is in it.
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If more than one variable is present, you will be told which variable to solve for.
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What does solving an equation mean? It means you are looking for the things that make the equation true.
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You are told that this side equals this side; the stuff on the left equals the stuff on the right.
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But they both have variables in them, or one side has variables in it, or one side has just one variable in it.
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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.
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So, what you need to do is make sure that this is true.
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You were told that it equals one another; so you have to figure out what variable, what value for my variable,
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or what values for my variables, will make this equation continue to be true.
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I was told it was true from the beginning; so I have to make sure that it stays true.
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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.
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You will isolate the variable on one side, and then whatever is on the other side must be the value of that variable.
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How are you going to do that in general?
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Normally, you are going to isolate the variable by doing algebra.
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You will ask yourself what operation would help get this variable alone.
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What would I have to do to this side to be able to get this variable on its own?
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Then, you do that operation to both sides; you continue to apply these operations, asking yourself,
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one time after another, "What could I do to get this variable alone?"
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You keep asking yourself, "What could I do?"; you keep doing operations to both sides.
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And then, you keep doing this until, eventually, the variable is alone on one side, and you have solved it.
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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.
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Now, keep in mind: sometimes you will not solve something by directly doing algebra.
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Algebra will probably be involved, but you might actually be doing something a little bit more creative.
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For example, we will see stuff like this when we work on polynomials.
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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.
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But the key idea is that we are figuring out what makes this equation true.
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What are all the possible ways to make this equation be true?
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That is the real heart behind solving an equation.
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It just so happens that it is very often a good way to solve it by doing algebra and getting the variable alone,
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because once you get the variable alone and on one side, that tells you what value would make that original equation true.
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Order of operations: it is critical to remember the order of operations.
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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.
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Certain operations take precedence over others.
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In order, it goes: parentheses (things in parentheses go first), then exponents and roots, multiplication and division, addition and subtraction.
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Always pay attention to the order of operations.
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If you forget to do the order of operations, and you do it in a different order, disaster will befall your arithmetic.
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So, always make sure you are working based on this idea of the order of operations.
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Also, I just want to point out something: exponents and roots are two sides of the same thing.
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x² reverses square root: x², √x...if you take something,
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and you square it, and then you take its square root, they reverse one another.
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Multiplication and division reverse one another: if we multiply by 3, and then divide by 3, it reverses.
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Addition and subtraction reverse one another: if we add 5, and then we subtract 5, they reverse one another.
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So, exponents and roots--the reason why they go at the same time is because they are really two sides of the same thing.
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They have some similar idea going on behind them.
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We will talk about that more when we get into exponents more, later in the course.
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Multiplication and division: they go together at the same time, because they are two sides of the same thing; they can reverse one another.
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Addition and subtraction go together at the same time, because they are working together; they are, once again, things that can reverse one another.
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So, that is why we have these things paired together.
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Parentheses, exponents/roots, multiplication/division, addition/subtraction: always make sure
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that you are working in that order, or at least that whatever you are doing goes along with that order.
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Sometimes, you might be able to do things where you don't have to follow this order absolutely precisely,
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because you might see something like 3 times 2, plus (7 - 5).
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Well, because there is this plus sign in the middle, we know that we can actually do what is on the left side
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and what is on the right side simultaneously, because they will never talk to each other
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until both orders of operations have completely gone through on their two sides.
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So, we can just skip right to 6 + 2 = 8; we don't have to do everything there.
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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!