WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to start looking more at linear equation starting off with the vocabulary of linear equations.
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There will be lots of new terms in here, it will definitely take some time to look at them all and what they mean and play around with them a little bit.
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Some of the terms that will definitely get more familiar with are variable, term.
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We will look at coefficients and we will definitely see how we can combine like terms.
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We will be able to tell the difference between equations and expressions and get into a linear equation.
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What we want to solve later on and solutions.
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When looking at an equation, we often see lots of letters in there, those are our variables.
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What they do is they represent our unknowns.
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One favorite thing to use in a lot of equations is (x), but potentially we could use any letter.
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You could use a, b, it does not really matter but most the time our unknown is (x).
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A term is a little bit more than just that variable.
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It is a number of variables or sometimes the product or quotient of those things put together.
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To make it a little bit more clear, I have different examples of what I mean by a term.
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All of these things down here are types of terms.
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The first one is the product of an actual number and a variable.
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Down here with the (k) it is simply just a variable all by itself.
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A coefficient of a term is a number associated with that term.
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If I'm looking at a term say 2m, the coefficient is the number right out front that is associated with that term.
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I'm looking at another one like 5mq, but again the 5 would be the coefficient of that term.
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Terms with the exactly the same variables that have the same exponents those are known as like terms.
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There are two conditions in there you want to be familiar with.
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It must have exactly the same variables and it must also have exactly the same exponents.
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Let us say I have both of those and you can not consider them like terms.
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Let us take a look at some real quick.
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I'm looking at 5x and I'm looking at 5y, these are not like terms.
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Why, you ask? They do not have the same variable.
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One has (x) and the other one has (y).
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How about 3x² and 4x², these are like terms.
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These ones are definitely good because notice they have exactly the same variable and they have the same exponent.
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They have both of those conditions.
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This one is little bit trickier so be careful, they both have an (x), that looks good.
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They both have a (y), that seems good but they have different exponents.
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This one has the y² and this one has nothing on its y.
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I would say that these are not like terms.
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An expression is the statement written using a combination of these numbers, operations, and variables.
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This is when we start stringing things together so I might have a term 2x and then I decide use may be addition and put together a 4xy.
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That entire thing would be my expression.
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In the equation, we take a statement that two algebraic expressions are actually equal.
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I can even borrow my previous expression to make an equation.
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I simply have to set it equal to another expression, maybe 2x².
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What a lot of students like to recognize in these two cases is that in an equation there better be an equal sign somewhere in there.
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With your expression there is not an equal sign because it is just a whole bunch of string of terms and coefficients, numbers, operations.
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Since we are interested in linear equations and eventually getting solved for those, what exactly is a linear equation?
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It is any equation that can be written in the form, ax + b = c.
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There is some conditions on those a, b, and c.
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Here we want a, b, and c to be real numbers, we would not have to deal with any of those imaginary guys.
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We want to make sure that (a) is not 0.
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The reason why we are throwing that condition in there is we do not want to get rid of our variable.
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If (a) was 0, you would have 0 × x and then we would have a rare variable whatsoever.
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It is an equation, not necessarily look like that but it almost can be written in that form, it is a linear equation.
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A number is a solution of that equation if after substituting it in for the value the statement is true.
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That means if you actually take out your variable and replace it with a number that is the solution.
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Then it is going to balance out, it is going to be true with that number in there.
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This first part we just want to identify the different parts of the equation so we can better feel of what we are looking at.
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First of all I know that this is an equation because notice how we have an equal sign right there.
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I have the expression 7x + 8 that is one expression one side and expression 15 on the other.
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What else do I have here? I have my 7x and I have the 8 both of these are terms.
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If I pick apart that one term on the left, I can say that the 7 is a coefficient and that the x is my variable.
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There are many different parts of the equation and you want be able to keep track of it.
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And probably terms are one of the most important for now.
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Let us look at this one, let us see 30x = (4 × X) - 3 + (3 × 3) + (x + 2).
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Again identify the parts of the equation, let us see.
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It has the equal sign, I know it is an equation, it is important to recognize.
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Let us see, over on this side I have a term 30x, I have this term, I have this term.
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I have a bunch of different terms.
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Terms are always combined using addition or subtraction.
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That is how we can usually recognize them.
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In my terms I have some coefficients but you know it might be easier to use my distributive properties to see even more of those coefficients.
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Let us use that distributive property, let us take the 4 multiplied by the x and 3 and do the same thing with 3.
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4x -12 + 3x + 6, not bad.
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Looking at that I have even more terms, I got my 30x, I get 4x, 12, 3x, 6 and lots of different terms now.
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Into those terms I can identify what its coefficient is and I can identify the variable, the x.
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This one says simplify it by combining like terms.
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Remember our like terms are terms that have exactly the same variable and they have exactly the same exponent.
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We have to be careful on which things we can actually combine here.
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Let us see I have 12w and 10w those are like terms, they both have a w to the first.
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Over here is 8- 2w, all three of those are like terms, we only write them next to each other.
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I know that I need to combine them.
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The 9 and the 3 I would also consider those like terms because both of them do not have a variable associated with them.
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I will combine those together.
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Let us take care of everything with the (w), 12w + 10 would be 22w.
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That would give me a 20w when combining those, now we will take care of 3 and 9, -3 + 9 = - 6.
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It is important to recognize that you should not go any further than here because the 20w and 6, those are not like terms.
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We are not going to put those together.
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On to example 4, this one we want to simplify by combining like terms.
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I have lots of grouping symbols in here so it is hard to pick out what my like terms are.
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I think we can do it though but we might have to borrow our distributive property first.
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I'm going to take this negative sign and distribute it inside my parentheses here.
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Now that would give me (2y - 3y) - 4 + (y² + 6y²), not bad.
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I can see there is a few things I can combine, let us see.
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Specifically I can put these y's together since both of them are single (y).
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I can combine these ones together over here because both of them are y².
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Let us put those together, 1y - 3y would be -2y, I have a 4 hanging by itself, -4.
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y² + 6y² would be a 7y², that looks good.
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Remember, I have not distributed my 2, it is not going to help me combine anymore like terms.
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It will definitely help me see my final results, -4y - 8 + 14y².
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My final answer would be -4y - 8 + 14y².
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I would not combine those anymore together because none of those are like terms.
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I have a single (y), I have an 8 that does not have any variables whatsoever.
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I have that y², definitely not like terms.
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