WEBVTT
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welcome to Unit Zero in our review for algebra.
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So, uh, and when we're doing these reviews
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, remember that we're doing them in the context of
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physics. Okay, So, for example, in
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algebra, when you first learned about variables, you
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just had these letters that showed up in your equations
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Suddenly. Well, in physics, those variables means
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something. Okay, They're attached to physical measurements in
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our world. So, for example, say we
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have a problem where a car starts at T equals
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zero seconds, it begins to accelerate. And then
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at T equals one second. It stops accelerating and
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continues at a constant speed. And then at T
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equals 10 seconds. It starts slowing down, and
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we want to find out how. What time does
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it come to rest? Okay, um, now
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you'll notice that I've given some variables, put some
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variables down here, and they're all called tea with
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the sub script In the sub scripts. Important tells
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you the order in which they happen, so t
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subzero or t not as we sometimes say is the
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first important time. T one is the second important
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time. T two is the third important time.
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Then T final is our last important time. I
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could have also called this t three. What I
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wanted to illustrate is that for your sub script,
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you can use numbers. You can use letters.
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You can even use words. I could have had
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tea. Initial is not an uncommon thing. To
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see t initial equals zero seconds. Okay, um
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, on. That's perfectly fine to say. The
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important thing is to define what your variable means.
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Okay to tell me that. Okay, this thing
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is that t equals zero seconds. This thing is
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at one second, this thing is at 10 seconds
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. The reason you have to do this is because
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if you're solving a problem on a tasker on your
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homework and you write down some random variable and don't
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tell your teacher what it is, they could take
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off points for that because they have no context or
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reason to understand what that variable might mean. Now
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, as you go on in time, that you'll
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have some basic variables that you'll know what they mean
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. Like little T. It's time V little V
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is velocity things like that. But you should definitely
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start out defining all your variables. Um, speaking
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of little V so Ah, lot of times,
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variables are written out in terms of function. So
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say we have little V as a function of time
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. And maybe it's equal to something like t squared
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minus one or something like that. Okay, um
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, now, this actually doesn't work in terms of
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dimensionality, because tea is in terms of seconds and
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the should be in meters per second. Um,
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but we'll get to that later. What I want
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illustrate here is that you're gonna have the is gonna
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be the dependent variable and T is your independent variable
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. Okay, The time doesn't care what the car
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is doing. Okay, but the speed of the
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car kind of cares about what time it is.
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Eso weaken, right? The dependent variable as a
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function of the independent variable. Okay, One of
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the best ways to illustrate this is often with a
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graph. Okay, where we have V f t
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the dependent variable on the y axis and t the
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independent variable on the x axis or the horizontal axis
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. In this case to follow the story problem,
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we say at T equals zero. It's not moving
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, but it starts accelerating, which means via gets
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bigger and then AT T equals one. It stops
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accelerating and it goes horizontal. It's constant, and
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then it starts decelerating. So we have t's of
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to t's have one and t subzero here. Okay
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, so now we've got a new actual graph that
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shows us a picture of what we're talking about.
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Okay, so this is one step closer toe solving
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numerically for what's going on with our system. Um
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, now there's a couple other things that we should
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remember because not only are we dealing with the graph
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, we're dealing with some sort of equation. You're
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dealing with equations with exponents and number or with variables
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and numbers. We need to be able thio use
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the basic algebraic properties, and we need to be
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able to remember what ordered to do them in.
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Remember, the order is pretty simple. It's this
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Penda sting that you learned oh so long ago.
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Okay, where you do the what's inside Princes first
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, then exponents, multiplication and division from left to
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right, then with addition and subtraction from left to
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right. So if I were to have something like
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two squared minus three times four minus five, I
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know I need to do What's inside the princes First
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I do. Two squared is four minus three.
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That's one. So we have one times four minus
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five four, minus five. That's equal to negative
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one. Okay, so you can find some videos
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online, reviewing pandas or go back to the basic
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courses on algebra here. Um, next. We
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also have the algebraic properties. We had the community
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of property, which simply says that if you have
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a Times B, it's equal to B Times A
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for a plus. B is equal to B plus
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A. Helps you reorganize your variables to help you
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find a solution. You also have the associative property
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. I understand. Okay, Associative Property says If
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I have a times B in parentheses, all multiplied
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by C, that's equal to a times B,
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times C in parentheses or again, there's an addition
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version that we could write down. Yeah. Oh
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, I forgot my plus. So a plus a
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plus B inferences plus C is equal to a plus
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. Princes B plus D uh, then distributive,
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which is one of my favorite in of all the
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properties, distribute you native property, Distributive says.
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We have a Times B C. It's equal to
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a B plus a sea. Okay. And notice
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this could go backwards, too. If I have
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a B plus a C, I can pull in
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a out of it. This could be really helpful
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for solving problems as you'll see over and over again
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as we get to those in the actual physics.
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We also have the identity properties have to give when
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you say things like eight times one equals a or
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a plus zero equals a These might seem really,
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really simple, but they can be really effective when
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it comes to solving problems, especially the multiplication identity
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property on and last of all, we also have
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the inverse properties which say that addition and subtraction are
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inverse operations and as our multiplication and division, a
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times one over a equals one. Okay, hopefully
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, this all sounds really familiar and basic to you
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. If not, you definitely want to go back
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and review one of the math courses. Um,
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last of all, we're also gonna be have to
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be able to solve systems of equations. Ah,
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lot of times in physics, we're going to come
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across not just one equation that defines relationships between variables
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, but we'll have two or three or more equations
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that do this. So we have to be ableto
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take care of this and those basic math sense.
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Usually you saw something like this to x minus Y
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equals three. And why, plus X equals six
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. You have two equations that tell you about how
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, why and extra related to each other. And
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we have multiple ways that you've learned in your math
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classes about how to solve for this, uh,
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the quickest way, perhaps in some cases, is
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graphing. Okay. Were you simply solve each of
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these for why? And then type him into your
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graphing calculator and see where the lines cross. Okay
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, that could be a really quick and effective way
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to do it, but it's not always necessary.
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And it usually results in some sort of decimal where
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you didn't need to have a decimal. Oh,
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um, on the other hand, you could use
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what's called substitution. Yeah, Okay, when you're
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doing substitution, I saw for one of the variables
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first, so I might say, Why is equal
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to six minus X, and then I'll take that
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and I'll plug it in up here, so I'll
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have to x minus six minus X. So I've
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plugged in. Why appear is equal to three.
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And now notice I have an equation with just excellent
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. So I could go back and use my algebraic
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properties to solve for X. I'm not going to
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show you the steps here because a lot of this
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is gonna be about you figuring out how to do
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the steps for yourself. That's what a lot of
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physics is. I can't e can't just tell you
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and have you understand it, you need to try
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it yourself. Um, also, we had the
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elimination method elimination in the elimination method. What we
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do is we combine the two equations. So,
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for example, in this case, I'm gonna add
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this first equation to the second equation. Okay.
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In this case, right hand sites, Easy six
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plus three. That's nine. Over here. I
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need thio. Add these together. Remember the similar
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variable Similar terms will add So two x plus one
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x that's three X, but then negative one y
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plus one y that's gonna be zero. It's very
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quickly. Then I have a solution. X is
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equal to three and then I can resort to substitution
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, saying X equals three appear, and I'll find
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that why is also going to be equal to three
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? Um, so these are the three basic techniques
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for solving systems of linear equations. We will use
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them over and over again, along with another one
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that involves matrices, so that won't show up until
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much later.