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RC
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Algebra - Overview

In mathematics, algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

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McMaster University

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

RC
University of North Carolina at Chapel Hill
Top Physics 101 Mechanics Educators
Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Jared E.

University of Winnipeg

Meghan M.

McMaster University

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