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# In mathematics, an equation is a statement that two mathematical expressions have the same value. Solving an equation is often the primary goal of algebraic solution. There are several types of equations that are used frequently in mathematics.

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##### Ahyeon H.

University of California, Berkeley

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##### Charles C.

Kennesaw State University  ### Recommended Quiz

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Okay, So in this video, we're gonna be talking about an introduction to equations. So let's first think about what even is an equation. Well, just like it sounds an equation is just something that relates to of the same things. Two or more of the same things. Okay, So, as an example, I could say that two plus one is equal to three. And I know that that works. Because if I do two plus one, that gives me three and three is equal to three, so we could make sure that the left side of the equation and the right side of the equation are equal. So then, um, in terms of the S A. T, we know that they'll typically introduce a variable. So you might find something like this. You might say X plus one is equal to five. Well, in this case, you know that in order to solve it, you can try. Your main goal will be to try to isolate the variable, which is X. So your main objective is to get this alone. Okay, So in order to get that alone, I'm just going to go ahead and subtract one from both sides of the equation. So then if I do one minus one, that gives me just X. So now we have it alone, which is exactly what we wanted. That's good. And then on the other side, I have five minus one, which is just four. So we know that X is going to be four Mhm. So then, in terms of checking my work, I could just take this four that I found and plug it right back into our initial equation To say four plus one is equal to five. Well, yes, indeed. Four plus one is five, and so five is equal to five. So that also works. So then let's kind of backtrack and think about what we did. Okay, Well, we had X plus one is equal to five. So in order to get this X alone in orderto isolate that all I have to do is the opposite operation of what is currently going on. So what's currently going on is we're adding one. So the opposite of addition is subtraction. So I'm that's the whole reason why I subtracted one from both sides. Okay, so then let's consider a slightly different equation. I'm going to say I have the equation three times. Why is equal to 12? Well, for a situation like this, in order to isolate my variable, why? Which again? We always want to try to get the variable alone. We wanna isolate that in order to solve for it. So in order to do that, we will always want to do the opposite operations. So I'm gonna go ahead and divide both sides by three. If I do that, this three on the top and the three in the bottom will cancel out and I'll leave me with y is equal to 12, divided by three and 12, divided by three is four. So why must be four? Then again, in order to check my answer, all I'm going to do is take this Y is equal to four and plug that right back into the original equation. And that's gonna give me is three times four is equal to 12. And we know yes, that does work. Because three times four is 12, the 12 equals 12. So, again, that works. Okay, so then let's think about Well, okay. So we know how to add and subtract and multiply and divide, and that all we have to do is the opposite operation. But how would we do something that is a little bit more complicated? So let's go ahead and to think about it. So then let's see. So then there's something that we could use called pen does. That kind of tells us in what order we have to do it. So let's think about it. So pen does, um, does so I'm just gonna go ahead and go through the letters of that acronym. So P stands for parentheses. Any stands for exponents? M stands for multiplication de stands for Division A stands for edition and S stand for subtraction. Okay, so basically, when we're solving through equations, we want to do it in this order. So Peco's first egos next, um D a and s mhm. So then, if we have an example like two times three minus one squared, um, Times five. So is equal to something. So let's think about what that would equal well again in order to think about which one we have to do first we have to do the parentheses first because we see that that's there. So inside of the parentheses is three minus one, and that gives us two. So now our new equation is too. Times two squared times five. Okay, so then we see an exponents there, so I'm gonna go ahead and take care of that exponents. So two squared is just four. So I have four times, two times five. So then, at this point, we just have multiplication, so it doesn't really matter which order you do it. You can do two times two, which is eight, and then multiply that by five, and that's going to give me 40. So the answer to this appear is going to become 40. So then let's think about Well, what if we don't have one of them? Well, that's totally fine. So then I will give you a totally different examples. I'm going to say three or 30 divided by it's too to plus 11 times too. Okay, well, we don't see a parentheses here, so we can go ahead and check our next thing, which is exponents. We also don't see an exponents here, but that's also okay. But then we do see a division and multiplication, so we're going to go ahead and do those first. So 30 divided by two is 15 and then 11 times two is 22. So I'm going to do those first. Then 15 plus 22 is going to be our next operation because we see addition and that's are also our last step. So that's just gonna equal 37. So the answer to this up here is just going to be 37. So that's just in general how you would do that. But then we sometimes also have something that also makes us deal with penned as but with a variable. So let's go ahead and look at that example. Then, if we have something with a variable that involves multiple steps to algebraic Lee solve, let's say let's do X minus three times two Um, plus seven over four is equal to two. Okay? And we're trying to solve for X. So for something like this, because we initially we talked about how we wanted to do the opposite operation. So for pem guys, we're also going to go backwards. So we're going to do the subtraction of the addition components first. The only exception to that is when we have a situation like this where we have a fraction, so I'm just going to write. Exception is for factions, so if we have a fraction and we're gonna try to get rid of that fraction first. So the very first thing that I'm going to do to this equation is I'm gonna multiply both sides by four. Because then that gets rid of this for on the bottom and the four on the top. And now we're left with two times X minus three plus seven is equal to it. And that's a lot easier to work with now because we don't have a fraction. So then again, like I said, we're going to do this pandas backwards. It's going to be more like sad MEP If you were to actually sounded out. So then the first thing we want to do is take care of the addition or subtraction components. So I'm gonna first take care of this component right here. So, in order to take care of that, get rid of that. I'm going to subtract seven from both sides because then that councils that act Then we're left with two times X minus three is equal to one from there, I see a multiplication. So I'm gonna go ahead and defy the entire equation by two. So then we're left with X minus three is equal to one half. At this point, the parentheses, um kind of go away on their own because there's nothing that we're having to do to the parentheses. So if we have X minus three, the opposite would be to add three both sides and we get that X is equal to three and one half, which is also 3.5. Or you can also express it as 7/2 as an improper fraction. So then let's think about it. What's another thing that I was kind of applying to this entire process? Well, another important thing to remember is to do the same thing to both sides. Because whatever you do to one side, you have to do it to the other side. So if you're adding free to both sides, one side you also have to do to the other. If you're multiplying by 7.5, you have to multiply it to the other side as well. So ultimately, it doesn't really matter what you do as long as you're doing the same thing to both sides, and that is a very key component of this. Okay, so then let's think about some different examples. So just to do one more example So then, um, let's think about if we have something involving exponents as well or square root Mhm. So then I'm gonna do Let's do X squared minus four. The whole thing. Times three divided by five First three is equal to Let's do seven. Okay, again, we do notice that there is a fraction, but this fraction is one unit on this side right here. So it's basically like this big chunk plus three is equal to seven. So we do want to get rid of this three first. So I'm going to go out and subtract three from both sides like we talked about, and that's gonna leave with me with With four is equal to three times X squared minus four. I'll divided by five. So now you can try to break apart that fraction by multiplying by five on both sides, because then, like I said, they will get rid of the five on the top of the five on the bottom and that'll leave me with three times X squared minus four is equal to 20. So then, from there I'm going to defy both sides by three. To get that X squared minus four is equal to 20/3. And then from there, I could add forward to both sides again because we want to do these subtraction addition type components first. So then, if I do that, I'm gonna be adding forward both sides, and that gives me X squared is equal to 6.7 plus four because 20 divided by three is roughly about 6.7. So then that gives me I'll just go over here. X squared is roughly about 10.7 now to get rid of the square. The opposite of a square is a square root, so we're gonna go in and square root both sides. If I do, that X is going to become plus or minus 10.7. And that's a key detail that will examine a little bit more in depth later on. But we have to realize that it's gonna be both 10 plus 10.7 and negative 10.7, because if you think about or the square root of that value other because we know that if we were to square these, for instance, if we have a three squared that's equal to nine but a negative three, that whole quantity squared is also equal to nine. So that's definitely something that we have to keep in mind. Mhm. Johns Hopkins University
AH
##### Ahyeon H.

University of California, Berkeley

CC
##### Charles C.

Kennesaw State University  