Common Core Algebra I.Unit 2.Lesson 4.Justifying Steps in Solving an Equations.by eMathInstruction
Algebra 1
Learning the Common Core Algebra I.Unit 2.Lesson 4. Justifying Steps in Solving an Equation by eMathInstruction
Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk Weiler, and today we're going to be doing unit two lesson number four justifying the steps in solving an equation. Before we get before we begin, let me just remind you that you can find the worksheet that goes with this video, as well as a homework set by clicking on the video's description. Don't forget also on the corner of every worksheet we've got our handy dandy QR codes that will allow you to take a smartphone or a tablet, scan the code and come right to the video. All right, let's begin.
One of the things that common core asks us to do is to be able to not just learn the procedures for solving equations, but to also justify why equation solving techniques work. It really all boils down to what are known as the properties of equality. Now, there are two main properties of the quality, and they probably make a lot of sense. The first one is known as the additive property of equality. Additive property of equality. And what it says is pretty simple. If we've got two expressions that are equal to each other, if a is equal to B, then when we add the same thing to both sides, the equality still holds. Now that should make a lot of sense, I would assume, right? If I know that 5 is equal to 5, then if I add, let's say, three to both sides. Right? I get 8 is equal to 8. Sure. In other words, if this step is true, if that's a true equation, we've talked about true equations before. Then when I add something to both sides, it remains true.
Please note, of course, if the original inequality is false, you know, if I say that three is equal to 7, and then I add 5 to both sides, then it remains false. False. And continues to be false. So everything that we do today is going to be based on the idea that we're starting with an equation that's true. Let's look at the multiplicative property of equality. It says basically the same thing. If I know that two expressions are equal, then I can multiply both sides of the equation by any constant I want by any real number I want, and it will maintain that true equality. Let's again start off with our true quality 5 is equal to 5, right? If I suddenly come along and I multiply both sides of this equation by two, let's say, right, then I get the very true equality ten is equal to ten, right? So if that equality is true, then multiplying both sides of the equation is true. What happens a lot of times is that teachers summarize both of these rules is basically one thing, which is do what you want to either side of the inequality as long as you do the same thing. No. That's a nice shortcut to understanding, but we really want to know these two properties. Because these two properties allow us to do manipulations on both sides of the equation. It's a little bit different than when we actually work with equivalent expressions on both sides.
There we're using things like the commutative property, the associative property, and the distributive property. All right? Let me clear out this text. And then let's jump right into justifying some equations. Or not just find the equations, but justifying the equation solving techniques. Okay. Exercise number one, consider the equation two X plus 9 equals 21. The steps in solving the equation are shown below, justify each step. So the first step that we would really do in doing this equation. Now we would probably typically show it kind of like this, is subtracting 9 from both sides. All right, that's actually the additive property of equality. Now you might object and say, wait a second. We're not adding there. We're subtracting. But we can get away with that, right? Because calling it the additive property of equality, because really we can think of this as simply doing two X and then adding a negative 9 to both sides. Adding a negative 9 is the same as subtracting 9. So we get our typical two X is equal to 12. All right. Now, in step two, we multiply both sides by one half, and we get our final true statement. X is equal to 6. And that is then the multiplicative.
Property of equality. All right, just like we can subtract from both sides, by the way, we could have easily divided both sides by two, and we would still call it the multiplicative property of equality. And I know that's a little bit confusing, right? But dividing both sides of an equation by some constant, or even by some variable, is the equivalent of multiplying by its reciprocal. So dividing by two is the same as multiplying by one half. Those two properties basically, those two properties form the basis of all linear equations solving techniques. Everything else, you know, manipulating the two sides to get equivalent expressions. Again, that rests on things like commutative associative distributive properties. But when we add things to both sides, subtract things from both sides, that's the additive property of equality. When we multiply or divide both sides by some constant, the multiplicative property of equality. Okay? So let's do some more elaborate justification.
All right, I'm going to clear the text out, so pause the video now if you need to. All right, let's do it. Next page. There's our next page. Oh my gosh. All right, so now we have a much more complex equation. Let's actually take a look at it for a second before we start to get into the justification. Now remember, today is not about solving these equations. You did that in math 8, right? You did that in 8th grade common core. You have, we've reviewed it in this course. So what we're doing today is justifying each step. All right? So let's take a look. Step one, what happened? We took this equation and we rewrote it as three X plus 6 minus two X -14 equals four X plus 7. Pause for a moment and pause the video if you need to to think about what what thing that we've learned already justifies that step. All right, hopefully you realized that it was the distributive property. So the distributive. Property. Okay? It's not the distributive property of equality. This actually has nothing to do with equality.
The plain fact is, if we're multiplying X by two, X plus two by three, we can distribute that multiplication and likewise we can distribute the multiplication by negative two. Now let's take a look at our next step in the next step, it looks like we took this 6 minus two X and rearranged it into negative two X plus 6. Again, think about what property allows us to just flip flop that. Hopefully you got it. That's going to be the commutative property. And specifically, it's the commutative property of addition. All right, the commutative property of addition allows us to flip that 6 minus two X and make it into a negative two X plus 6. Oh boy. This next one's kind of weird. This next one, we took the three X plus negative two X and we pulled that X out, right? What is that? That's kind of a weird one, right? You might not think that even needs justification. But what allows us to do that? Again, it's the distributive property. Many of you would just think, how all we're doing there is combining light terms. But the reason that we can combine like terms is because of the distributive properly property, no matter how poor my handwriting is, that's why we can do that. All right? Now we're finally down to this step.
Look at where we are. X -8 equals four X plus 7. Let's see what happens next. In the next line, it appears that what we're doing is that we're subtracting four X and adding 8 to both sides, tracking four X adding 8. What allows us to just subtract the four X and add 8 to both sides of the equation. All right, finally we get into sort of the topic of the day. That is the additive property. Of equality. Additive property of equality, right? Added a negative four X to both sides and we added an 8 to both sides. Okay? Now let's take a look at the next step. Next step, we took this negative 8 minus four X and we made it into negative four X -8. Likewise, we took this 7 minus four X and turned it into a negative four X plus 7. I think we've seen that one before. Think about it for a moment. Yep. That is again. The commutative property. Okay to abbreviate properties prop. Of addition. All right, this is starts to seem a little redundant now. Then we took this X minus four X and we factored in X out. And that is, again. The distributive property. Again, not of equality, just the distributive property. Technically, by the way, it's the distributive property of multiplication over addition, but who wants to write that. And finally, we're at negative three X -15, we divide both sides by negative three, we get negative 5, and it's weird as this is, that is going to be the multiplicative.
Property. Of equality. And again, you might say, well, why not, why shouldn't we call it the division property of equality? Or likewise, in step, I guess it was four, why not the subtraction property of equality? And again, this just boils down to the fact that if we have something like negative three X is equal to 15, instead of bringing in a whole nother property, what we're really doing is the equivalent of multiplying both sides of the equation by negative one third. So we are still using the multiplicative property of equality. Wow, look at that. But as weird as this may be, as crazy as this is, at the end of the day, these are the reasons that you use to justify solving a linear equation. It's one of the standards in the common core algebra curriculum is to be able to justify the steps in solving an equation. So we want to get some practice on it in this lesson. All right, there's a lot on the screen. I'm going to clear it out. So pause the video now if you need to. All right, here we go. Okay, let's keep going.
Now, one of the things that I like the most in math is when strange things happen. When things happen that make me go, wow. What? What happened there? Right? And we're going to see that exercise three. So what I'd like you to do is I'd like you to look at this equation. In letter a, we've got this kind of justification thing going on. And I've even filled in some of the reasons. What I'd like you to do right now is pause the video and try to fill in the other reasons, okay? Take as much time as you need to go back and look at the last exercise if you have to. Rewind the tape if you have to or not the tape, but the video. All right, pause the video now and try to work through those three justifications. All right, let's go through it. So the first one was already done for us. In other words, we distributed a negative three and a positive two and got these two quantities. In the next step, what we did is we subtracted 8 from both sides. Hopefully, you wrote that that was the additive.
Property. Of equality.
Any time you add or subtract the same quantity from both sides of an equation, it's the additive property of equality. All right? We then factored in X out of both the 5 and the three. Anytime we factor, we're actually bringing in the distributive property. It's tricky because it seems like we're doing the opposite of the distributive property. But we're just using it in sort of the reverse order. That's the distributive property. The next thing that we did once we were down at this step is we subtracted a two X from both sides. That's the additive property of equality again. So we've got that already written in. And then the last thing we did is that we flip flopped this 11 negative 11 minus two X and made it into a negative two X -11. All right, and hopefully you said that that was the commutative property. And done. All right. But the weird thing about this equation, the thing that makes me go, oh, that's kind of cool. Is that we don't get a value of X? We don't get a value of X. Take a look at B the final line of the set of manipulations is very strange statement. Negative 11 equals zero.
Now, I think the first question here is pretty easy. Is this a true statement? No. This is false. Right. And the second thing could any value of X make it a true statement? Is there any value of X that I can put into this equation that will result in negative 11 being equal to zero? No. Negative 11 is never equal to zero. No matter what X is. All right, last question. And I want you to think about this a bit. What do you think this tells us about the solutions to this equation? In other words, the values of X that will make it true. What I'd like you to do is pause the video right now. Go back to the basic idea that we saw in the first lesson, which is that solutions to an equation are those values of X or whatever the variable is that make the equation true. So think about it. Pause the video and try to answer C. All right, now this is actually quite important. This kind of problem will come up time and time again as you move up in mathematics, but let her see is very simple, right? It basically means there are no solutions to this equation. All right, and that is really, really critical to understand.
Normally, when we solve a linear equation or other types that you'll see this year, you know, you go through a set of manipulations and eventually what happens is your final line is X equals block X equals 5, X equals ten, X equals negative two thirds, things like that. But here are our final line is negative 11 equals zero. And because that statement is false and always false, and it doesn't matter what value of X we put in. It's false. The plain fact is no value of X will make this equation true. And because no value of X will make it equal, it will make it true there are no solutions to this equation. Kind of cool, huh? All right, I'm going to clear this very important exercise out. So pause the video, really think hard about it right down anything you need to, and then we'll move on to another one. That's really rather weird. Okay, here it goes.
Let's move on to the next page. All right, same deal, a little bit of a complicated equation. It says consider the equation 7 X plus two X two times X plus 5 equals 9 X plus ten. Letter a says show that X equals negative 5 and X equals two are both solutions to this equation. Both solutions pause the video now at this point you should know how to show that a value of X is a solution to an equation. See if you can remember how to do that. All right, let's go through it. I want to draw like a little line down so that we've got some space for each one of these. And we'll start with X equals negative 5. All right, remember to show that something is a solution to an equation. What we have to show is that when we substitute it into the both expressions, the expression on the left and the expression on the right, we find that those two expressions are equal and make the equation true. So let's take a look. We're going to get 7 times negative 5 plus two times negative 5 plus 5. Equals 9 times negative 5 plus ten. That's going to be negative 35 plus two times. Oh, that's zero.
Equals negative 45 plus ten. I think I kind of used a little bit of extra room on this, but we'll live. We'll get negative 35 plus zero is equal to negative 35. And if you don't mind, I'm just going to scooch it over here. Negative 35 equals negative 35. That's true and therefore. It's a solution. Remember, that's what a solution is. A value of X that makes the equation true once we substitute it into the left expression and the right expression. Let's do two. Whoops X equals two. Let's put it right here. All right, so we get 7 times two. Plus two times two plus 5 is equal to 9 times two plus ten. So we'll get 14 plus two times 7 equals 18 plus ten. We'll get 14 plus 14 equals 18 plus ten. Again, not so great on the room management. But we'll get 28 equals 28. So that's also true. Um. Well, all right. So what does this really tell us? It tells us that this equation has at the very least two solutions.
Now, let her be says solve this equation by manipulating each side of the equation as we did above. In other words, just like we can do. Let's solve the equation. What does the final strange result tell you? So here's what I'd like you to do. I'd like you to pause the equation. And then I might get a pause the video. Try to solve the equation. I'm not sure how you pause equations. But I'd like you to pause the video, solve the equation, and something strange is going to happen. Okay? And then if you can figure out what that strange result tells you, try to write something down. All right, let's go through it. Okay. Well, I suppose probably the first thing I would do on this equation is I would distribute that too. That's going to give me two X plus ten on the left hand side. Not much I'm going to do the right hand side. I'm going to get 9 X plus ten. I think I'm going to combine light terms now and I'll get 9 X plus ten. Is equal to 9 X plus ten. I don't know. If I'm just kind of rotating going along, I would probably subtract 9 X from both sides right now, and I'd get ten equals ten.
Now a lot of students will look at that and they'll say, okay, X is equal to ten. That is certainly not the deal. In fact, this result is kind of the opposite of the one we had last time. That is a true equation. Okay, it's always true. Let me put that down. Always true. Always true. Ten is always equal to ten. So what does this result tell us? It tells us that every value. Of X solves this equation. All real numbers. Every real number will be a solution to that equation. So let's take a look at C test your conclusion by picking a random manager or really any number, and showing that it's a solution to the equation. So what I would like you to do, and then we'll do maybe just one example. But I'd like you to pick one example. Just one, and show that this equation is true. Don't pick negative 5 or two. We already did those two, but grab one for yourself. Pause the video now. All right. Now, of course, the one that I'm going to randomly take probably isn't going to be the one that you're going to randomly pick. But I think I'm going to go with X equals, let's see. We did two. We did negative 5. Let's go with let's go with 7. X equals 7. Let's see if 7 is a solution. All right, so again, as always, a solution is any value of X that makes the equation true. So let's substitute it in 7 times 7 is 49. 7 times 7 plus 5 is 12. 9 times 7 is 63.
Hopefully, you're doing this without a calculator to strengthen your arithmetic skills. 49 plus 24, 63 plus ten. 49 plus 24 is 73, 63 plus ten is 73. And that's true. And therefore, 7 is a solution. Any number we picked, we would get a true statement at the end, which is really rather cool. All right? So when you solve equations, really, one of three things can happen. Either you do manipulations and at the end of all your manipulations, you get something like you get something like X equals three. And then if that's the case, then there's your solution, right? Or you get something like negative ten equals four, in which case you say there are no solutions. Or you get something like 8 equals 8, in which case, every number is a solution. Sometimes we'll even say that it has an infinite number of solutions. Okay? That's it. So I'm going to clear out this text. All right, but really pause the video now and think hard about this problem. Very important one as well.
All right, here we go. Okay, let's finish up. So today, what we saw is we saw the additive property of equality and the multiplicative property of equality. And how we can combine these two properties of quality, along with the commutative associative and distributive properties to justify the steps in solving an equation. How we can start from the idea of a true statement, manipulate manipulate manipulate until we get the value of X and we can justify each step. We also saw how doing this equation solving process can give a strange results, things like eliminating the X's and just ending up with something like negative ten equals two, in which case no solutions, or we get something like ten equals ten, in which case we say, every number is a solution.
There's an infinite number of solutions. So this is pretty heavy stuff, and it's going to take a little bit, but it's good, because if you go on to take common core geometry where you do a lot of proof and proof is all about explaining your thinking and explaining your steps, then this will help. All right. Well, I want to thank you for joining me for another common core algebra one lesson by email instruction. My name is Kurt weiler. And until next time, keep thinking. I keep solving problems.