Welcome everyone to our lecture on variables and equations. We're going to start replacing some of these numbers with letters. Of course these letters can be a good old letters, a, b, c, d. We could do x, y, z, w, we could even do greek letters, alpha, beta. It doesn't matter, but we're going to use these variables to solve equations, and that's what we after. And so before we start getting into the technical aspects of it, I want to talk about the more theoretical aspects of it. If I write some expression, w+x=7, what does this mean and how are you supposed to approach this thing? The idea behind this for true equality is that equals really does mean equals, the two values on the left have to be the same as the value on the right. This is the golden rule of equality. When you start treating and manipulating the left side, you have to do the same thing to the other side. Back in the day in school, the common picture that they used to give you was a scale, they used to give you a scale and whatever you put on the left side. However, w To find what everyone has to do would have to balance. This notion of balance and you put a big block over here, that's 7 and the scale can't tip, it has to be exactly equal. This notion of equality of these two scales balancing out is what we want to keep in mind as we go through and go through the algebraic manipulations. What happens to students when they start seeing equations inequality is they become symbol pushers. You're not thinking about why you're doing what you're doing. Instead you just know that I move this line over there or I do somehow direct notion. I really want you to think about one side of the scale tipping up in one direction and then you adjusting to balance it back out. So we're going to go through some of these properties and of course I'm just going to list a couple down. If I start with something like 7=7, well you're certainly allowed to add or subtract things to both sides. So you're always allowed to manipulate the equation by adding let's take 7+x. Now I just added plus x to one side. The scale has tipped in one direction, so I have to add plus x to the other side. You can certainly subtract as well, no harm no foul, do whatever you need to balance this out. And of course you can multiply, if I had 7=7, I could do 7x equal 7x. I can multiply these things multiply both sides by x. And of course you can divide, just make sure you're not dividing by zero. So let's divide by 2 or something like that, you can divide by 2. Each one of these expressions, it's not because I'm writing the thing on the left, I'm writing the symbol on the right. I'm unbalancing the equation temporarily and so I have to equate the equation, I have to solve the equation. I've seen people do beautiful Algebra on one side of the equation and then all of a sudden they leave the other side blank. So we're going to work with lots of variables and play around with these things and I just want you to be comfortable with any variable, x Is obviously pretty common but it doesn't have to be used. They call it of course dummy variables. So we can use anything we want. And I'm going to intentionally mixed up just so you don't get too comfortable with one. As we play around with equality, just keep in mind that not every equation has to be equal. There's also coming our way is inequalities. These are greater than, less than, strictly greater than, strictly less than. So equality is just one way to do things. When I have two numbers, let's say a is equal to b. I can play around with equality and we can always write it backwards. So you always have three properties and the first one is that you can write it backwards. So if a is equal to b, then you're perfectly allowed to say b is equal to a. We're kind of writing the rule book here so that you know what rules you break in case you get a wrong answer or soming like that. So we have phase equal to b, then b is equal to a. This is called symmetric. Another property is if a is any real number, then of course it is equal to itself. a Is equal to a, this is called reflexive and last but not least if a is equal to b and b is equal to c, then a would be equal to c. I don't want you to think about these justice and numbers. I want you to think about these is really expressions. Think about simplifying, maybe know a little bit of algebra all ready factor foiling but you're manipulating expressions, it takes multiple lines. So if you have the first line and you start doing intermediate steps, this last property gives you that the first line will always be equal to the last line after you do all your work. The fancy way to say this, of course is transitive. So here's a couple more definitions or things that I just want you to be aware of, that you can do nothing perhaps surprising here, but I just want to write the rule book out so that we have it and then we know what to do as we go forward. Let's do just a couple of simple examples as we go through this. So, if I tell you that 3x+x is the square to 10, let's solve for x. Okay, so we're going to try to look for things here. We have a nice equation, I have numbers, same degree, like terms as they call it. So 3x+x, well, this becomes 4x. Let's combine the expressions and this becomes square to 10. Now, square to 10 is a nice irrational number. Don't get scared by the square to 10, I don't know what it is, I don't really care to get the calculator at this point. I'm just going to leave it at some number. If I need to work it out as a decimal, then I'll grab the calculator. But for now square to 10 Is square to 10. I have 4x equals 10, I want to get rid of this 4, I want to undo the process of multiplications, so I divide by 4. Whatever I do to one side, I must do to the other side. So x then becomes, cancel the force of course, square to 10 over 4. It's some number, you're going to get two questions with the answer is not always so pretty like x equals 2. Be ready for numbers that look funny and if again, if your calculator has no problem, if you can go over to a calculator and compute square to 10 over 4, then you should have no problem with it either. I don't really care for a decimal at the moment, so I'm just going to box this one up and say good enough for now. I care more about the steps just to solve for x and play around with this. Let's do another one, how about 4x+5x is 99, add like terms 4x+5x, well, that would be 9 x's. And remember that's still equal to 99, don't lose the right side. Now we're going to divide by 9. When I do that, I get x is 99 divided by 9. Better way to say that of course is 11. So this one's a little nicer than before. We're just playing around reviewing some basic algebra skills, making sure we're on the same page. We're multiplying adding, subtracting both sides. Let's do another example, feel free to pause just to make sure you're on board against early in the course. So nothing too crazy so far but make sure you know how to do this x+3 using all our properties, solve for x. All right, there's a lot going on here. There's a few ways to do this one. I'm going to take the long road for a minute. Four times an expression by the distributive property allows me to bring in the 4, for all the same reasons I'm going to bring in the 5. This becomes 4x plus 4 times 3 is 12 plus 5x+15. And that is still 99, notice I haven't changed the scale on the left side. I'm just simplifying the expression on the left side, 4x+5 Combine some like terms, those are like terms, they both have a number of the coefficient times a variable x. That's of course 9x, 12 and 15, well that's of course 27 is 99. Now there's a few things we could do here, we could subtract 27 from both sides. Maybe we'll do that but I want to show you another thing we could do. 9, 27 and 99, they all have a 9 in common, 9 goes into all these things. So maybe what I'll do is I'll divide both sides by 9. You may not have done this step, I would imagine most people subtract 27. That is perfectly fine, let's just see a different way, let's get the right answer. I want to divide everything by 9. That means I have 9 x over 9 + 27 over 9 = 99 over 9. I'm intentionally being nontraditional, just to show you some alternatives of things you can do. Now the 9's cancel and I get x + 27 over 9, well, hey, that's 3 and 99 over 9, well that's 11. Now I'll subtract the 3 to the other side, when I do that, of course I get x = 8. There's more than one way to do this, I could have subtracted 27 or maybe you realize you could have even factored out the x + 3 at the top, I don't care how you do it. There's no right way or wrong way to sort of pick your method, pick the method that's the most comfortable for you, and as long as you get the right answer, we're fine with it. By the way for all of these, at least what the numbers are nice enough, you can always go back and sort of check your work. This is a really good skill to develop, as the questions get longer and you do more steps, it's very easy to make small algebraic mistakes. There's a minus sign, say 2 + 2 is 5, it's a very human thing to do. So I don't care if you make mistakes, I just care if you go back and fix them. So I got x = 8, I filled up you don't have to slide here, let's go back. Is it really true that 4, 8 + 3 + 5, 8 + 3 is 99? I put a little question mark over the equal sign, just to see if that's true. Then let's check, so we have 4 times 8 + 3, 9 ,10, 11 + 5 times 11 is that in fact was 99, let's figure it out. Well that's 44 + 55, and of course that is equal to 99, so that does work out. If I didn't get it to work out, I would go back and try to solve from my mistake, go back and fix the problem. These problems here, when equation just appears and there's no backstory, there's no reason why we're doing it we're just practicing, in the business, we call these a little bit of naked math like the numbers are just there. There's no reason, there's no rhyme. Of course as you specialize and go into some degree program or do these, these equations then pop up in applications, based on whatever it is you're studying. So now let's look at actual word problems, to prepare you for that moment. So here's the problem, Nico has 5 outstanding bills from a recent home renovation project, that are each accruing late fees at $10 per day. The bill for the cabinets was due one week before the other bills which were all for utility work. If his total fine was $870, how long was each bill overdue? Pause the video, read this again, go slow and let's go through and see if you can solve it, pause the video. Okay, you ready? The question that they're asking for, why this is harder is because they don't give us the equation, they're asking for how long. So that's the unknown, let's call that x. We can certainly call it any other variable you want if you love y, then call it y, it does not matter. So I have 5 bills, 4 of them are for utilities, so I have 4 utility bills, and I have one for a cabinet, and they're accruing $10 per day. So the 4 utility bills, so I have 4 bills at $10 per day, what I don't know is the number of days, that's how long each bill was do. So at $10 for the mystery amount of days. The 4 is there because there's 4 utility bills, and then $10 a day, that I don't know. Now I have this cabinet, the one cabinet bill. The cabinet came in one week before the other bills, so whatever the days of the utility bills are, I have still $10 a day, but I have x -7. It's an additional nice to say x + 7, sorry, it's an additional 7 days, so x + 7 to the fee, the fine on the cabinet. And I know that when I add these up, when I add up the total, I get 870, my total fine is $870. All right, let's play around with this and solve this here. So let's simplify this a little bit. The first line for the for utility bills becomes 40x, the cabinet bill, we can bring the 10 into both pieces, this becomes + 10x + 70, all of that is equal to 870. It's all dollars here, so I'm not forgetting that, but I'm going to stop writing the dollar sign and I'll clean up the units at the end. So the skill of course we're after now solving, for how long each bill was due. We have 40 and 10, add them up, like terms you get 50x + 70 is 870. Let's subtract 70 from both sides, and we get 50x equals 800, and therefore x's 800 divided by 50 which is 16. Now be careful here, it's very easy to answer the question but then sort of forget about why you're doing this. This is the challenge of word problems. What does this mean? What's the units? When students leave off the units, I come in and put cats for doughnut question mark, just to pick some ridiculous unit. What does this thing actually mean? How long with our time here, are units of time or days? So x is in 16 days. Now we gotta answer the question. They asked how long was each bill overdue? So the 4 utility bills, they were each 16 days overdue and the 1 cabinet bill, remember there was came in a week earlier, so that's 16 + 7, that's 23 days overdue. So the answer to this is not actually 16, it's 16 for the utility and 23 for the cabinet. They're asking for all the bills, so make sure you answer the question using the units. Let's do another example. Marco is 10 years older than Ralph and 2 years ago Marco was 3 times as old as Ralph. How old are they all now? All right, so I have some information hidden, little cryptic, but I can find their ages. So let's introduce some variables, we'll let M of course be the age of Marco, so this will be age of Marco. You have to be careful because we're talking about the present and the past. So I'll just specify this, but it's today's age and then Ralph will be the age, so use R for that, I could call it x and y it doesn't matter. Again, I want to introduce new variables, you don't get comfortable just only working with x and y, so I have my M in my R. So Marco is 10 years older than Ralph, whatever Ralph's age + 10, that's Marco's age, this my first equation. Now, 2 years ago, be careful M -2, right? That's in the past, whatever's ages today, he was 2 years younger and they're telling me that he was 3 times as old as Ralph. So he's 3 times as old as Ralph, but be careful Ralph that were in the past 2 years, this is also R -2. I have 2 equations and I really need to solve for both variables now. So what I'm going to do, is I'm going to use my first equation, M is 10 + R, I'm going to take this entire equation and just substitute it right into the second equation. When I do that, I get 10 + R- 2 = 3 Parentheses R -2. This is a nice equation I can solve for R, let's go ahead and do that. So we have 10 minutes to of course is 8 + R and this is 3R, I'm going to bring 3 into both pieces 3R- 6. I have an R on both sides, when you have that, you want to simplify and combine like terms. Let's subtract an R from both sides of the equation. When I do that I get 8 = 3R- R of course is 2R -6. Now let's add 6 to both sides, we've gotta isolate that R, and I get 8 + 6 which is 14 is 2R, divide both sides by 2 and you get that R is 7. Marco who's 10 years older than Ralph, With Ralph seven. While that puts Marco at 17, you should absolutely check. We just did a whole bunch of work. I really love to guarantee that this is correct. So why don't we just go off on the side here in check is Marco at 17, 10 years older than Ralph who were claiming is seven. Is there a 10 year difference? Yes, so that's good and two years ago. So what would their ages be two years ago? Marco would be 15, 2 years ago and then Ralph two years ago would be five and so would Marco be three times as old as Ralph that works out as well. So I can feel pretty good about my final answer that Ralph's seven and Marco is seven. So hopefully you got that as well before we move on to just other things, you're going to see once in a while, this notion of absolute value, I'd like to remind you what it is. When I write down for any real number X, the absolute value I put these bars next to it and there's a couple ways to think about it. I think the wrong way to think about this and I just this is like the way that I've seen people be taught is it's not a magic eraser, it's not this like erasure of minus signs, it is the distance. Don't you think about is a distance? You'll thank me later. The distance from X 2 to 0. Why watch this? So here's our number line and I put zero on here and let's put like five. And if I want to ask you what the absolute value of five is. So you say well how far away am I from five from zero? What's the distance? Well of course that's just five. I don't think the right way to think about this is to say well like it's positive, so erase the absolute value is really doing something, it's calculating something. It's not a magic eraser because that also helps explain why if I have negative five and I ask you what's the absolute value of 512345 this would also be five. So distance is always positive, we're not negative, I should say it is perfectly fine to ask what the absolute value of zero is. That of course is just zero. Why the distance from zero to itself is zero. So if you think about in terms of a distance, this will help later if you go off and do geometry or you're trying to find distance formula or something that it all makes more sense when you understand the absolute value as a distance. That will also help you to answer questions about it. To solve expressions or or equations using it will do some of those in a second. But I really want to think about it as a distance and not a magic eraser. The way that it works in addition to just calculating things, so if X is non negative then the absolute value of X is always itself. That's why we got the absolute value of five is five, absolute value of 10 is 10 absolute value of pie is pie. If you're giving me a non negative number positive or zero, then the absolute value of X returns itself. And if the absolute value is negative five or whatever the absolute value, how does it work? Yes, it could sort of hop and count distances but we want an algebraic way to do this. We're here to learn algebra after all to turn a negative number of positive again, don't think of it like a magic eraser. When I learned this as a kid like the teacher brought in this novelty enormous eraser and it was erasing negative science not the right way to think about this. The way that it works is, it actually multiplies by the number multiplied by -1. This is multiplication by negative one. So why is the absolute value of 55? Because it's a distance. And what is the absolute value? Do how does it turn negative five into positive 5? Well it takes negative one times negative 5. That's the algebraic idea behind it. The geometric reasoning is just because it's five units away from zero. But as a negative number it Multiplies it by negative 1. To keep that in mind as we go through. Let's do a couple here. So let's simplify some of the following. So let's do one. Let's simplify I'll give you three. You pause and see if you can do it. Okay so actually it's due for it. So I have the absolute value of negative five, absolute value 10 negative of the absolute value negative five and negative the absolute value of five. Take a second, go through them. Just remember how this works. I'm inputting a negative number to my absolute value so as we saw before this is negative negative five of course this is five. How do I get there that are multiplied by negative one absolute value of 10. 10 is positive. I just Keep it as is no change you get 10. Now be careful we're going to start to put these inside of other operations so now I'm taking the absolute value negative five and negating it. So I take my five from Friday and I negate it and so the answer here is minus five, and one more time finding negating the absolute value of five. I'm negating five again and I get just back negative 5. Let's do a couple more. Hopefully you went for four on that one. I'll write them out and then you pause the video and see if you can solve them. Give you some varying difficulties. Okay pause the video, try to work these out. 1st 1 is absolute value of X minus 6 6. He says the absolute value of exports to is 10. See says the absolute value of negative X is X. And last but not least D says the absolute value of X is minus 3. Pose video so if you can solve some of these ready. All right, let's go ahead and solve some of these things here. So we have absolute value of X minus 6 is six. Now remember there's really two ways to look at this absolute value. If X minus 6 Let's think about this for a second. If X minus 6 is non negative, say like X is seven or something that if it's all positive so or zero then the absolute values do nothing there are just the same thing X minus 6 is equal to six. And of course from that move the six over and you get access 12. So that is a good answer but that's not the only possibility. So the other way to think about this is if X minus 6 well what if it's negative? What if the thing going inside the absolute value is negative? How does the absolute value work? Doesn't erase the negative sign and multiplies it by minus 1. So I take the whole thing and I negate it and that's going to still equal six. And when you clean that up you get negative X plus six is six subtract six from both sides you get negative X zero and the X is equal to zero. So you actually get to answers here and that is not surprising. Why is that not surprising if you think about it in terms of a distance on the number line, if you put six right in the middle you could say well if I subtract six from any number from what numbers do I get the number 6? A little weird because they're both going six year but six on the outside is saying I want the distance to be six and six on the inside is where I'm starting. So if you sort of jumped six to the right or jumped 6 to the left, subtract six, what numbers you land at will be 12 and zero. So that's why it's 12 and zero in terms of a distance. Let's keep going and solve some of these things here. I think the rest are maybe a little simpler. If we do B we have the absolute value of X plus two is 10 subtract suitable sides and of course you get acid value of X is eight. Now be careful for all the same reasons the other one there are two numbers, two numbers that have distance away from zero to be eight. So when you do that, do you get X is eight or minus 8? You can write this of course is plus or minus eight If you want to be concise but really just recognize there are two answers. Let's keep going for C if I want negative absolute value of X is equal to the absolute value of X, I'm asking what two numbers give me the same. One way to look at this is sort of break it up into both pieces and say, well what if X is positive? So if X is positive or non negative, what happens to the absolute value? So imagine X is seven or something like that. The left side becomes what's the absolute value of negative X. Well that's just going to equal X. And on the right side we have X. So if X is positive, these two things turn out to be the same. What values of X are equal to itself? All values here, we have infinitely many solutions. That's kind of nice. Every single non negative number is an answer to this. You can test it, plug in to plug in three, plug in four. They all work out. So the absolute value here is really nice. You get all of them and you can check if X is a negative number, say like negative two then the absolute value of negative two becomes positive. So you get X on the left and the acid value on the right is also positive. So the answer to this one is all rials. This I think is our first example of an equation where we have infinitely many solutions. So it's good to point out that some equations can have one solution as we saw, some equations can have two solutions as our A B in our A and B showed us and then some solutions can have infinitely many. So the answer here is all real numbers. All real numbers satisfy this equation. Pick any number you want plug it in and I guarantee that it's going to work. That's really neat to see. Last but not least if the outside value of X is minus 3. Just in the interest of time, we're going to get to this one quickly. What values can I plug an asset value? Remember it always outputs a zero or non negative positive. Can I get zero? Can I get a negative three? Never going to work, there are no solutions, no solutions whatsoever. So we saw a couple of types of questions here on this video. One solution, two solutions infinitely many solutions and of course no solutions. So all of these are possible, and so sometimes when students get these answer choices, they think that something went wrong. No, I got no solutions. That's okay, assuming on your math is right. I got infinitely many solutions, that is also okay. So just be confident with your work. Check your answers, and then, it's okay to get these different types of answers. Alright, great job in this video. We'll see you next time.
