It's time, we got to start. All right. As we know, some friends says it should be A, some of them says it should be D, and other answers, no one says anything, maybe it's none of them. As you see, guys, Karl opened his dictionary and said, if I add the number of the page I am looking for and the number of the page following it, it will be 345. Remember, we are talking about algebra, algebraic thinking and creating algebraic equations today. That means, let's say, the page Karl looking for is represented by X. Remember, solving those type of questions, we will most likely create equation. So, and they say, if I add the number of the page I am looking for and the number I am literally following it. Anyway, we have X and the next page, which is X plus one. And they said, when they add, here is another keyword, as you see. I am going to add another one to the X plus one. And we should get 341, guys, okay? We have one of the page and the following page, then we will get 341. Then we are going to solve for it. If we have some of the like terms, like like terms are all of the numbers, either integers, traditional numbers, whatever. Only numbers are like terms. Also, only given letters. If you are talking about X, yes. X, 2X, 5X, these are the like terms. If we have 7Y, 5Y, 1Y, whatever, these are also like terms. If we have same letter, we can call them like terms. To be able to solve that, first, we got to add those like terms. We have 2X plus one. Remember, there is invisible 1X here, invisible one here, 1X plus, 1X, 2X. Anyway, at the end, we get 341. So I assume, since it's a warm-up question, I'm not going to ask you anything yet, guys. I assume since we have 2X plus one, and when we solve equations with inverse of PEMDAS, well, remember, when I say inverse of PEMDAS, I am going to start with addition and subtraction first. Since we have one of the multiplication, two times X, and one of the addition, I am going to cancel that plus sign by subtracting. Minus one to both sides. Remember, solving equation means isolate that variable, that letter, or just literally make it alone. So we have two times X is 340, and since we have two of them, we got to divide both sides by two, guys. We get X equals 170. Well, remember, the page Carl is looking for, you know, is going to X, and X is 170. It's a little bit tricky, because remember, they said, if I add the number of the page I am looking for, and the following page, yeah, well, the page number he is looking for, 170, and following page should be 171. So answer shouldn't be 171, guys. Answer is 170, all right? If there's any question, Suhan, please just check from the chat box. If we have any question about that warmup, I am going to check one more time for you guys, then I'm going to move on. Any question do you have guys? I assume you don't have any question for me. Okay. Today guys we are going to talk about algebraic thinking, creating algebraic equations and so on. Please remember the four step problem solving strategy. First make sure you understand the problem. If it's necessary, circle, underline, whatever those keywords, plan how to solve them. You can create some sort of chart, table, graph, anything works or actual algebra equation. We got to work on creating algebraic equation a lot today. Carry out your plan. Do the actual calculations and check if it makes sense. Look back and reflect and check, okay? And the next. Algebra use a variable like typically a letter. Or I use X in the previous question to stand in the place of an unknown or changing value. Determining numerical values for the variables helps us to solve mathematical problems. You will see without generating X for the number of page they are asking, we couldn't just find or we couldn't just guess the answer, you know, it's not that simple. Variables can be used in inequalities or equations. Frequently multiple variables and equations can be created to describe a problem statement and must be solved together to meet all the required conditions. We got to talk about that in the future with the questions. In most cases, adding to both sides of an equation or multiplying both sides or applying powers to both sides will not make an equation untrue. That means, guys, equation kind of represents a balance. Whatever you are doing on the right side, you need to do the same thing on the left side. Then because our goal is to keep the balance, remember. All right, next. Here we have like a little example for you guys. When you solve the equation, we have 2X plus 5 equals 21. As you see, even PEMDAS says you are going to deal with the multiplication first. No, you got to work with the inverse. You got to cancel that plus 5. OK, basically adding minus 5 both sides or subtracting 5. Anyway, so when you do that, as you see from here, we will have 2X equals 16 left. Second step should be multiplying both sides of an equation by 1.5 or dividing by 2 basically becomes X equals 8. This is the example, OK? We first get rid of 5, then get rid of 2. Combining equations can help eliminate variables and allow solving them one at a time. So we are going to use that method when we talk about system of equations, solving system of equations. Sometimes you can combine, you can add. It depends on the case, OK? Since I like terms like 7 equals 7X, from that equation actually you just add those two equations, guys. And you get rid of those negative 3Y, positive 3Y, then you just keep continuing, guys. 7X equals 35, X equals 5. Then you just plug it in. X equals 5 in any value to find Y equals 2 as well. Remember, if we have two unknown or two variables or two letters, whatever you call, we need to find both of those variables. As the solution said, we should put X first, comma Y, like ordered pairs. Anyway, there are numerous ways to solve, ways to use variables and algebraic operations to solve problems. All right. Let's check some problems together. I said together. Remember, look, I want you to give us some answers, at least how to solve them. Sofam is going to check our chat box actively. Whenever we get one or some answers about number one, even how to solve, please share with us, guys. Then I'm going to start timer. Let's see. All right. Your time starts. I'm going to give you one minute. all right guys it has been 40 seconds i believe we have one answer right now i don't know who is that okay it has been one minute guys so do we have some answers yeah i got one person said a and one person said b oh interesting a and b look at that all right what about cd or a i mean cd or e no one all right so in that case guys either i am asking everyone either the friends who gives us answer or some of the others what is going to be first step here what do you do or how can you create equation here or what type of equation you get remember i don't necessarily want you to give me answer choice i want you to give me what do you think our first step should be Okay, I believe anyone else give us answer so you can check and let me know for this So guys they say some well some means you gotta add them. Remember you need to work with keywords as well seven consecutive numbers If we are talking about consecutive numbers, they are just increasing by one remember, but since it says seven consecutive odd numbers Let's just check some of the Consecutive all numbers it does it doesn't have to be seven of them. Anyway, I can just start with one then keep three five and so If I say the smallest number is x Then next number should be x plus two guys as you see Then the next number is I am not gonna put x plus two plus two as you see I need to convert every single number in terms of x x Since I know one plus four is five. I'm gonna put one plus i'm x plus four well, the next number should be Seven, by the way, not five. I don't know why it's repeating It should be x plus six guys As you see if you are talking about consecutive even or odd numbers You gotta start with x and keep going with that type of sequence algebraic segments correct They said some of the seven consecutive numbers is equal to 119 We gotta add seven of them to get 119. Then we gotta find the smallest smallest of all of them If you like guys you can just start And you can say the smallest number. Let's just say smallest number is x And just go from the x plus x plus two plus x plus four And so far then you just get the seventh number I'm, not sure. What's that x plus whatever x plus 12, I believe Then you would get 119 or guys For some of the cases You do not even need to give x for smallest number. There is easier way As you see this question from 2008 and number eight well Remember there are going to be like 30 questions. This question is one of the easier ones compared to others What if guys? If we have seven numbers like if we have odd number of terms One of the number is going to be in the middle Number fourth the fourth term. Let's just make this one x The number next to x is x plus two The number before x is x minus two as you see remember from the relation between consecutive numbers After x plus two we get x plus four before x minus two we get x minus four Interesting after x plus four we get x plus six And before x minus four we get x minus six guys If you count as you see we are going to have seven numbers, I mean you can just check if you like But when you add them That's four Two plus x Plus two Plus x plus four plus x plus six is equal to 119 So right now guys, I believe you all know that We can do some simplification even Before we start solving or add them. Okay As you see we have minus two plus two Please don't start adding them minus four plus four and minus six plus six They all are gone. If you count those x's you will see there are seven x 7x is 119 When you divide both sides by seven You would see that guys x is going to be Should be 17 But remember the only thing they don't ask us to find the middle number They want us to find the smallest number, which is x minus six 17 minus 6 is 11 guys. This is your final answer. Okay Is there any question guys? So please check check the chat box and let me know if there is any question about number one, please Any question you see soham So i've got a question is Why is he doing even numbers when he says odd numbers? I think this might be a reference to x and x plus two So Here guys if you choose x as an odd number X plus two has to be odd number as well You may think that guys Those plus two plus four plus six, whatever. These are the even numbers. But again think about that. I gave you that example Let's say you start with nine and you just continue with the next consecutive odd numbers, which is eleven What is the relation between nine and eleven guys? If I say nine represents x eleven has to be x plus two Remember, I cannot make x plus one x plus three and x plus five. Whatever. It doesn't look that way It was just trying to show the relation between those numbers any more questions on There is no question I assume Middle yeah, this is the middle number Okay guys then i'm gonna move on Soham says there is no question that means no question All right, check number two, please And share your answer or share some of the steps you get with us you can type anything chat remember Um And so the answer for your question answer is yes, I got that You asked me question on the chat box. I believe so answer is yes. Thank you so much Oops, I forgot to start the timer. Okay, it has been four seconds. Come on You Do we have some answers so far or not yet? I got one for E. Okay, one person says it should be E, okay. Others, it has been 47 seconds. All right, guys. It has been one minute. Do we have anyone else give us some other answers? Soham or... No. Okay. We have only others guys at least can give us what should be the first step here. What would you do as a first step here when you need to solve this question? Remember, I don't care about the answer right now. I want you to think about and give me the first step only. How would you start that question? Tell me, please. Via chat. Then Soham is gonna let me know. so kiddos you are saying you don't necessarily know the first step either that's sad man All right, so I believe we are getting some more chats do we get some answers for So can you hear me? Oh, someone says multiply both sides by y. Okay, what else? So then x minus 3y equals 12y? No, no, I got this part. So we can do that. So we basically cancel y first. x minus 3y over y. Since y is division, we can just cancel this one. Here like that. Then we get x minus 3y equals 12y. Add 3y both sides. I believe you already remember that. x equals 15y. And when they say x over y, basically 15y over y. You know I just plug here x, I mean plug 15y for x. It's going to be 15. But is there another solution way guys? What do you think? Okay, so you are saying nothing. That means I believe there is no different ways. Guys, sometimes you can use the properties of the fractions. Can I rewrite that expression as x over y minus 3y over y? Is it possible? What do you think? Is that right? Or I'm just making up? It's just yes, no question guys. If you think it's true, just click, just type t. If you think it's false, just type f. Someone says yes. That means they say it's true, okay. Then we just make it 12. Guys, here as you see we have 3y over y. Well, 3 multiplied by the same number and divided by the same number. Just ignore that. We have x over y minus 3 is equal to 12. I really think that you already know what's the last step here. You know, x over y is equal to 15 guys. Is there another way? It's all up to you. I believe there are a couple of more ways. I'm not aware of them yet. Any questions for number 2? Or does it make sense? So, if there is a question, please let me know, then I can just wait a little bit more. No questions. Thank you so much. So, okay, we can move on to next one, which is number three, guys. This is your new challenge. And I'm gonna start timing. Okay. all right we get one minute do you have any answers on none no one answered that's certain okay all right guys instead of getting the answer can you please let us know via chat what should be the first step here what am i supposed to do like how do you start with that type of questions look you see that I am asking a lot of questions so learning and teaching requires a lot of complication guys I cannot just say like act like regular lecturing lecturer you know I mean I cannot just keep talking yeah we gotta do this and that and create equation and so that I don't care what you're listening or not I mean it shouldn't be that way you know I am asking you only the first step how would you start that please share it with us just a first step guys come on you got this I guess so would you like to act like a student so finally we got an answer from someone yes someone says oh they got answer interesting can i ask that someone but i'm sure i cannot provide your name so what was your first step here please type on the chat box i mean others can that too i mean don't be jealous man yeah just type your first step here in the chat box writing the equation with three variables which three variables okay let's just make your life easier i'm gonna a b c okay a is here b is here and c is here you know we have three boxes okay guys if you read the question there are three equations i should get can you tell us those three equations you see there is definition some number something is going on here what's that okay let's just make your life easier the second sense of you know the third sentences she wants the sum of all the numbers equal to 35 what does it mean what do you understand me you got 3 plus a plus b plus c plus 4 equals 35 excellent thanks guys look we have 3 plus a plus b plus c plus 4 is equal to 35 as you see guys we have 3 and 4 extra if I take out if I get rid of those numbers a plus b plus c equals what come on you can do this you can do this but you get only cancel 3 and 4 come on I'm gonna cry here, man Remember it's positive to me. There's invisible plus sign here You gotta do minus three both sides then minus four both sides. What would you get? 28 Okay Thanks, so is that you or someone give us answer actually. No, there was someone in the chat. Okay. Thanks Come on man, okay. Okay guys, so we check the third sense and we found something we just continue They say the sum of the numbers in the first three box equal to 22 Please convert that expression to algebra What type of equation would you create from that? Yeah You No one Let's talk me Look the sum sum means you add remember some of the numbers in the first three boxes to equal to 22 What does mean? You have an equation 3 plus a plus B equals 22. Yes. Okay Plus a plus B is equal to minute sir. Then you solve this equation guys What is going to be a plus B? what is a plus b guys come on 19 all right we get a plus b is 19 okay please read the last part of this sentence they say and the sum of the numbers in the last three boxes equal to 25 maybe i should change the color here please tell me what we get here b plus c plus 4 equals 25 excellent b plus c plus 4 equals 25 guys if you know me enough you already know i'm gonna ask b plus c get rid of 4 and tell me what would you get If b plus c plus 4 is 25, what is b plus c? It means b plus c is 21. Okay, we have b plus c plus 21. Alright guys, so here I want you to take a look at those three equations. We have a plus b equals 19, b plus c equals 21, and a plus b plus c equals 28. When you solve those equations or just plug in something, let's see, are you able to find a, b, or c? Tell me something, please, via chat. A equals 7. How do you find that? That's smart, Ferhan. I am asking you. How do you find A equals 7? A plus B plus C minus the quantity B plus C. Okay. Excellent, guys. As you see it. Thank you. I don't know your name, by the way, but I appreciate you. So, here, guys, we have A plus B plus C, as you see. Here, I just transferred 28. Also, B plus C here is 21. Only this part is 21. We already know that from that equation I am showing you right now. 21. So, A plus 21 is equal to 28. That gives us A equals 7. You know, you got to take out 21 from both sides. A is 7. Well, do you think we can use A plus B equals 19? And A plus B plus C equals 28. To get something else, you know. Look, A plus B plus C is equal to 28. And since I know first two, sum of first two is 19. That gives us 19 plus C is equal to 28. And C equals what? I believe you'll get 9. Alright, guys. Here, do you think... Please read the last sentence. They said, what's the product of the numbers she writes in the gray boxes? Do you think we need to find B? Do we need to find B, guys? What do you think? No. Alright, someone says no. That's right. If you give us just a number, I'm going to ask you, how do you find that? First question, what's the answer? Second question, how do you find that? Come on. 9 times 7. Yes. That gives us 63, guys. Any question for number 3, then? Alright, that means, guys, this is your next challenge. And have fun. Let's just start the timer. Well, well, well. It has been 3 seconds. Come on. You should already find the answer. You're right, and I'm just messing with people on my screen, you know, I cannot see anyone besides you, so no. All right, 40 seconds. We got one answer for D. Oh okay, someone get D, all right. Cool. It is 54 seconds. And it's more than one minute, guys. It's just yes, no question. If you think you need more time, like 30, 40, whatever, more seconds, I can give you that. Please type Y represents yes in the chat box. If you do not need that time and prefer to start solving, please say or type no, I mean N represents no in the chat box. If you need more time, type yeah, type Y. Somebody says C as an answer. C as an answer, okay. What else? And people need more time. Oh okay, sorry. Then I'm going to show up and wait for them, all right. Hey, we got one answer for E and another answer for C. All right. So, so far we get two C and one and one D and E. Am I right? If you kind of create polls, you know, two person, I mean, two people says two, one person says D and E. Okay. All right. So, and one person says nothing yet. I believe that person is still working. All right, guys, let's just start giving you a hint. As you see, that area here from the smallest square gives one. That means, can I say each side length of that small square is one? Am I right? This is the time you need to say yes, but you cannot anyway. All right. So starting from the smallest square, we already know the side length. Can I just put here X represents side length of the next smaller square shape. From here, what am I supposed to do next, guys? What do you think? Hmm. All right guys, after I kind of color those squares just for you, I'm going to start. As you see, we have one of the little square which has side lengths are 1, another little square which side lengths are x. So when I focus that yellow square, as you see, can I say one of the side lengths is x plus 1 here from the height? I'm going to put here x plus 1 as well, and put here x plus 1 as well and so on. So right now, if I focus that little tiny blue square, this piece is x plus 1 and we have another 1, can I say side length of that square is x plus 2 guys? I am talking about blue square. I believe you get what I meant. At the end, if I focus that kind of pink color square, if this piece is x plus 2 from blue color and we have another 1, can I say each side length of that pink square is x plus 3? You know, 2 plus 1 is 3. And top part is also x plus 3. So since we have all of the regular shapes, you know, regular means we always use the straight lines, anyway, and these are all of the squares are regular shape, can I say from that shape, the sum of the side lengths at the top is equal to sum of the sides at the bottom? Well, we have x plus 1 plus x plus h. x plus h is equal to at the bottom, we have x plus 2 plus x plus 3. Okay, any question until that time, guys, until that moment, if you kind of lost in anywhere, if you have no idea for some of the parts, if you have a question about how I find particular side length or equation, please let me know. I mean, not let me know, let Soham know because he's going to check the chat. Is there any question after that step, Soham? Okay. So, guys, here, if you look at the left side of the equation and the right side of the equation, I believe you're going to realize that we have one of x here and another x here. We can just cancel those, you know. As you see, if it's possible, guys, possible to simplify something, I don't even recommend you to just combine them. It's not necessary. We can just keep canceling those. It doesn't... change anything. I mean, it doesn't change too much of the steps anyway, but makes our job easier. So we have plus one here. We have one plus h is equal to two plus three is five. Well, I believe every single of you right now figured out that we need to get rid of that extra one, plus one. Why are you doing? Minus one both sides, guys, okay? Then it gives us h is equal to five minus one is four. So two people said answer should be four and they get right. I am proud of you guys, those two friends. Whoever gets D and E, you still try. Thanks so much, guys, but maybe you did little mistakes. I don't know. Okay, number four, any questions so far? Please ask us, don't be shy. I'm not going to say that person ask question. It's going to be private, no worries. So I assume no friend ask any question. We got no questions. All right. Thank you, sir. Then we can move on next one. All right, guys. Next one is number five from 2003. Number 18 of that year. Then I will start timer. Let's see how many seconds it's going to take for someone to give us some answer or some hint at least. All right, it has been one minute, I guess there is no answer yet. so if there is no answer i'm gonna start we got one for me okay we get one person says b okay i am asking to that person buddy how do you get b please share some of the steps with us explain a little bit you know briefly all right so i am asking our friend who says b if that person give us some sort of explanation three equations okay bottle plus glass equals pitcher okay we get we get b plus g is equal p okay what else bottle equals glass plus mug okay what else three mugs equals two pitchers all right guys so combined capacity how many guys combined have the capacity of one month they ask us to find or compare so guys here if you look at the second equation b is equal to g plus m can i just plug it here in the first equation guys what do you think g plus m plus another g gives us p as you see guys all right you can say sir why are you doing that think about that our goal is kind of get rid of some extra letters or variables and focus especially glass and mug as you see we have glass and mug here but right now we need to get rid of p like those pictures well as you see the equation the last equation gives us 3 m 3 mark is equal to 3 2 p if we divide both sides by 2 to get p can i say 3 over 2 m is equal to p so can i just literally plug it in here that expression goes here guys so we are going to have 2g as you see it here and here 2g plus m is equal to you see instead of p i just plug 3 over 2 m 2 m remember i told you we are going to focus only g and m glass and mug because we gotta compare those well we have 1 m here and 1.5 m here 3 over 2 represents 1.5 so we get 2g is equal to 1 over 2 m well if you multiply both sides by 2 to be able to cancel that division of 2 guys we get 4g is equal to m as they said how many glass combine have the capacity of one mark yeah it says 4g equals m 1m that means we have answer is 4 guys any question for that guys remember here i plug the second equation into the first one look i could say well b is also equal to you know p minus g i can just plug here p minus g you would do that but if you keep working with the b values here in second equation or whatever you might not get the answer quickly because remember they give us a hint in the last sentences you will only compare glass and mug you would always work with g and m okay then you will go from there i assume there is no question so i'm gonna move to next one all right all right guys so we have number six please check number six as you see this is the hardest question they asked in 1998 number 30 literally the last question on the exam If you are here, either your camera is off or you have technical problems, okay, you are here, thanks. So do we have someone who would like to answer for number six or... I haven't received anything yet. Okay, we can get some answers, please share that with me. We got one for E. Okay, one person says E. Okay, what about others? No one says A, B, C or D? That's tough, man. Come on. Give me something. I hear five students so far, so they should give me something. We got no other answer. And I'm about to cry. Okay, so look guys. So for those cases, they say find the sum of x plus y if x and y fulfill the conditions here. Well, as you see, we have sum of the two square expressions and the result is zero. We used to get that guys. Think about like almost every single case the number squared is positive number, you know. Well, but if you think let's say the first part is positive, then second part has to be negative to be able to get zero or vice versa. If first part is negative, second part is positive, still we get zero, you know, additive inverse. Like 5, negative 5, negative 10, 10, whatever. But we know that we can never ever get negative value as a square of a number. As you remember, I hope you remember that there are another condition, which some of the cases a number square might be zero guys, okay? So that gives us guys x minus y minus 1 equal to 0. Also oops also we get x plus 7. No, not 7, x plus y. I will be able to find it. Plus y minus 7 is equal to 0 guys. So from the first expression, if you take out 1, we get x minus y is equal to 1. But we are not looking for x minus y as you see. We gotta find x plus y. x plus y when you add 7 to both sides, x plus y equals 7. And that is going to be answer. As you see guys, when you look at the answer choice, they both include 1 and 7 because they want to trick us. They want someone to choose one order and get it wrong. Because it's last question and you already got tired anyway. But x plus y was 7 guys. This is the answer. Please do not worry about this part, okay? Any question for number 6? So I believe no one says anything that I am gonna move on No questions, let's move on excellent. Thank you, sir. Okay number seven guys Check number seven please she writes a positive integer on each edge of a square Try each vertex the product of the numbers The sum of the numbers on the vertices 15 was the sum of the numbers on the edges I try to underline the key part just for you guys We got one person who says E 15. Oh interesting. Okay, anyone else? Not yet Let's just wait a little bit more Ok, I am asking to that person. That person, why were you sent 15? Can you tell us what would be the first step here? How do you get 15? What first step you follow to get 15? Well, so I wonder if that first person, whoever gave us 15, tells something about that or not. So they say, write x that goes up by 2 because it is even. I don't understand that. I'm sorry, I don't either. Okay, Sanya writes a puzzle in each edge of a square. She also writes each vertex. So vertices are those corners, guys. Each edge, basically each side, let's just say, you know, A, B, C, and D, guys. Okay, look, I am only giving information and creating the chart. That's it. And when you multiply those numbers, A times B here, A, B, as you see, we have B, C here, we have C, D, and A, D here. And they said the sum of the numbers at the vertices, you know, those A, B, B, C, A, D, whatever, is 15. What is the sum of the numbers on edges of the square? Well, guys, here is the deal. Let's just add those terms, like A, B, plus B, C, plus C, D, plus A, D. Actually, you know what? If I just make this one A, D, it's going to make our job easy. C, D, and B, C. Well, when you add all of them, you get 15, as you see. This is what they say. Then you have A, B, and A, D, guys. I can work accordingly and factorize those. I get A, and open parentheses, we get B plus D left over. I hope you remember that rule. When you see we have C and C, these are the same terms. You can get C, open the parentheses, D plus B. Remember, factoring is kind of opposite of distribution, so A times B, A, B. A times D, A, D here, you know. Or C times D, C, D, C times B, C, D, or B, C. The result is 15. I want you to look at those parentheses. D plus D, D plus B, same thing. At the end, we have A plus C left over, times B plus D. Is equal to 15. So guys, after that, as you see, we have A plus C, times B plus D. Basically, two numbers you multiply to get 15. What two numbers you would choose to get result 15? What do you think? Look, if you think about 15, guys, we don't have too much of pairs. We have 1 times 15, or 3 times 5. But what is the answer of that? Are we going to use 1 times 15 as a factor pair, or we should use 3 times 5, and why? Okay, we got someone who says 8 as the answer. Can you please explain why? We are still waiting for that friend. Meanwhile, I'm going to fix my ugly handwriting. Where do we get something so more I'm gonna keep going buddy I didn't get anything the chat that's all right guys we have two answers so far 15 and 8 look guys we can either choose 1 times 15 here or 3 times 5 here but think about this if you choose 1 and 15 we are assuming a plus C is 1 we already know we have positive integers on each edge if a and B are positive integers there is no way we can get a plus C is 1 guys you know they are all integers it should at least 2 even they both are the same number in that case kudos I cannot choose 1 and 15 unfortunately we have to choose 3 and 5 as a factor pairs of 15 yet we will assume one of them is 3 another one is 5 it doesn't matter 3 times 5 give us 15 and they see what's some of the numbers on the edges basically what's the a plus B plus C plus D they are asking us what's 3 plus 5 we get 8 one of the friends find 15 maybe that friend thinks we got them out by 3 and 5 to get 15 I have no idea what the answer was 8 guys answer was C whoever give us C I am proud of you guys thanks so much all right any question about this part it's all the time we just missed 4 minutes already sorry okay any question for number 7 if you have something ask me right now what I'm gonna end the session you so I will do we get anything buddy on the chat or now no all right guys I hope you enjoy the session it's end of our another weekly session and I'm gonna not see you but hear from you next week okay thanks bye take care

algebraic equations

problem-solving strategies

logical problem-solving

algebraic expressions

variables

PEMDAS

interactive participation

multiple solutions

algebraic techniques

critical thinking