This eighth grade mathematics lesson focuses on using the elimination method to solve simultaneous equations. It is the sixth session in a sequence of fifteen lessons working on solving simultaneous equations with two unknowns. The lesson is taught in Cantonese and is 42 minutes in duration. There are 40 students in the class.

00:00:05Stand up.
00:00:18Good afternoon, class.
00:00:25While I am taking roll, put away all the things you are passing around as soon as possible.
00:00:39Lo Pak Man? Chan Ka Cheong, Lo Pak Man, report to Room 201 on the twenty-first of this month, Wednesday, at 7:30 a.m.
00:00:48Otherwise the two of you will have disciplinary punishment. Is Chan Ka Cheong here?
00:00:53Pass the paper to him to see.
00:00:59Yeung Wan Chi.
00:01:00Is absent.
00:01:01Is absent? Who else?
00:01:04We have only one absentee?
00:01:13Can any of you turn in the new book today?
00:01:17Yes you hear.
00:01:18If yes, you can... how many are there?
00:01:21Next time I'll tell oh!
00:01:23Only the two of you?
00:01:24Two. Okay, just bring it in tomorrow. Just the two of you...
00:01:30Why are you so late?
00:01:34I was washing the glasses.
00:01:38Are you ready? Open your books.
00:01:48Before we start on a new topic today, we will take a look at what we have done yesterday.
00:02:00Yesterday, we should have stopped on 22, right?
00:02:05How can it be number 24?
00:02:08There is a star next to number 23. I did not tell you to do it.
00:02:13First of all, are there any special questions?
00:02:17Are there?
00:02:19Raise your hand if you have questions.
00:02:21If not, I will ask some of you to come up and try some of the questions.
00:02:28Are there any questions on question number 20 with the star?
00:02:30There is nothing special about it, right? It did not look too special.
00:02:33[ Bell ]
00:02:34Uh, let's do number 20 then.
00:02:40Twenty-one and 22 are both very easy, right?
00:02:54Twenty... how about doing number 19 and 20?
00:03:02Fung Cheuk Yue, do number 19.
00:03:22Cheung Lung Man, do number 20.
00:03:42Our class is being taped today, therefore, the light is turned off to avoid reflections. You may have a hard time looking at the blackboard.
00:04:26Are you in a hurry?
00:04:29Which is the last question you have finished?
00:04:30This one.
00:04:31Which question did I tell you to do?
00:04:41I am thinking.
00:04:42Don't, don't tell me that you don't know how to do them.
00:04:44Okay, I am thinking.
00:04:46Don't do them when I am teaching later, okay?
00:04:49Stay here when class is over.
00:04:51Did you hear me? Stay after class. Finish them for me when you stay after class.
00:05:02I don't know how to do number 20. Can you teach me?
00:05:08How to do number 20? There is nothing special about this question.
00:05:13Um... that will work too, but you will have an extra sign. Don't you think it will make it look terrible with that extra sign?
00:05:25Alternatively, you don't have to take the two T as the subject.
00:05:28You can take the T as the subject. Take the S as the subject. Then, your negative two will become a positive two when you move it to that side.
00:05:33Then 28 plus two T. It equals to three S. Divide the three from that side. That will look better.
00:05:39It will look better if you divide three from that side. There will be no signs.
00:05:43Then, substitute it to the formula number one above. Yep, don't move the S, move the T instead.
00:05:50When you move the T to here, it will become positive. Yep. Divided by three. Erase this sentence and do it once again.
00:07:16Number 19 should be okay. Is the answer correct?
00:07:19I think it is okay.
00:07:23Number 20.
00:07:36The things that I taught yesterday were quite good. All of you seem to be getting it.
00:07:42The whole equation is multiplied by three. Be careful when you are multiplying and do not miss any terms, or else, it will be wrong. Three times three...
00:08:09Is the answer correct? No problems, right?
00:08:15Do you want to do any of the problems? Or do you want me to talk about any of them?
00:08:21Any questions?
00:08:22If not, I will- this method- this will be it for the method of substitution for now. We will start to learn another method. Let's turn to the next page. Who will clean up the blackboard?
00:08:41Turn to the next page. Eleven point three. Method of elimination [ in English ].
00:09:22Elimination in Chinese is elimination. [ in Chinese ]
00:09:29Elimination. Elimination. To eliminate something.
00:09:36Well, um... I will not talk about example number four in the book right now. I will talk about another one that is simpler, one that I have mentioned before. I have mentioned it before.
00:09:46Um, do you remember I mentioned before- in the last two classes- that you should have learned it- seen it in primary school.
00:09:53Um, it involves a big and a small number and you don't know what the numbers are.
00:09:56The sum of the big number is, for example, 20. No, the sum of the big and small numbers is 20.
00:10:04Um, the difference between the big and small numbers is, for example, eight.
00:10:08Then, you are asked to find out the big and the small numbers. Do you remember?
00:10:12At that time I said that you, um, after learning this chapter,
00:10:15you will understand why your primary school teacher told you to find the big number by adding the sum to the difference and divide it by two.
00:10:21To find the small number, you subtract the difference from the sum and divide it by two. The main point is that these are related to what I will be teaching right now.
00:10:27Okay, I will be using this example to teach. This is not included in the examples of your textbook. However, this is simpler.
00:10:33Suppose I have, um, a sample question here. You can copy it down in the book, um, in your notebook or textbook.
00:10:40Solve A plus B is equal to 20. A minus B is equal to eight. Like that.
00:10:48Well, now there are two numbers, A and B. The sum of these two numbers is 20 and their difference is eight. Is that right? It is obvious for you to see which is bigger and smaller.
00:11:00Can you see it? Which one is bigger and which one is smaller?
00:11:02Uh, Ho Wai Kwan?
00:11:06A is bigger because the second equation- in the lower equation, A minus B is equal to eight. Therefore, A should be bigger than B.
00:11:10Okay, I named the first equation as one and the second as two. Give them names. Well, we are not going to use the earlier method- the method of substitution right now.
00:11:22The method of elimination is- um, like it is named, is to decrease the number of unknowns- to decrease the number of unknowns- to eliminate the unknowns.
00:11:33In fact, this is closely related to the previous method, um, the main point that I was talking about earlier.
00:11:38Didn't I mention before that, um, when there are two unknowns in a question, it is impossible to calculate it, right?
00:11:44Well, do you remember from the previous- the previous method of substitution, you were trying to simplify two equations with two unknowns into one equation with one unknown.
00:11:56Is that right?
00:11:57Well, you can do it. In fact, the present method is using the same technique, too.
00:12:03It is also trying to create an equation with only one unknown. Okay? But the question is, how do we do that?
00:12:11The problem is how we put the two equations together. Okay, now, here is equation one and equation two. Well, you- you can see that here is two A.
00:12:22Positive A here and here. Positive B here and negative B here. Right?
00:12:27Okay, if now I want to find out A but B is blocking the way, right? So, what should I take away?
00:12:41The best is to take B away, right? Take away B and the remaining A will be easier to calculate, right?
00:12:45The problem is how do we take A away? No, sorry, how do we take away B?
00:12:48How to take away B? Okay, let's see. B is a plus sign here and a minus sign here. We have to pay attention to the plus and minus. Let me write them in color.
00:13:00Okay, it's plus and minus. A plus and A minus. In this case, let's think that there are a positive B and a negative B. If you have to eliminate it...
00:13:09If you have to eliminate it, you will add the two numbers together to eliminate it. Why? What is the sum of three plus negative three?
00:13:18Kam Mo Lam?
00:13:25Yes, zero, right? Right. Four minus- four plus negative four is equal to zero, right? So, now, positive B plus negative B is equal to what?
00:13:33Should it be zero? If it is zero then there is no B, right? Therefore, I will add these two equations together.
00:13:39Okay, how do we add these two equations together? Adding two numbers is possible, but how do we add two equations together? We add equation one into equation two.
00:13:49How do we add it? Here are two equations. There is an equal sign- equal sign in each equation. Add the left side of the equal sign to the left side of the other equal sign.
00:13:56Add the right side of the equal sign to the right side of the other equal sign. Okay? So, 20 plus what?
00:14:03Twenty plus eight, right? Well, on this side, 20 plus eight is equal to 28. Okay? Okay, here, this one is added into this one. What is equal to A plus A?
00:14:12Two A.
00:14:13A plus A is?
00:14:14Two A.
00:14:15Two A, right? Well, but- I said earlier that positive B plus negative B is how many?
00:14:20Siu Hiu Ching.
00:14:22Stand up. You should be accustomed to it.
00:14:27Right. Positive B plus negative B equals to zero, right? Do we need to write zero out?
00:14:29No. If you want to write it out, write positive B and negative B out. It should be understood, right?
00:14:33Well, you can see in this situation that what is left after the summation is?
00:14:37Two A equals to 28.
00:14:39What is the remaining unknown?
00:14:42A, right? Is there a B? No, it's gone. In this process we have eliminated the B. Do you understand?
00:14:50This process has eliminated the B. That is, to eliminate B, okay? This method is called elimination. Okay?
00:15:04Well, now that, um, B is eliminated, we can immediately find out A. A is...
00:15:10Fourteen, right? Okay?
00:15:11So, you will realize the reason why when we are looking for the bigger number- A is the bigger number, right? We have said earlier that it is the bigger number.
00:15:18When it is the bigger number, we need to add the sum to the difference- this is the sum and this is the difference- the sum added to the difference, is this right? What is the next step?
00:15:26The sum added to the difference and then... divided by two, right? Can you see that? That will turn out like this. Can you see that? Get it yet?
00:15:33Okay, next, how do we get the smaller number?
00:15:40We have already found one of the unknowns.
00:15:41Substitute it back into the equation.
00:15:42Right, substitute it back into the equation, right? So, fourteen will be substituted back into the equation. It is up to you to substitute it into either equation.
00:15:48Put A equals 14 into this, um, either one of the equations. Um, let's say equation two. Well, 14 minus B is equal to eight. B is equal to...
00:16:11Any questions?
00:16:45Here I used- I added- I used summation. The B is eliminated when I used summation. Okay?
00:16:52You may ask- you may say what if I want to eliminate the A? What should I do? What should I do to eliminate A?
00:16:59Instead of eliminating B, I want to eliminate A first. Then what should I do?
00:17:02Maybe I should teach you by using example number four in the textbook. Okay?
00:17:10The student on duty, please clean up the blackboard.
00:17:22Erase it. Erase it.
00:17:42Example number four in the textbook.
00:17:43Can I go wash my hands?
00:17:44Go. Go.
00:18:05I have taken role already.
00:18:14Now the equation this time is similar to the one I have previously stated. One of them is A plus B and the other one is A minus B, right?
00:18:20Now, it is three X plus two Y and three X minus two Y, right?
00:18:24Just now, I eliminated the B- to eliminate B. So, now... if I eliminate the two Y- is there a plus and minus before Y? Is that right?
00:18:37So, I should- based on the method I have mentioned before, I should add the two equations together.
00:18:43When they are added together, positive Y and negative Y will become...? Will become zero, right?
00:18:49Okay? Well, if- I said earlier that if I don't want to cancel this one- to eliminate this one, but rather, I want to eliminate this one- this one written in yellow- I want to eliminate this.
00:18:58What should I do?
00:19:02This time, I don't want to find out X first. I want to find out Y first. If I want to solve Y, I will need to eliminate X. I will need to...
00:19:11Come in.
00:19:12I need to eliminate X first. So, in order to catch your attention, I have purposely written it in yellow.
00:19:18Well, how do we do the elimination? Pay attention. In the previous question, one is plus and one minus. In that case we add them together.
00:19:25However, are this three X and this three X positive? Okay, what do you do when you cancel the X?
00:19:34Three X and three X. How do we do it?
00:19:36Move three X...
00:19:37Move three X to the other side.
00:19:38Cheng Ka Ho, what do you think we should do to eliminate X? Do you- do you have any ideas?
00:19:45Three X minus three X.
00:19:46Three X minus three X will be zero. Subtraction. So we- that is correct. It will work out by subtracting the two equations from each other.
00:19:58Now, when you are doing the subtraction- this is easy- 19 minus 11 is equal to what? Eight, right? Right?
00:20:04Well, what about here? Three X minus three X will be nothing- it will become zero, right?
00:20:09Now, pay attention. Will it be zero if this is subtracted from this? This one minus this one. Will it become zero?
00:20:14No, this one is plus and this one is minus. The two are different. One is plus two Y and the other is minus two Y.
00:20:19So, if you really subtract them from each other, it will become two Y minus negative two Y, right? You should take- take the whole positive two Y minus two negative Y, right?
00:20:28You will find out that after the subtraction is- here, negative times negative is positive- it becomes four Y is equal to eight. Therefore, Y is equal to... two. Okay?
00:20:37So, pay attention to this. This will be mistaken easily- you will get this wrong easily. Therefore, put Y equals two into... either one of the equations, um, let's say... equation one.
00:20:52So, three X plus two, then times two equals 19. Three X is equal to- move four to the other side is equal to 15. X is equal to five. Therefore, the solution...
00:21:10X equals five. Y equals two. Okay?
00:21:17Any questions?
00:21:21Any questions?
00:21:26So, you can see that- in concluding the two methods- two- two, um, methods- these two examples that I have done. At the first time, I add the two equations together.
00:21:35At the second time, I subtract the two equations. It all depends on which unknown we want to eliminate. Okay? Let's say we want to eliminate Y- in fact, I can eliminate Y now.
00:21:47So we can add them together since one is positive and the other is negative.
00:21:51If both of them are positive numbers and we want to eliminate the X. In that case, we use the two- the two equations by subtracting each other. Okay?
00:22:03In fact, you can control it yourself- which one you want to calculate- which one to calculate first. Okay? Any questions? Well, it is now your chance to try it out.
00:22:13Open your class workbook. Turn to...
00:22:21Exercise 11 B. Page 119. Well, try out number one to four.
00:22:27[ Bell ]
00:22:28One to four first.
00:22:32Check the answers when you are done.
00:23:57You have to circle- to circle equation one and equation two or else it will be confused as numbers one and two, right?
00:24:22Chan Lap Man, come up and do question number two.
00:24:26Did you hear?
00:24:27Number two.
00:24:28After you have finished.
00:24:59You haven't given it a name.
00:25:00Oh, yeah, that's right.
00:26:34Chan Lap Man, hurry up.
00:26:56Question number three (inaudible).
00:26:57Come up and do question number three later.
00:27:05(inaudible) X is (inaudible).
00:27:06Name equation one and equation two first.
00:27:15You have to tell me which equation minus that equation first. How do you work this out? Two X is equal to 12.
00:27:21Addition. In that case, write one plus two. Equation one plus equation two. Right. Colon.
00:27:29Next, um, it is equal to six. Substitute X is equal to six back into the equation.
00:27:33This X is equal to six, so... it becomes six minus Y is 10...
00:27:42So, you move six to here and why can't you subtract them? Y can be a negative number, right? Can Y be negative four?
00:27:49Um, that is right.
00:27:53Can I erase this?
00:27:55Erase it. Erase it. Erase it.
00:27:58Erase the ones I wrote on the top too.
00:28:03Erase the ones I wrote on the top too.
00:28:05Erase example four too.
00:28:55Chan Lap Man, it is a colon after your equation one minus equation two- it's a colon, don't write it like an equal sign.
00:29:03Conversely, your equal sign is like a colon. Write the equal sign in a better way.
00:29:37Question number two is correct. Pretty well done. However there are some places that are not well written- written- written. Not clearly written.
00:29:44What is that thing down there? Is that a one- one under the (inaudible)?
00:29:48X equals to one.
00:29:49No. Here. Why is it- here- here, what is this?
00:29:54You did not write that?
00:30:04Ten A equals 30.
00:30:22Any questions?
00:30:26Are all four questions completed?
00:30:34Check your answers with the ones on the blackboard. Both of them are correct. Pretty good.
00:31:11Here. Of course you cannot find the answer by adding the two equations together. What do you expect to be eliminated by adding the two equations together? X plus X is two X.
00:31:20Six Y plus two Y is eight Y. You cannot cancel them. With six Y and two Y, no matter whether you use addition or subtraction, you cannot cancel it.
00:31:27Look, six Y plus two Y is eight Y and subtraction will leave four Y. The Y will still be there.
00:31:34On the other hand, there are two X. Both X are positive.
00:31:37If you use subtraction, the X will be cancelled. Therefore, you should never use addition- addition to calculate this question.
00:31:44If you use addition, nothing will be cancelled- nothing will be eliminated. If and only if you use subtraction, then you will make it.
00:31:51Right? Do not mindlessly use addition all the time.
00:31:53You have to see if anything will be cancelled, right? Here, it is for sure that only X will be cancelled, right?
00:32:09Finished? Raise your hand if you have not finished. Raise your hand if you have not finished.
00:32:18Raise hand.
00:32:29Take a look at the example if you have finished. Example five.
00:32:51Student on duty, clean up the blackboard.
00:32:56No, the upper one times the lower one.
00:32:59Are there any questions?
00:33:00The one on the bottom...
00:33:02Are there any questions?
00:33:14Finished yet?
00:33:20Why are you so slow?
00:33:30Hey, why are you lying on the table?
00:34:13I will not wait for you due to the time constraint. Stop for a moment. Calculating number four- for those of you who have not finished, are you doing number four?
00:34:22I am doing number five.
00:34:23Stop for a moment. Please take a look at example five. Page 117. Page 117.
00:34:32Now, this equation- this question is slightly different from the previous ones.
00:34:38It is because, you will find out that when you add the two equations together. Two X plus X is three X. It cannot be eliminate- eliminated- cannot eliminate the X.
00:34:51This one plus this one will leave a Y here. Three Y plus negative two Y, right? The biggest question is that this is a three but that is not a three, right.
00:35:00This is a two but that is not a two.
00:35:01This is a two but that is a one, right. If you subtract the two equations, the unknown will not be cancelled.
00:35:10When this is the case- which is quite common- of course, you can ask me whether you can use the previous method of substitution. Of course you can.
00:35:19However, we are learning the method of elimination now. So, if we have to use the method of elimination in this case, how do we do it?
00:35:26Here, the biggest question is that this is two, this is one, this is three and this is two. They are all different, right?
00:35:33If they were the same number, we could find some ways to use addition or subtraction to eliminate, is that right? Is there any way to make them the same?
00:35:40I said it yesterday. What I said yesterday is mainly for today's lesson.
00:35:44What we learned yesterday can be- of course- used to solve mathematical problems, but today we can use it too.
00:35:50If the numbers are not the same, we can think of some ways to make them bigger, right? For example, if I want to eliminate the X but this is two and this is one.
00:36:00If I turn the one into two, then, both of them will be two, right? In another words, if I turn this into two, we have multiplied it by two. Therefore, the rest of the equation will be...?
00:36:09Multiplied by two.
00:36:10Multiplied by two. Okay? So, I need to solve equation number two first. Multiplied it by two first. Okay?
00:36:18Equation two times two. After multiplied by two, the first part will be two X and what about the second part? Chan Kwan Wu, after multiplied by two- after multiplied by two...?
00:36:29The second part... uh, positive four Y.
00:36:34Four Y and then?
00:36:37Twenty-four, right. Okay? Now, we have formed a new equation, right? Let me give it a name. Equation number three. Okay?
00:36:43So, now, I will not look at equation number two anymore.
00:36:45Since it is turned into equation number three, okay? So this is two X and that is two X too. What do we do next?
00:36:51Lo Pak Man. Should we use addition or subtraction for the two equations?
00:36:59Addition or subtraction?
00:37:03Positive two X and positive two X.
00:37:05If you want elimination, shouldn't we use subtraction? Use subtraction, okay? Two X minus two X is...?
00:37:06[ Bell ]
00:37:16It became zero. Two X minus two X is zero, okay? If you add them together, what will it be? Two X plus two X is what?
00:37:21Four X. In that case, the X will not be cancelled. Understand? Since both of them are positive, you will need to use subtraction. Sit down.
00:37:27So, we use this- equation one minus equation three. We will no longer use equation two.
00:37:33Equation one minus equation three. This one minus this one. After that, two X minus two X will be zero.
00:37:40So, three Y- positive three Y and this is negative four Y. Subtraction. Three Y minus negative four Y, okay? This is negative four Y and that is positive three Y.
00:37:53Equals to three minus 24. Three minus 24 looks really strange. It is a negative number, but it is okay to have a negative number. Three minus 24.
00:38:01So, let it be three minus 24. Well, next, this will become seven Y, right? How about this one?
00:38:09How many?
00:38:10Negative 21.
00:38:11Negative 21.
00:38:12Negative 21, right? So, Y is equal to...
00:38:13Negative three.
00:38:14Negative three, okay? Next, put Y equals negative three into any one of the equations above.
00:38:20I will not suggest you putting it into equation three, because it had been multiplied.
00:38:24It is more complicated. So, I will put it into equation one or equation two. Put it into equation two, okay?
00:38:29So, X minus two times negative three equals 12, okay? So, X is- this is six and move it to the other side, it becomes six. Okay?
00:38:45Negative two times negative three equals positive six. Move the six to the other side, it becomes negative six. Six.
00:38:49So, the solution...
00:38:56Y equals... what is Y equal to? Negative three, right. Okay? Any questions? Now, the most important point is- I think- this step here.
00:39:08How do we use the method of elimination? In the process of elimination- before that, we may need to multiply some, um equation.
00:39:17By doing that, we will make both numbers of the equations the same.
00:39:20This one multiplied by two will become two and that one is two. So, they can be subtracted by each other, okay?
00:39:24If you say, if I want to eliminate the Y- earlier I said I want to eliminate the X but if I want to eliminate the Y, one is three and the other is two, what should I do?
00:39:34Chong Fong Wai. If I want to eliminate the Y, I will need to make them the same. So how do we make three and two the same?
00:39:43Find out their LCM.
00:39:44Find out their LCM. The LCM of three and two is...?
00:39:48Six. So, three times what is six?
00:39:51Times two. If three time two, then, the whole equation- equation one will need to be multiplied by two. Okay? How about this one? Two times what?
00:39:58Two times three. It is six again. Okay? So, we need to multiply the lower equation by three, okay? Understand?
00:40:02If you want to eliminate the Y, you will need to do more things.
00:40:06If you multiply the upper equation by two and the lower equation by three, that will create equation three and equation four. Equation three and equation four.
00:40:12Then, we can add the two equations together. One is positive six Y and one is negative six Y, okay? Understand?
00:40:19Try it out at home. Since we only have one class today, (inaudible).
00:40:24Well, um, go back and try out number five to eight. Okay? Go back and do number five to eight. Okay? That is not too many.
00:40:35It should be the day after tomorrow- do we have class on the day after tomorrow? Day six. We have class on day six, right?
00:40:41Okay, we will talk about it more on day six, because we will have a test tomorrow. Do you remember?
00:40:46Test. The chapter on slopes.
00:40:47Day five, right? Distance formula. The chapter on slopes. Any impressions?
00:40:55This equation.
00:40:58Parallel. Straight. I have just talked about it.
00:41:02There is also the distance formula. A B equals- the distance between two coordinates.
00:41:11Tomorrow- tomorrow, you will be tested on chapter eight. It seems like all of you have amnesia.
00:41:21Tomorrow- um, are there any single lined papers?
00:41:31Who is in charge of buying the papers?
00:41:35It is you? Chan Lap Man? Help me buy one more pile of F four- long paper, single lined, white colored, tomorrow. Long paper.
00:41:44Is the test going to be difficult?
00:41:45Not difficult. Study for the test. Stand up.
00:41:48Do we need to bring our books?
00:41:50You don't need to bring books tomorrow, okay? However, do you remember you have to bring two new books tomorrow?
00:41:54I remember.
00:42:02Goodbye class.
00:42:12Sit down and be quiet. The student on duty, come up and clean up the blackboard.