Simultaneous equations

追加: 2026-05-10

[00:00:00]Uh we'll be basically looking at simultaneous equation solving. In our last class we looked at simultaneous equation by substitution method. But today we are going to look at simultaneous equation by elimination method. To solve simultaneous equation by elimination method, it is very very simple. Okay. Please let us know where you're watching from. Let us know where you're watching from. So to solve simultaneous equation by elimination method, it is simple. First thing you have to do is to check if the coefficients of x or y are the same. So for instance, if I have something like uh 2x - y is 6 and I have again to be 2x + 4 y is 1. This is a simple example of a simultaneous equation. You know, simultaneous equation involves solving multiple equations uh together at once. Okay, where the values of the unknown satisfy both equations. So first thing I have to do here is to assign this to be equation one and assign this to be my equation two. So on getting this I will now check the coefficients. If I look here, the coefficient of x here is 2 and the coefficient of x here is two. Why? The coefficient of y here is minus1 and the coefficient of y here is four. So the easiest way to eliminate is to use the one that has the same coefficient. So that if I subtract them, I can now have what? Zero. Hence, it is eliminated. It is out. So what do I do? I will subtract this.

[00:01:54]So if I subtract equation one from equation 2, I'll now have 2x - x is 0.

[00:02:02]So hence x is what? eliminated.

[00:02:06]Then I have it to be - y. You get - y - + 4 - +. So we read it as - y - - 4 y would be what? - y - 4 y is what?

[00:02:24]Write it in the comments. - y - 4 y is what? - y - 4 y is what? What is - y - 4 y is what? So - y - 4 y is basically - 5 y. Okay. And that is equal to 6 - 1 is 5. So we have it as what? Five. So for us to get in or for us to get the value of y we are going to put what? We are going to divide through by -5. So if you divide here by -5 and you also divide here by -5. So you see that -5 cancels out -5. We see that our y is equals to write in the comments what's the value of 5 / -5.

[00:03:20]What's the value of 5 / -5? Fastest finger please. 5 / -5. What's the value for 5 / -5? Correct. -1. Thank you. So we have it as what? -1. So we have that y is -1.

[00:03:42]Then for us to get the value of x it is quite simple. Okay. For us to get the value of x it is very very simple. All we have to do is to put in the value of y as minus1 in any of the equation. So for me for it to be easier I'll put y to be -1 in equation one. So I have equation 1 to be 2x - y = 6. So we have to anywhere we see y we replace it with -1. That is what we mean. So what we have it as what? 2x - okay I put my -1 and that's equals to what? Six.

[00:04:31]So what's the value of write in the comments what's the value of of -1?

[00:04:38]What's the value of Thank you. What's the value of -1 - What's the value of -1 - -1? What was the value?

[00:04:49]That will be a plus. So we have it at 2x + 6 + 1 sorry is equ= to 6. Okay. So the next thing we do is that we take away this one from both sides minus one. If you subtract one from both sides, you see that 2x is equal to 6 - 1 is what?

[00:05:13]What's the value for 6 - 1? Please let's make it interactive. What's the value of 6 - 1? Who is following? What's the value of 6 -1?

[00:05:25]What's the value of 6 -1?

[00:05:30]6 - 1 is 5. So we have that 2x = 5. To get the value of x, you just have to divide here by two and divide here by two. So if you divide by two, you see that x is = 5 / 2 is what? What's the value for 5 / 2?

[00:05:53]What's the value for 5 / 2? 5 / 2 is what? Yeah, I'm live. Thank you for joining. What's 5 / two?

[00:06:04]What's the value for 2.5? Correct. Thank you for the support.

[00:06:09]So, you see that x is 2.5? Very very easy. Okay. So, to solve all this problem using this method, it is very very very very easy. Okay. So the next one we have to look at is another problem we have here. So here we are given that uh let's try this. We have that x - y is 8 and again we have 4x + y is 42.

[00:06:47]x - y is 8 and 4x - y is 42. Remember for us to solve this we are actually trying to find the value of x that can satisfy the >> equation. Okay, please help us by liking the life. Okay. So here first thing I have to do is to assign this to be my equation one and assign this to be my equation two. So if I have these two equations first thing I have to look at is the coefficient of the variable. For the first equation x is one, y is minus1. For the second equation x coefficient is four and y coefficient is + one. So here what I have to do is that if I look well at the coefficients of y they are almost the same. It's just that this is minus one and this is one. So to get rid of this I can simply eliminate the y. So to eliminate the y I will add the two equation because you know that 1 + - 1 + 1 is giving you zero. So y is eliminated. So, what I have to have now is x + 4x. What is the value for x + 4x?

[00:08:10]What's the value for x + 4x? Fastest finger, please. What's the value for x + 4x?

[00:08:18]What's the value for x + 4x? Please help us.

[00:08:23]What's the value for x + 4x? It is 5x.

[00:08:28]Thank you. So, we have it as what? 5x then - y + y will give us what? What's the value for - y + y? What's the value for - y + y?

[00:08:43]What's the value? It is what? Z. Thank you. - y + y is 0. So have it as zero.

[00:08:51]And that's equal to what's 8 + 42?

[00:08:55]What's the value for 8 + 42?

[00:08:59]What's the value for 8 + 42?

[00:09:03]What's the value for 8 + 42? 8 + 42 is 50.

[00:09:08]So for us to get the value of x, we now divide the left hand side by the coefficient of x which is five. And we also have to divide the right hand side by the coefficient of x which is five.

[00:09:24]Are you following? So if you divide five cancels out five. So I have that x is equals to 50 / 5. What is the value for 50 / 5? What is the value for 50 / 5?

[00:09:41]Are you following? 10. Thank you. 50 / 5 is 10.

[00:09:47]So I have it as 10.

[00:09:50]Okay. So for us to get the value of y, we are going to put x to be 10 in any of the equation. For it to be faster, I always like putting the value of x to be 10 in the equation that has little coefficients. So I'll go with equation one. So the equation one is x - y = 8.

[00:10:14]So for us to get the value anywhere we see x in this equation, we put it to be 10. So if you have here to be 10 - y = 8 then for us to get the value all we do is to take away 10 here and take away 10 here. So we have it that this is zero.

[00:10:37]Now - y is equals to what is the value of 8 - 10? What is the value of 8 - 10?

[00:10:46]What is the value of 8 - 10?

[00:10:50]What is the value of 8 - 10? 8 - 10 is -2. So we have it as what? - 2. So to get the value of y, we divide by a -1 and a1. So if I divide y will now be a positive and that's equals to what's the value of -2 / -1?

[00:11:15]What's the value of -2 / -1?

[00:11:20]What's the value of -2 / -1? Fastest finger please. It becomes what? A 2. So y is = 2. So simply say that y is x is 10, y is 2. So this is the simple way to solve a system of equation by elimination method. This is very very what? Easy.

[00:11:46]So let's take another sample please.

[00:11:50]Uh let's take another sample and do uh that's how we actually solve the equation. But do you know that you can actually solve simultaneous equation when the they are coefficients are not actually matching. Okay, when they have variable coefficient that you cannot just simply eliminate. Okay, let's look at this before we go to that type.

[00:12:19]x + 2 y is 8 and x - 3 y is 6.

[00:12:28]x + 2 y is 8 and x - 3 y is 6. Let's try to find the value of x that can solve this. You can be solving it in your book so that we can know the actual answer we got. So to do that first thing we have to do here is simple we assign this to be equation one and assign this to be our equation two. So if you have this thing this system now it is very very easy to solve. First thing you see is that for equation one the coefficient of x here is one and coefficient of y here is + two. For equation two, cofficient of x is still one and cofficient of y is minus2. So for us to solve this, the easiest way to do is to eliminate the x.

[00:13:18]So to eliminate the x, I have to subtract equation one from equation what? Two.

[00:13:26]Subtracting equation one from equation two is very very simple. So all we have to do to do such subtraction is very simple. We have it. X - X is what? Let's fastest finger. Let's see. X - X will give you what?

[00:13:45]X - X is going to give you what? Let's see that in the comments. X - X is going to give you zero. X - X is 0. So, X is eliminated.

[00:13:57]Then 2 Y - - 3 Y is what? - is plus. So we read it as 2 y + 3 y is what? What's the value of 2 y + 3 y? What's the value of 2 y + 3 y? What's the value of 2 y + 3 y? You are saying five. We say 5 y, not just five. So we have it as what? We have it as what? 5 y. And that's equals to what's the value of 8 - 6? What's the value of 8 - 6? What's the value of 8 - 6? What's the value of 8 - 6? 8 - 6 is 2. So we have it as what? 2. Then to get the value of y, we divide through by five and divide through by five. So when you divide through by five you see that y is = 2 / 5 is what? What is 2 / 5?

[00:15:06]Okay we just leave it as 2 over 5. So for us to get our value of x we have to plug y to be 2 over 5 in any of the equation. So I'll just use equation one which says that x + 2 y is = 8.

[00:15:28]So we have it that x + our y is 2 over 5. We have 2 into putting it 2 over 5 is equals to 8.

[00:15:39]So 2 * 2 over 5 is what? What's the value for 2 * 2 over 5?

[00:15:46]What's the value for 2 * 2 over 5?

[00:15:50]What's the value for 2 * 2 over 5?

[00:15:54]What's the value for 2 * 2 over 5? 2 * 2 over 5 will be x + 2 * 2 4 over 5 is = to 8. So what do we do? You see that x is now equal to 8. You have it at 4 5 crossing in equality sign it becomes a negative it becomes -4.

[00:16:22]So 8 - 4 5 is going to give you the x = 7 1 / 5. So x is 7 1 5 y is 4 over 5. If you have any question on this please let me know. If you have any problem on this, please let me know.

[00:16:45]Okay. So this is how we do that.

[00:16:51]X is 7 over 5.

[00:16:55]So uh let's look at this again. I have that 2x - y is 6 and 2x + 4 y is one.

[00:17:13]Look at this system of equation.

[00:17:16]Remember we are solving this system of equation by elimination method. So for us to solve this by elimination method all we have to do is to assign the variable. So this my equation one and this is my equation two. Okay. So since I have this let's study the coefficients. For equation one the coefficient of x is 2. The coefficient of y is minus1 and the coe and for the two x again is two y is four. So it's easier to get the one that has eliminate the one that has similar coefficients.

[00:17:55]So to do that we subtract.

[00:17:58]So let's go together.

[00:18:01]Help me. 2x - 2x is what? What's the value for 2x - 2x?

[00:18:08]What's the value for 2x - 2x?

[00:18:12]What's the value for 2x - 2x?

[00:18:16]2x - 2x is going to be zero.

[00:18:21]Then the next one is - y - 4 y. So - plus is minus. So we do - y - 4 y. So what's the value for - y - 4 y? Fastest finger, please. - y - 4 y. What's the value for - y - 4 y?

[00:18:46]What's the value for - y - 4 y? What's the value for - y - 4 y? Is what? is basically - y - 4 y is - 5 y and that's equals to what's the value for 6 -1?

[00:19:07]What is the value for 6 -1?

[00:19:11]What is the value for 6 - 1? What's the value for 6 - 1? What's the value for 6 -1? 6 - 1 is 5. So we write it as 5.

[00:19:30]Then the next thing is to divide by the coefficient - 5 / -5. So -5 cancels out -5. We have y to be equals to what's the value of 5 / -1.

[00:19:46]What's the value of 5 / -1?

[00:19:51]What is the value of 5 / -1?

[00:19:56]What's the value of 5 / -1?

[00:20:00]So 5 / -1 is going to min -5 sorry is going to give you -1. So y is -1.

[00:20:12]So remember we are looking for we are going to look for x again. So to look for x again we put y to the minus1 in equation in any of the equation. So I can simply do that in my equation what 1. Okay.

[00:20:30]So to do that it is very very simple. So equation one is that 2x - y is 6.

[00:20:38]So I'm putting y to be minus one. So what I have to do is that 2x - y is now -1 = 6.

[00:20:50]So what's the value of - -1? What's the value for - -1? Let's simplify the bracket. What's the value for - -1?

[00:21:02]What's the value for - -1?

[00:21:07]What is the value for -1? It becomes + one. Correct. So we have it as 2x + 1 is 6. To get the value of x, we see that 2x is = 6 + 1 crossing in equality it becomes minus.

[00:21:26]So we have it as 6 - 1 is 5. So I have it to be 2x = 5. To get the value of x, we have to divide by two and divide by two. We divide by two. You see that x is equals to what's the value for 5 / 2?

[00:21:45]What is the value for 5 / 2?

[00:21:49]What is the value for 5 / 2?

[00:21:52]What's the value for 5 / 2?

[00:21:56]What's the value of 5 / 2?

[00:22:01]It becomes 2.5. So x is what? 2.5.

[00:22:07]x is 2.5.

[00:22:10]So you see how we solve this system of equation easily and very fast. So solving system whenever you hear system of equation it is a form of equation that is very very easy to do. So you don't need to panic anytime you see that. Let's look at this. We have 3x + y is 2 and we have 2x - y is 3.

[00:22:37]So let's solve this. To solve we are trying to find the value of x that can satisfy the what equation. Okay. So to do that all you have to do again is to assign the first one is equation one and the second one is equation two. So after assigning this we study the coefficient.

[00:22:59]For here this coefficient is three. For x for y it is one. For x here it is two.

[00:23:07]For y here it is minus one. So you can see that the coefficients are a bit similar for y. So we find as much as possible to eliminate our y.

[00:23:18]So to eliminate y, what do we do? What do we do? We just have to do what? Add.

[00:23:26]So if you add the two equations, what are we getting? So you help me. 3x + 2x is what? What's the value for 3x + 2x?

[00:23:37]What is the value for 3x + 2x?

[00:23:42]What's the value for 3x + 2x? It is what? 5x. So we have it as 5x. Then what's the value of y + -? So read it as y - y. What's the value for y - y?

[00:24:01]What's the value for y - y?

[00:24:06]y - y is 0.

[00:24:09]y - y is 0. And that's equals to 2 + 3 is what? What's the value for 2 + 3?

[00:24:18]What's the value for 2 + 3?

[00:24:23]What's the value for 2 + 3? 2 + 3 is 5.

[00:24:26]So write a five. So to get the value of x we divide through by five and divide through by five. So when I divide by five you see that x is equals to 5 / 5 is what?

[00:24:42]What's the value for 5 / 5?

[00:24:47]5 / 5 is 1. So you see that x is one. So since we got x to be one, we are going to put x being one in any of the equation.

[00:24:57]So let's choose equation two. Equation 2 is 2x - y is 3.

[00:25:05]So anywhere we see x we are going to put one. So this is where x is. So we have it as 2 lots of 1 - y is 3.

[00:25:18]So we do 2 * 1. What's 2 * 1? Let's know down in the comments. What's 2 * 1?

[00:25:26]What is 2 * 1?

[00:25:29]2 * 1 is 2. So we have it as what? 2 minus y is = 3. So to get the value of y, we subtract two from both sides. So -2 here and -2 here.

[00:25:46]So if you minus two here, this is out.

[00:25:48]So what do we have left? We have it as - y = 3 - 2 is what? What's the value for 3 - 2?

[00:25:59]What's the value for 3 - 2?

[00:26:03]What's the value for 3 - 2? 3 - 2 is 1.

[00:26:09]So we have it as 1. So we have to divide through by -1ide by -1. You see that y is equals to 1 / -1 is what?

[00:26:23]What's the value for 1 / -1?

[00:26:28]What's the value for 1 / -1? 1 / -1 is what?

[00:26:35]Is -1. So x is 1, y is -1.

[00:26:41]Very simple.

[00:26:50]So we go to the next one.

[00:26:53]The next one says that x - 4 y is -2 and x + y is 8. So let's try to solve this. Remember to solve system of equation all we do is very all we do is very very simple all we do is very very simple we just have to what eliminate the variable so this is x one coefficient y is -4 here this is one this is one okay so it's easier to eliminate x by just subtracting so what we do is that we subtract the equations.

[00:27:40]Okay, to subtract equation it is very very simple. Okay, to subtract equation it is very very simple. What you have to do is just x - x is zero. So we do - 4 y - y is what? It will now be - 4 y - y.

[00:28:04]What's - 4 y - y? It becomes - 5 y and that equals to what is -2 - 8. What is -2 - 8?

[00:28:19]What is -12 - 8?

[00:28:23]What is -12 - 8?

[00:28:27]It becomes -20.

[00:28:31]Then the next thing is to divide through by -5 and divide here by -5. So -5 cancels out. You see that y is equals to what's the value for -20 / -5?

[00:28:49]What's the value for -20 / -5?

[00:28:54]What's the value for -20 / -5?

[00:29:00]What's the value for -20 / -5?

[00:29:04]You thought - 8 is 8 - 12 - 8 that y that's why we got the minus 20. Okay. So you have it to be what? Four. So y is 4.

[00:29:18]So to get x, we put y to be four in any of the equation. So we go with equation two. x + y is 8.

[00:29:29]So our y is already four. So we have that x + 4 is 8. To get the value of x, we take away y to be four here and take away y to be four here. So 4 - 4 is what? What's the value for 4 - 4?

[00:29:47]What's the value for 4 - 4?

[00:29:51]What's the value for 4 - 4?

[00:29:54]You don't understand in in linear where you having problems. So 4 - 4 is zero.

[00:30:02]So what's 8 - 4? What's the value for 8 - 4?

[00:30:09]8 - 4 is 4. So x is 4, y is 4. x is 4, y is 4.

[00:30:19]x is 4, y is 4.

[00:30:24]Thank you guys for today.

[00:30:27]This is just a breakfast. I'm testing the internet. So, I really appreciate what you guys are doing. Thank you.

[00:30:36]Thank you so much. Hope to see you guys next time.