Video: Pack 1 • Paper 1 • Question 16

Pack 1 • Paper 1 • Question 16

08:22

Video Transcript

One shuttlecock and three table tennis balls weigh 13.3 grams. Three shuttlecocks and six table tennis balls weigh 31.8 grams. Find the weight of one shuttlecock and one table tennis ball.

So the two things that we’re looking to actually find out here are the weight of one shuttlecock and the weight of one table tennis ball. And what I’ve done is I’ve actually called the weight of a shuttlecock 𝑥 and the weight of a table tennis ball 𝑦. So now, what I’m gonna use is my 𝑥 and 𝑦 and the information from the question to actually set up two simultaneous equations. So first of all, we’re gonna use the fact that we know that one shuttlecock and three table tennis balls weigh 13.3 grams to set up our first equation. So that’ll be 𝑥 plus three 𝑦 is equal to 13.3.

The reason we just have 𝑥 plus three 𝑦 equals 13.3 is that we don’t actually have to write the one. I have just written it here in orange just to remind us that actually we’ve got one shuttlecock. So the coefficient of 𝑥 is just one. So now, we can actually use the next bit of information to set up our second simultaneous equation cause we know that three shuttlecocks and six table tennis balls weigh 31.8 grams. So what this does is that it gives us the equation three 𝑥 plus six 𝑦 is equal to 31.8.

So what we have now are a pair of simultaneous equations and what I always do is I label them just to help us in the next stages to be nice and clear. So we got equation one and equation two. Another two methods that we usually use to solve this kind of problem elimination or substitution. I want to start by showing you how to use elimination. But then, I’m gonna go on and actually show you how to use substitution just because there might be one method that you prefer.

So now with elimination, what we want to do is we first want to see if we’ve got the same coefficient of our 𝑥s or 𝑦s in both equations. In this case, we haven’t yet, which means we have to put in one extra step. We’re going to need to multiply one of our equations to actually make it so that we have the same coefficient of 𝑥 or the same coefficient of 𝑦. It doesn’t matter which you can choose.

In this case, what I’m going to do is I’m going to make our coefficients of 𝑦 the same. And in order to do that, what I’m gonna do is I’m going to multiply our first equation by two. And that means every term in our first equation by two because then we’ll have six 𝑦 and we’ll have the same coefficient of 𝑦. So when I do that, I get two 𝑥 plus six 𝑦 equals 26.6. And remember you’ve got to multiply every term because one of the common mistakes here is that people may multiply the 𝑥 and 𝑦 terms, but forget about the numerical term as well. And as I said, just want to reiterate here I could have done the same for 𝑥. So I could have multiplied maybe the top equation by three. It doesn’t matter. It’s just that this time I’ve decided to work out 𝑦.

Great, so we now we got two equations, equation number two and equation number three, that have the same coefficient of 𝑦. So now, our next step is actually I’m gonna do equation two minus equation three. And that’s because that’s going to eliminate our 𝑦 terms because actually we’ve got the same coefficient. And how do we know that? Well, it’s because we’ve got the same sign. So we’ve got the same sign for our 𝑦 term which is what we’re trying to eliminate. They’re both positive. If they’re both negative, we’d also do the same — we subtract. However, if we had different signs — so one positive, one negative — then what we do is we actually add.

So first of all, we do three 𝑥 minus two 𝑥, which just gives us 𝑥. And then, we have positive six 𝑦 minus positive six 𝑦 which just gives us zero. So we don’t need to write that term because we’ve actually eliminated it. And then, we have 31.8 minus 26.6, which gives us 5.2. So great, we’ve actually found our 𝑥-value. So we got 𝑥 is equal to 5.2. Right, so now let’s use this to find 𝑦.

So now, what we’re gonna do is substitute our 𝑥-value that we found. So we’re gonna substitute 𝑥 equals 5.2 back into equation one. We can actually substitute it into any of the equations. It doesn’t matter. We’ve just chosen to put it into equation one. So when we substitute it back in, we get 5.2 plus three 𝑦 equals 13.3. So therefore, if we subtract 5.2 from each side, we’re gonna get three 𝑦 equals 13.3 minus 5.2 which gives us three 𝑦 is equal to 8.1. And then, we divide both sides of the equation by three and we get the answer 𝑦 is equal to 2.7.

So therefore, we can say that the table tennis ball is equal to 2.7 grams because 𝑦 was the weight of the table tennis ball and the shuttlecock is equal to 5.2 grams and that’s because we said 𝑥 was the weight of the shuttlecock. Okay, great, so we’ve solved the problem. But like I said, what I’ll do now is actually show you how you could solve the same problem using substitution. And again, this is just so you have two methods that you could use and also because you might find the other method the one that suits you better.

Okay, now, for the substitution method, we start in exactly the same way. So we have our two simultaneous equations and what we do is we actually labelled one, one and one, two. So the beginning is exactly the same. So now to actually enable us to use the substitution method, what we want to do is actually make 𝑥 the subject of equation one. And that’s why we’ve actually chosen to use this method as well with this question cause it lends itself easily to that because it lends itself because what we have is a single 𝑥. So the coefficient of 𝑥 is one. And similarly, if we had a single 𝑦 or a coefficient of 𝑦 that was one, it’d be a good method to use because we can actually make one of them the subject easily.

So therefore, to actually make 𝑥 the subject of equation one, all we need to do is actually subtract three 𝑦 from each side. And when we do that, we get 𝑥 is equal to 13.3 minus three 𝑦. So now we’ve actually made 𝑥 the subject. What we can do is substitute 𝑥 into equation two. So when we do that, we get three multiplied by 13.3 minus three 𝑦 plus six 𝑦 equals 31.8. And we get that because we’ve substituted in our 𝑥-values. So 𝑥 is equal 13.3 minus three 𝑦 instead of our 𝑥. And this is the second equation.

Okay, great, so now what we’re gonna do is expand our brackets which gives us 39.9 minus nine 𝑦 plus six 𝑦 equals 31.8. So now, if we simplify our 𝑦 terms, we get negative nine 𝑦 plus six 𝑦, which gives us negative three 𝑦. And this is gonna be equal to 31.8 minus 39.9. So therefore, we get negative three 𝑦 equals negative 8.1. And then, we can divide each side by negative three and we get 𝑦 is equal to 2.7. So now, we’ll check our answer from the previous way, so from the elimination. And yes, great, it works because a table tennis ball weighs 2.7 grams.

Okay, so now, let’s find 𝑥. So now to actually find our value of 𝑥, what we’re gonna do is to substitute our value of 𝑦 into what I’ve called equation three. But it’s 𝑥 is equal to 13.3 minus three 𝑦. And when we do that, we get 𝑥 is equal to 13.3 minus then three multiplied by 2.7 which gives us 𝑥 is equal to 13.3 minus 8.1, which will give us a final 𝑥-value of 𝑥 is equal to 5.2. And therefore, as 𝑥 is equal to 5.2, we can check against our earlier method, which was the elimination method. And we can see yes, it’s correct cause we had 5.2 as well.

So therefore, we can say with confidence that the weight of one shuttlecock and one table tennis ball are a shuttlecock weighs 5.2 grams and a table tennis ball weighs 2.7 grams. And as I said, you can use either method and we’ve given you both because actually there will be questions in the future that lend themselves better to each of these types of solution.

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