Video: GCSE Mathematics Foundation Tier Pack 1 β€’ Paper 3 β€’ Question 8

GCSE Mathematics Foundation Tier Pack 1 β€’ Paper 3 β€’ Question 8

05:04

Video Transcript

π‘‹π‘Œπ‘ is a straight line, where the length π‘Œπ‘ is seven times the length π‘‹π‘Œ. Given that 𝑋𝑍 equals 120 centimeters, calculate the length of π‘Œπ‘.

Well, the first thing we can do is actually start to form some equations using the information that we have. So we know that π‘Œπ‘ is seven times the length π‘‹π‘Œ. So we can rewrite this as π‘Œπ‘ equals seven π‘‹π‘Œ. And what I’m gonna do is actually label this equation one. So that’s our first equation.

And next, what we can do is we can say that 𝑋𝑍 is equal to π‘‹π‘Œ plus π‘Œπ‘ and that’s because we know that π‘‹π‘Œπ‘ is a straight line and π‘Œ is a point on that straight line. So what I’m gonna do is call that equation equation two.

Okay, so now, what’s my next step? So the first thing we can do is actually substitute equation one into equation two. And we can do that because we know that π‘Œπ‘ is equal to seven π‘‹π‘Œ and π‘Œπ‘ also appears in equation two. And what we’re actually doing here is solving a pair of simultaneous equations using substitution.

So therefore, when we do that, we get 𝑋𝑍 is equal to π‘‹π‘Œ plus seven π‘‹π‘Œ. And we got that because actually we substituted in seven π‘‹π‘Œ for our π‘Œπ‘. And then when we simplify this, we get 𝑋𝑍 is equal to eight π‘‹π‘Œ. And that’s because if we have one π‘‹π‘Œ and we have seven π‘‹π‘Œs and add them together, we will get eight π‘‹π‘Œ.

So that’s great because we now have 𝑋 in terms of π‘‹π‘Œ. So what we do now is actually substitute in the fact that we know that 𝑋𝑍 is equal to 120. And we get that from the second part of the question. So when we do that, we get 120 is equal to eight π‘‹π‘Œ.

Okay, now, what we can do is actually solve this to find π‘‹π‘Œ. So to do that, what we’re gonna do is actually divide each side of the equation by eight. And that’s because we have eight π‘‹π‘Œ and we want single π‘‹π‘Œ. So if you divide eight π‘‹π‘Œ by eight, we get just π‘‹π‘Œ, which is what we’re looking for. But remembering that whatever we do to one side of the equation, we got to do to the other side of the equation. And when we do that, we get 15 is equal to π‘‹π‘Œ. So we now know that π‘‹π‘Œ is equal to 15 centimeters.

Okay, great, so what do we need to do next? Well, next, we need to look at the question and see well what is it looking for. The question wants us to calculate the length of π‘Œπ‘. So in order to actually find out what π‘Œπ‘ is and calculate that, what we’re gonna do is actually substitute our value of π‘‹π‘Œ β€” so π‘‹π‘Œ equals 15 β€” into equation one. And when we do that, we’re gonna get π‘Œπ‘ is equal to seven multiplied by 15, which will give us a value of π‘Œπ‘ of 105.

So therefore, we can say that if π‘‹π‘Œπ‘ is a straight line, where the length π‘Œπ‘ is seven times the length π‘‹π‘Œ, and given that 𝑋𝑍 is equal to 120 centimeters, the length π‘Œπ‘ is 105 centimeters. So we’ve got the answer, great! But what I wanna do here is just a couple of things, just a sort of check.

First of all, I’m just gonna show you how we could actually check if that is the correct answer and then give you an idea of another sort of visual method you could have used. So what we’re gonna do first is actually look at a little check. And to do that, we’re gonna substitute π‘‹π‘Œ and π‘Œπ‘ into equation two.

So basically, what we know is that π‘‹π‘Œ plus π‘Œπ‘ should be equal to 120. So what we’re gonna do, we’ll put our values in. So we’ve got obviously 105 π‘Œπ‘. And we found that π‘‹π‘Œ is 15. So we can see that actually yup, when we add 15 and 105, we will get 120. So we’ve checked it. So it works and that’s equal to our 𝑋𝑍 value.

Okay and the last thing I wanted to do is actually show you how we could have thought about this question, maybe with more visual method. Well, if we think that we know that π‘Œπ‘ is seven times the length π‘‹π‘Œ, then we could have thought of our line, broken down into these sections. So we have one which is between 𝑋 and π‘Œ and then seven between π‘Œ and 𝑍. So therefore, we would have had one plus seven, which is eight parts. So we know that actually the full line 𝑋𝑍 would have eight parts.

So therefore, if we wanted to find one part, we’d do 120, total length, divided by eight, which is given as 15. So we know that one part will be worth 15. So therefore, if we wanted to find π‘Œπ‘, well, we know π‘Œπ‘ is gonna be equal to seven parts. So we do seven multiplied by 15 because 15 was the value for one part, which gives us 105, which is the same value we got using the first method. So this is just another method you could use if you preferred kinda of visual representation.

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