Here is a number machine. So when we’ve got an input of 𝑥, we then multiply by three, subtract three, and we have an output of 𝑦. Part a) Work out the output when the input is 20. Part b) Work out the input when the output is 24.
There’s also a part c) that we’ll come on to. So if we want to work out the output when the input is 20, we can say that the input 𝑥 is equal to 20. So then, what we do, first of all, is multiply this by three. So we do 20 multiplied by three, which is 60. And we can work that out because two multiplied by three is six. And then, we’ve got a zero because of the 20. So it’s 60.
So we take this value, which is 60. And then, we complete the second part or second step, which is two subtract three. So therefore, when we subtract three from 60, we count down three. So we go to 59, 58, then 57. So therefore, we can say that the output when the input is 20 is 57. Okay, so now let’s move on to part b).
For part b), we want to work out the input when the output is 24. So that means if we want to find the input when we’ve been given the output, we’re gonna work backwards. So we’re gonna complete inverse operations to our original number machine.
So therefore, we’re gonna start with our output. And then, getting back, we’re gonna add three because it’s the inverse operation of subtracting three. Then, we’re gonna divide by three, again cause this is the inverse operation of multiplying by three. And then, once we’ve done that, we’ll end up with our input, which is 𝑥. So if we say 𝑦 is equal to 24 cause 𝑦 is our output, so then we’re gonna complete our new step one, which is add three. So we’re gonna have 24 add three, which is equal to 27.
So we’re then gonna complete our second step, which is divide by three. So we’re gonna take the number we had, which is 27, and divide it by three, which is equal to nine. And you can think about how we did that. Well, we know that 10 multiplied by three is 30. So that means three less is 27. So therefore, it must be nine multiplied by three is 27. So 27 divided by three is nine. So therefore, our input if the output is 24 is going to be nine. And we could check this by using our number machine, the forward way.
So we take nine. We’d multiply it by three. And then, we’d subtract three. Well, this would give us 27, because nine multiplied by three is 27, minus three, which is what we started with, cause that was the output that we’ve started with. So therefore, this is correct. So the input would definitely be nine. Okay, great. We’ve solved part a) and b). Now, let’s move on to part c).
So for part c), what we need to do is “Write an expression for 𝑦 in terms of 𝑥.”
So what we’re gonna do is write 𝑦 is equal to because we wanna an expression for 𝑦 in terms of 𝑥. So let’s have a look at our number machine. So we do 𝑥. Then, we multiply it by three first of all. Well, if you multiply three by 𝑥, the way we’d write this is three 𝑥. And what we need to make sure we do is we just multiply the 𝑥 by three at this point. We’re not doing anything else to it at this point because that’s in the second stage.
So now that we have our three 𝑥, the next stage is going to be subtract three. And we’d write that as three 𝑥 minus three. So therefore, the correct expression for 𝑦 in terms of 𝑥 is 𝑦 equals three 𝑥 minus three.
And just remembering to be careful because a common mistake with this type of question is to multiply three by 𝑥 minus three. And this is not correct because 𝑦 equals three 𝑥 minus three is not the same as 𝑦 equals three multiplied by 𝑥 minus three. And that’s because if we expanded the bracket with 𝑦 equals three multiplied by 𝑥 minus three, we get 𝑦 is equal to three 𝑥 cause three multiplied by 𝑥 is three 𝑥. Then, three multiplied by negative three gives us negative nine. So this would not be the same thing.
So therefore, the correct expression for 𝑦 in terms of 𝑥 is 𝑦 equals three 𝑥 minus three.