Rearrange 𝑦 equals two 𝑥 over five minus two to make 𝑥 the subject.
So when we’re looking at a question like this, we need to think what is making 𝑥 the subject mean. Well, it means getting 𝑥 on its own. So we have 𝑥 is equal to something else, so an expression. So if we start off with the equation that we have, which is 𝑦 equals two 𝑥 over five minus two, our first step is going to be to add the two. Because we want to leave the term that has the 𝑥 involved, which is two 𝑥 over five on its own. And if we add two to negative two, then we’ll get zero. And with any equation, whatever we do to one side of the equation, we must do to the other side. So we add two to the left-hand side as well. So when we do this, we get 𝑦 plus two is equal to two 𝑥 over five.
Now it’s worth noting at this point we added two because we wanted to leave the term with 𝑥 on its own. We could, however, have removed the denominator from two 𝑥 over five. But this would have been more complex because we would have to multiply each of the additional terms by five. I will, however, demonstrate this method at the end just to show you how it could have been done this way. Okay, so now what’s the next stage?
Well, now, as we want to leave the 𝑥 on its own, and we’ve got 𝑥 involved in a fraction because we’ve got five as the denominator. What we want to do here is do the inverse operation. So instead of dividing by five, we’re gonna multiply by five. And that’s because two 𝑥 over five multiplied by five will just give us two 𝑥, remembering though whatever we do to one side, we do to the other. So now we have five multiplied by 𝑦 plus two is equal to two 𝑥. And the way that we write it is like this: five and then we’ve got a bracket then 𝑦 plus two. And this means that five is multiplied by each term within the brackets.
So now for the next stage, what we could do is we could expand the brackets by multiplying the five by each term. However, we don’t need to do this because what we need to do is just make 𝑥 the subject. So therefore, if we’ve got two 𝑥 on the right-hand side of the equation, then the inverse of multiplying 𝑥 by two is dividing by two. So we could do that, which will give us five multiplied by 𝑦 plus two over two is equal to 𝑥. Now, we can rewrite this in a better form by writing the five over two as a coefficient outside of our bracket. So now we’ve got 𝑥 is equal to five over two multiplied by 𝑦 plus two. And I’ve put 𝑥 on the left-hand side just it’s easier to see what 𝑥 is when we’ve made 𝑥 the subject.
It’s worth noting that if we had expanded the bracket, the answer wouldn’t have been incorrect. It would have just given us a different way of writing it. And I’ll show you what that would have been. Well, if we’d expanded the bracket, I would have multiplied the five by the 𝑦 to give us five 𝑦 and the five by the positive two to give us positive 10. So we get five 𝑦 plus 10 is equal to two 𝑥, which would have given us five 𝑦 plus 10 over two is equal to 𝑥. So again, this is perfectly acceptable. It would just be a different way to write the answer.
Okay, now, I said that I was gonna show you how we could have rearranged in a slightly different way at the beginning. So this way, we start with 𝑦 equals two 𝑥 over five minus two. But then to remove the denominator, first of all, what we’re gonna do is multiply each side of the equation by five. Now, this is where the mistake can be made. Because when we multiply by five, it means you have to multiply every single term. The common mistake would just be to write five 𝑦 equals two 𝑥 minus two, but they have not multiplied the minus two or negative two by five.
So if we do it correctly, we get five 𝑦 is equal to two 𝑥 minus 10. And that’s because two 𝑥 over five multiplied by five gives us two 𝑥 and negative two multiplied by five gives us negative 10. So next, what we’re gonna do is add 10 to each side of the equation to leave our term two 𝑥 on its own. So when we do that, we get five 𝑦 plus 10 is equal to two 𝑥. Then, we’d divide each side of the equation by two to leave us with single 𝑥. So we get five 𝑦 plus 10 over two equals 𝑥, which is the same as what we showed on the left-hand side. But as I said, this is the same answer we’ve got on the original answer. It’s just written in a slightly different way. But I’ll show you how it’s the same answer. We can show that by factorizing.
Well, first of all, if we look at five 𝑦 plus 10, we can factorize this because five 𝑦 and positive 10 have a common factor. And that common factor or the highest common factor is five because five goes into five once and it goes into 10 twice. So therefore, if we take five outside the bracket and we have inside the bracket where we have need to multiply by five to get five 𝑦 plus 10. So it’s five multiplied by 𝑦 plus two. That’s because five multiplied by 𝑦 is five 𝑦. Five multiplied by two is 10. So we get five multiplied by 𝑦 plus two over two is equal to 𝑥, which we can rearrange to get it in the same form as the original answer: 𝑥 is equal to five over two multiplied by 𝑦 plus two.