### Video Transcript

Solve ๐ฆ equals two ๐ฅ plus one over three ๐ฅ plus four with an expression for ๐ฅ in terms of ๐ฆ.

And the equation we were given in the question gives us ๐ฆ in terms of ๐ฅ: ๐ฆ on its own on the left-hand side and some expressions involving ๐ฅ on the right-hand side. So the question is asking us to take that equation and rearrange it so that we got ๐ฅ on its own on the left-hand side and some ๐ฆ terms on the right-hand side.

Now, the first thing we notice is that the ๐ฅ variable appears in this denominator. And we should also remember that all the terms on the numerator need to be grouped together and all the terms on the denominator need to be grouped together. Now, I want to eliminate that denominator from the right-hand side. And I can do that by multiplying the right-hand side by the multiplicative inverse of one over three ๐ฅ plus four because one over three ๐ฅ plus four times three ๐ฅ plus four over one is equal to one.

Basically, dividing the top by three ๐ฅ plus four gives us one. Dividing the bottom by three ๐ฅ plus four plus four gives us one. So weโve got one times one over one times one, which is just one. So these two terms cancel each other out. But the problem is weโve now unbalanced our equation. Weโve multiplied this side by three ๐ฅ plus four over one, so itโs no longer equal to this side. We need to use the multiplication property of equality. And that means that if we multiply one side of the equation by something, we need to multiply the other side of the equation by the same thing if that remained equal.

Now, that means weโve got to multiply the left-hand side by three ๐ฅ plus four over one as well. Now, when we do that, weโll notice that ๐ฆ doesnโt have a denominator and that three ๐ฅ plus four over one is just the same as three ๐ฅ plus four. And this means on the left-hand side, weโve got ๐ฆ times three ๐ฅ plus four. Now, we can use the distributive property on the left-hand side of the equation to give us ๐ฆ times three ๐ฅ and ๐ฆ times four.

Now, ๐ฆ times three ๐ฅ can be written as three ๐ฅ๐ฆ. And ๐ฆ times four can be written as plus four ๐ฆ. And of course that is just equal to two ๐ฅ plus one on the right-hand side. Now, letโs just remember what weโre trying to do. Weโre trying to isolate the ๐ฅ on the left-hand side of our equation. A good next step will be to get rid of the two ๐ฅ on the right-hand side. And I can do that using the subtraction property of equality; I can subtract two ๐ฅ from both sides.

And then on the right-hand side, two ๐ฅ plus one take away two ๐ฅ. Well, two ๐ฅ take away two ๐ฅ is just nothing. So that just leaves us with positive one. And if we look at the left-hand side, there werenโt any simple ๐ฅ terms that we could subtract two ๐ฅ from. So weโve got three ๐ฅ๐ฆ plus four ๐ฆ minus two ๐ฅ.

Now, we need to leave ourselves with only terms involving ๐ฅ on the left-hand side. So weโre gonna need to try and eliminate this ๐ฆ term from the left-hand side. And we can do that again using the subtraction property of equality and subtracting four ๐ฆ from both sides. And that means on the left-hand side, three ๐ฅ๐ฆ plus four ๐ฆ minus two ๐ฅ minus four ๐ฆ. Well, the four ๐ฆ take away four ๐ฆ gives us nothing. So weโre just left with three ๐ฅ ๐ฆ minus two ๐ฅ. And on the right-hand side, weโve just got one minus four ๐ฆ.

So now weโve got all of the ๐ฅ terms on the left-hand side and no ๐ฅ terms on the right-hand side. But we only want ๐ฅ terms on the left-hand side, and thereโs an ๐ฅ๐ฆ term there at the moment. But remember three ๐ฅ๐ฆ means three times ๐ฅ times ๐ฆ and two ๐ฅ means two times ๐ฅ. So we can see weโve got a common factor of ๐ฅ in both of those terms. So we can use the distributive property to factor ๐ฅ out of both of those terms. So that means weโve got ๐ฅ times three ๐ฆ minus two is equal to one minus four ๐ฆ.

Now remember one over three ๐ฆ minus two is the multiplicative inverse of three ๐ฆ minus two. So if I multiply both sides by one over three ๐ฆ minus two, then these two things cancel out to give us one. So the left-hand side just becomes ๐ฅ. And on the right-hand side, I just got one minus four ๐ฆ times one. So thatโs just one minus four ๐ฆ over three ๐ฆ minus two. And in fact, I donโt even need those parentheses there. And there we have our answer: an expression for ๐ฅ in terms of ๐ฆ. ๐ฅ is equal to one minus four ๐ฆ over three ๐ฆ minus two.

Now just before we go, I just wanna mention if weโd made some slightly different choices along the way, weโd have ended up with a different version of our answer. For example, in this stage here, if weโd have subtracted three ๐ฅ๐ฆ from both sides rather than two ๐ฅ from both sides, weโd have gathered all of the ๐ฅ terms over on the right-hand side of that equation. Then we could have subtracted one from both sides to eliminate this term, then factored out this ๐ฅ again using the distributive property, and finally multiplied through by the multiplicative inverse of two minus three ๐ฆ to give us an answer of ๐ฅ is equal to four ๐ฆ minus one over two minus three ๐ฆ.

Now, this looks very similar to the answer that we got over here. But weโve got four ๐ฆ minus one on the numerator instead of one minus four ๐ฆ and weโve got two minus three ๐ฆ on the denominator instead of three ๐ฆ minus two. So which one of our two answers is correct? Well, theyโre both right; theyโre both equivalent versions of the same answer.

Look if I multiplied both of those answers by one, I think youโll agree Iโm not changing either of them. But in fact, if I multiplied this one by a different version of one, negative one divided by negative one. Negative one divided by negative one is just one. And then I use the distributive property to multiply the terms on the numerator by negative one and likewise the terms in the denominator. Then negative one times one is negative one; negative one times negative four ๐ฆ is positive four ๐ฆ. So the denominator is negative one plus four ๐ฆ. And negative one times three ๐ฆ is negative three ๐ฆ, and negative one times negative two is positive two. So the denominator is negative three ๐ฆ plus two.

Then, if we remember the addition is commutative, it doesnโt matter which order I add these things together in, so negative one plus four ๐ฆ is the same as four ๐ฆ plus negative one or four ๐ฆ take away one. And on the denominator, negative three ๐ฆ plus two is the same as two add negative three ๐ฆ or two take away three ๐ฆ. So whether you got this answer or this answer, we can see they amount to the same thing.

Now, itโs well worth remembering this trick of multiplying your answer by one โ well, the version of one that is negative one over negative one, if you need to rearrange it into a slightly different format. Sometimes, youโll be given different types of questions. For example, โshow that ๐ฆ equals two ๐ฅ plus one over three ๐ฅ plus fourโ can be expressed as ๐ฅ equals four ๐ฆ minus one over two minus three ๐ฆ.

So if your working out gives you the answer in this format, you might need to use this technique in order to rearrange that answer into the format that theyโve asked for in the question. So even if it seems a little bit strange at first, I would practice using this technique on some questions of your own.