Narrator: A number of hints have been introduced in Chapter 3 to help you when rearranging equations. This video sequence illustrates the use of these hints in rearranging five equations of increasing complexity, and you will soon realise that the principles involved remain the same, no matter how complex the equation appears to be.

You may find it helpful to view the sequence more than once, and to pause from time to time to check your understanding. You may like to stop part-way through, at the time indicated, to try some questions for yourself .

We'll start with a very simple problem. Alison has to rearrange an equation and Shelagh, her tutor, is helping her.

Alison: So, er, I am asked to rearrange the equation in terms of f but I don't even know what the symbols mean. [Laughter].

Shelagh: Well, don't worry about it. Um, E is the energy of a photon, denoted by the subscript, so we'll call that E photon.

Alison: Right. Okay.

Shelagh: And h is something called Plank's constant and f is the frequency of the light.

Alison: Right.

Shelagh: But don't get hung up about the symbols for the moment.

Alison: Right.

Shelagh: Um, let's just work with the equation.

Alison: Okay.

Shelagh: So what does h f actually mean?

Alison: Um, h multiplied by f.

Shelagh: Yeah.

Alison: So, so I could, I could write it like that.

Shelagh: You could.

Alison: Okay.

Shelagh: You can write it either way. We would normally miss out the multiplication sign but, if you are more comfortable putting it in, that's fine as well.

Alison: Okay.

Shelagh: So what's the subject of the equation at the moment?

Alison: E photon because it, it, because it's on the left and it's by itself.

Shelagh: Right. And what do we want to make the subject?

Alison: f.

Shelagh: Okay. So let's just write that there so we know where we are heading.

Alison: Yeah.

Shelagh: So a really good way to start would be to try and at least get f on the left-hand side where it belongs as the subject.

Alison: Yeah.

Shelagh: How could you do that?

Alison: Um, could you just write the equation that way round?

Shelagh: Yes, because it's the reverse.

Alison: It's the same isn't it, yeah.

Shelagh: It's exactly the same, yes. So at least now you've got f on the left where you want it.

Alison: Yeah.

Shelagh: But you've still got it multiplied by this h.

Alison: Yeah.

Shelagh: So how could you get rid of the h?

Alison: Divide by h.

Shelagh: Yes, divide what by h?

Alison: Divide both sides of the equation by h.

Shelagh: Indeed, because you are always trying to keep a balance. Whatever you do to one side of the equation, you must do to the other side and then the balance is preserved.

Alison: Yeah. Yeah.

Shelagh: So why don't you try that?

Alison: Okay, so I've divided this side by h and then do E photon over h and then ...

Shelagh: Yes, um ...

Alison: ... the hs on this side cancel so that leaves me with f equals E photon over h.

Shelagh: And you've done it.

Alison: And I've done it. I've made f the subject.

Shelagh: Fantastic.

Narrator: Let's just look again at the hints Alison used in rearranging this equation. She started off by reversing the equation. This enabled her to get the f , the term that she was trying to make the subject, into place on the left-hand side. After that she was applying the important rule that whatever you do to one side of an equation you must also do to the other side. But Alison had to decide WHAT to do to both sides of the equation. In this case, the term she wanted, the f, was multiplied by h - so she divided both sides by h.

Rearranging Equations 2

Shelagh: Well here's another equation for you to have a go at. This time q is another energy. m is a mass.

Alison: Yeah.

Shelagh: c is something called the specific heat capacity. T is temperature.

Alison: Yeah.

Shelagh: And this delta denotes a change in temperature, so you can't separate the delta from the T, it's just one term ...

Alison: Right.

Shelagh: ... delta T.

Alison: Right. Right.

Shelagh: And this time why don't you try and find c, make c the subject.

Alison: Okay.

Shelagh: So what would be a good way to start.

Alison: Um, I could, er, reverse the equation, er, like this, to get ... c on the left-hand side.

Shelagh: Yes.

Alison: It's not on its own though. [Laughter]

Shelagh: [Laughter]. No, and this time it's multiplied by m and by delta T so you've got two things to get rid of.

Alison: Yeah. Yeah.

Shelagh: But you don't need to get rid of them both at once. Do, do one first and then the other.

Alison: Okay.

Shelagh: So, choose.

Alison: I could start by dividing by m.

Shelagh: Yes.

Alison: Both sides by m.

Shelagh: Yes.

Alison: Okay, er, so if I do that over m and q over m.

Shelagh: Yes.

Alison: Then those two ms cancel so I'm left with c delta T is equal to q over m.

Shelagh: Great. Okay, so we're half way there.

Alison: Yeah.

Shelagh: Now you've got to get rid of the delta T.

Alison: Yes, so I will divide by delta T on both sides as well.

Shelagh: Yeah.

Alison: So I'll do c delta T divided by delta T ...

Shelagh: Yes.

Alison: ... is equal to q over m divided by delta T.

Shelagh: ... divided by ... Excellent. Yes and you put the delta T on the bottom of the fraction ...

Alison: Yeah.

Shelagh: ... which is exactly what you want because it's doing the dividing.

Alison: Lovely. So then I can cancel these two delta Ts out and I'm left with c is equal to q over m delta T ...

Shelagh: Brilliant. You've done it.

Alison: ... which is what I wanted. Great.

Shelagh: Excellent.

Alison: [Big sigh].

Narrator: Let's look again at the hints used this time. Once again she started by reversing the equation to get the subject, c on this occasion, into place on the left-hand side.
After that, she had to get rid of two terms, m and delta T, and did this one step at a time. You should never be afraid of using several small steps to rearrange one equation.
To get rid of both the m and the delta T, Alison had to remember that the c was multiplied by those terms and so divide by them. And she had to divide BOTH sides of the equation by first m and then delta T.

Rearranging Equations 3

Shelagh: So the next equation is, power equals energy divided by time.

Alison: Yeah.

Shelagh: And, er, I'd like you to try and make t the subject. Now how have we started on the previous equations?

Alison: Well, we've reversed equation to put the subject on the left-hand side, the subject that we want.

Shelagh: Yeah, but if we reverse this one ...

Alison: Yeah.

Shelagh: ... so we have E over t equals P.

Alison: Well.

Shelagh: Where is the t?

Alison: The t is on the bottom and we need, er, the t to be on the top to make it the subject.

Shelagh: That's right. It's on the left but it's not on the top.

Alison: Yeah.

Shelagh: So, that's a perfectly valid equation but it doesn't get us where we want to go.

Alison: Yeah.

Shelagh: So we need to have a different strategy this time. So let's go back to the original, P equals E over t. So what could you do to get t on the top on the left?

Alison: Well as we're dividing by t on that side, if we multiply both sides of the equation by t that would put, um, t on the top on this side so it would be ...

Shelagh: Yeah, go on.

Alison: So it would be P times t and then E over t times t.

Shelagh: t, yeah.

Alison: And then the two ts on the right hand side will cancel.

Shelagh: Indeed.

Alison: So we've got P t is equal to E.

Shelagh: Great.

Alison: But we've still got t multiplied by P.

Shelagh: Yes, so you know what to do to get rid of the P, don't you.

Alison: Yes. I still need to divide both sides by P. So divide the left-hand side by P.

Shelagh: Yes.

Alison: And then those two Ps cancel and t is equal to E over P and it's now the subject.

Shelagh: Excellent. Brilliant. Well done.

Narrator: Let's look again at the hints that Alison used this time.
Hint 3: If the term you want is multiplied by another expression, divide both sides of the equation by that expression.
Hint 4: If the term you want is divided by another expression, multiply both sides of the equation by that expression.
Hint 7: Don't be afraid of using several small steps to rearrange one equation.
Hint 8: Aim to get the new subject into position on the left-hand side as soon as you can.
It's worth thinking a bit more about hints 3 and 4. These are special cases of a more general rule, namely, that to undo any operation, you should do the opposite. So to undo a multiplication you divide, to undo a division you multiply, to undo a subtraction you add and to undo an addition you subtract.

Rearranging Equations 4

Narrator: The next rearrangement looks more complicated. Although you won't meet this equation in S104, it's been included to show how you can use the same principles to rearrange any equation, however horrible it might look.

Shelagh: Right, our next equation is describing the speed v s of a wave going through a rock of rigidity modulus mu and density rho.

Alison: [Laughter]

Shelagh: [Laughter] And I would like you to find, er, an equation for rho.

Alison: Right.

Shelagh: Um, the thing to note is that I've drawn this square root sign so that it goes around the whole of the fraction.

Alison: Yeah.

Shelagh: You could put brackets around the fraction but, because it's drawn that way, it's not necessary.

Alison: Right.

Shelagh: Where do you think you might start with this one?

Alison: Well, um, the rho is on the bottom on the right-hand side and I need it on the top on the left-hand side but there's a great big square root symbol in the way and I don't know what to do with it.

Shelagh: Well, what's the opposite operation to taking a square root?

Alison: Squaring.

Shelagh: Yeah. So ...

Alison: So if I square both sides of the equation. I'll get rid of the square root.

Shelagh: You will.

Alison: So, um, if I do v s squared, er, is equal to mu over rho because I've cancelled that square root out by squaring.

Shelagh: Yes.

Alison: Yes.

Shelagh: But now you've still got ...

Alison: ... rho on the bottom

Shelagh: ... on the bottom on the right.

Alison: Okay, lovely, so, um ...

Shelagh: So how can you put rho on the top on the left?

Alison: I need to multiply both sides by rho.

Shelagh: You do.

Alison: So I'll just do that and that and then the rhos on the right-hand side cancel out and I'm left with v s squared rho is equal to mu.

Shelagh: Okay. So where were we ...

Alison: I've forgotten what I was supposed to do. [Laughter].

Shelagh: [Laughter]. Where were we heading? We were trying to find rho.

Alison: Okay. Right.

Shelagh: And now we've got rho multiplied by vs squared. So you know what to do with that?

Alison: Yes, divide both sides by v s squared, divided by v s squared so the v s squareds cancel on the left-hand side and I'm left with rho is equal to mu over v s squared.

Shelagh: You got there, well done. [Laughter].

Narrator: As the equations become more complicated, it becomes more important to tackle them in several small steps, as Alison did in this example. Also, we have now extended the general rule, that to undo any operation, you should do the opposite. In this case Alison needed to get rid of a square root, so she remembered that squaring something and finding a square root are opposite operations. So to get rid of the square root, she squared both sides of the equation. Before attempting the final problem you may like to take a break and try parts (a) and (b) of Question 3.3, immediately after this activity in the course book.

Rearranging Equations 5

Narrator: Before Alison tackles the final equation, let's look again at the two general rules she's been using. Most importantly, remember that an equation is like a set of scales, so that you can do almost anything you like to an equation, PROVIDED you do the same thing to the other side, to keep the balance. The second rule helps you to decide WHAT it is that you should do to both sides of the equation. So to undo an operation you should do the opposite, to undo a subtraction you add, to undo a multiplication you divide, to undo a square root you square and so on. The final equation can be rearranged using exactly the same principles, remembering to keep going in a series of small steps. It's like peeling away layers of an onion. But it needn't end in tears!

Shelagh: So this is your last equation, looks a bit horrible. The delta, like before, is a change in something ...

Alison: Right.

Shelagh: ... so delta E k is a change in the kinetic energy ...

Alison: Okay.

Shelagh: ... of an object of mass m that increases its speed from u to v.

Alison: Okay.

Shelagh: And what I'd like you to find is an equation for u.

Alison: [Big sigh].

Shelagh: You are.

Alison: [Nervous laugh]

Shelagh: We don't go for the u straight away, we'll go for this whole term here, and let's try and at least get that on the left-hand side.

Alison: Yes.

Shelagh: So what would you have to do to get that on to the left-hand side?

Alison: Um, er, well it's subtracted here ...

Shelagh: Yeah.

Alison: ... so if I add half m u squared to both sides ...

Shelagh: Yes.

Alison: Er, I'll just, so if I do delta E k add a half m u squared, that is equal to a half m v squared minus a half m u squared plus a half m u squared.

Shelagh: That's fine. So you, you've added that to both sides.

Alison: Yeah.

Shelagh: So you've preserved your balance again. And so what can you cancel?

Alison: Those two cancel each other out.

Shelagh: They do.

Alison: So I'm left with, er, ...

Shelagh: That's fine. You're left with ...

Alison: ... delta E k ...

Shelagh: ... E k

Alison: ... plus half m u squared is equal to half m v squared.

Shelagh: Right. So now you've got half M U squared where you want it.

Alison: Yes.

Shelagh: But you've still got it added to delta E K. So what can you do next?

Alison: Subtract that from both sides.

Shelagh: Yes.

Alison: Um, so delta E k plus half m u squared minus delta E k is equal to a half m v squared minus delta E k.

Shelagh: Yes.

Alison: And these two terms cancel each other out.

Shelagh: E k minus E k is zero.

Alison: Yes. So that leaves me with half m u squared is equal to half m v squared minus delta E k.

Shelagh: Good, okay. So now we're really making progress ...

Alison: Yeah.

Shelagh: ...'cause you've only got one term on the left, at least.

Alison: Yeah, yeah.

Shelagh: ... but you've still got half and you've got M in it.

Alison: Yeah.

Shelagh: So what could you do next?

Alison: Well a half m u squared is the same as m u squared over two, divided by two so if I multiplied both sides by two ...

Shelagh: Yes.

Alison: So, I'll just, actually I'll just write this out, if I've got m u squared over two.

Shelagh: Right. That's exactly equivalent to that, yes.

Alison: Yes, yeah. So if I multiply that by two.

Shelagh: Yes.

Alison: Then I have to multiply the whole of this side by two. So that's everything.

Shelagh: Indeed. Very good.

Alison: Sorry, I'm just going to get some more paper [laughter].

Shelagh: Yeah, I think you need another sheet.

Alison: Oh, it goes on. Um, so ...

Shelagh: Now, we're nearly there.

Alison: I can cancel these two because I'm dividing by two and multiplying by two.

Shelagh: Yes.

Alison: So, I've got m u squared is equal to, and I have to multiply each term in the bracket? ...

Shelagh: Indeed.

Alison: ... by two. So that's the same as m v squared over two multiplied by two. So that just gives me m v squared ...

Shelagh: Yes.

Alison: ... minus two times delta E k.

Shelagh: Great. Yes.

Alison: But ...

Shelagh: You've still got an m.

Alison: I've still got an m. Um, so I just need to divide both sides by m.

Shelagh: Yes.

Alison: And that's the whole of that side by m.

Shelagh: Yes.

Alison: And those ms cancel and I've still got a u squared.

Shelagh: [Laughter]. That's alright. We're homing in on it.

Alison: Aaargh.

Shelagh: Okay, so you've got to get rid of the squared.

Alison: Yeah.

Shelagh: And what is the opposite of squaring?

Alison: Taking the square root.

Shelagh: Yes.

Alison: So, if I take the square root of u squared I'm just left with u and that is equal to, er, m v squared minus two delta E k over m and that's ...

Shelagh: You have to take the square root of the whole thing ...

Alison: ... the whole thing.

Shelagh: You've done it.

Alison: [Laughter]

Shelagh: Well done. And actually, particularly well done because that is more difficult than anything you'll be asked to rearrange in the course. But just by going at it step by step it wasn't actually that bad, was it?

Alison: No, just got to break it down.

Shelagh: Excellent.

Alison: Thank you. [Big sigh].

Narrator: As Shelagh said, the final equation is harder than anything that you'll be asked to rearrange in S104. Rearranging equations is all about taking things slowly and steadily, undoing operations by doing the opposite and remembering that whatever you do mathematically to one side of an equation you must also do to the other side. Using the same rules you should now be able to deal with any equation that you have to rearrange in S104, and plenty that are more difficult. It really is that simple.