Lesson video

Hello, Mr. Robson here, and welcome to mathematics.

What a lovely place to be.

We're changing the subject of simple formula today.

This is a really useful skill.

You wanna use it in a lot of places in your mathematical learning journey.

So what are we waiting for? Let's get learning.

Our learning outcome is that we'll be able to apply an understanding of inverse operations to a simple formula in order to make a specific variable the subject.

There'll be lots of important vocabulary throughout this lesson, and the word I've just said, subject, might be a new one to you.

The subject of an equation or formula is a variable that is expressed in terms of other variables.

It should have an exponent of one and a coefficient of one.

For example, A is the subject of this formula.

A equals the square root of C squared minus B squared, but A squared is not the subject of this formula, A squared equals C squared minus B squared.

The subject should have an exponent of one and a coefficient of one.

Look out for this language throughout the lesson.

Two parts to our lesson today and we're going to begin by identifying the subject of the formula.

Rearranging equations and formula is a valuable skill in mathematics.

You know that 4 plus 6 equals 10.

This is 10 expressed in terms of 4 and 6.

What would 4 expressed in terms of 10 and 6 look like? Well done.

It would be 4 equals 10 minus 6.

What if I said, what would 6 expressed in terms of 10 and 4 look like? What do you think? Well done.

6 expressed in terms of 10 and 4 would be 10 minus 4.

We can generalise the rule for rearranging additive relationships.

If A plus B equals C, then it's also true that A equals C minus B.

This is A expressed in terms of C and B.

It's also true that B equals C minus A, that's B expressed in terms of C and A.

In this form, A is the subject of the equation.

And in the second example in that form, B is the subject of the equation.

The subject of an equation or formula is a variable that's expressed in terms of other variables and it should have an exponent of one and a coefficient of one.

You'll see this in various contexts in mathematics.

We might see it in solving equations.

If we were asked to solve y plus 35 equals 78, we can rearrange this additive relationship.

We could write it as 35 plus y equals 78.

You could give me two subtraction facts that relate y, 78, and 35, y must be 78 minus 35 and 35 must be 78 minus y.

In this form, y is now the subject of the equation.

In the context of this equation, we can immediately calculate the unknown because it is the subject.

We've got y equals 43 because we can simplify those like terms on the right-hand side, having isolated that y having made it the subject.

Here's the generalisation for additive relationships again.

In this form, A is the subject of the equation.

In this form, B is the subject of the equation.

So what is the subject of the original form, A plus B equals C? Andeep says, "I think A is the subject because it is the first thing on the left-hand side." Do you agree? Pause and have a conversation with the person next to you or a good think to yourself.

See you in a moment.

Welcome back.

What did you think? Sofia steps in to help Andeep.

She says, "Look at the definition of subject again, Andeep." Our definition was that the subject of an equation, a formula, is a variable that is expressed in terms of other variables.

It should have an exponent of one and a coefficient of one.

Andeep corrects himself, which is an awesome thing to do if you make an error in maths.

He says, "The subject must be C because this is C expressed in terms of the other variables." Sofia says, "Well corrected, Andeep.

C is the subject.

The subject can be on the left or the right-hand side of an equation or formula." Andeep says, "I was learning about right angle triangles recently and saw this formula, A squared plus B squared equals C squared, so I can rearrange this relationship like this." "Yes, that's right.

You can rearrange this like you would any additive relationship," confirms Sofia.

And Andeep says, "So I can have C squared as the subject, as in C squared expressed in terms of A squared and B squared, or I can make A squared the subject.

It depends what I want to work out." Sofia steps in and corrects him.

"Let's check that definition of subject again, Andeep.

Once again, the subject of an equation or formula is a variable that's expressed in terms of other variables.

It should have an exponent of one and a coefficient of one." Andeep has seen the error of his ways.

"I see now, we can't call A squared the subject because it doesn't have an exponent of one." "Well done, the subject should have an exponent of one and a coefficient of one," says Sofia.

I'd like to check you've got everything so far.

In which of these rearrangements of the equation X plus Y equals Z is Y the subject? Four to choose from there.

Where is Y the subject.

Pause, tell the person next to you, or have a good think to yourself.

See you in a moment for the answers.

Welcome back.

It was not A, nor was it B, nor C.

It was D.

Y equals Z minus X.

It's the only rearrangement where Y is expressed in terms of other variables.

This looks like a very similar question, but with one key difference.

In which of these rearrangements of the equation X plus Y equals Z is Z the subject? I'd like you to identify now where Z is the the subject.

Pause, have a good think to yourself, or a conversation with the person next to you.

See you in a few seconds for the answers.

Welcome back.

I hope you said A, X plus Y equals Z.

Z is the subject.

Z is expressed in terms of other variables.

The subject can be on the right-hand side of an equation or formula.

It doesn't have to be on the left-hand side.

B was not an option.

Z is not the subject there.

C worked.

This again is Z expressed in terms of the other variables and the subject again is on the right-hand side.

D was not a case of Z being the subject.

True or false.

In the equation, 5Y squared equals X plus Z, the subject is 5Y squared.

Is that true or is it false? Once you've decided whether it's true or false, could you justify your answer with one of these two statements? 5Y squared is not the subject because it does not have an exponent of one or a coefficient of one or five.

Or 5Y squared is the subject because it is expressed in terms of the other variables.

Pause and have a think about this one now.

Welcome back.

Let's see how we got on.

I hope you went with false.

Now, why would it be false? It's false because it's not the subject, because it does not have an exponent of one or a coefficient of one.

Y has an exponent of 2 and a coefficient of 5, so it can't be the subject.

We can also identify the subject in multiplicative relationships and we know that 4 and 8 make 32 and we also know that 8 and 4 make 32 and that 32 divided by 8 equals 4 and 32 divided by 4 equals 8.

A generalisation for multiplicative relationships would look like that.

If A times B equals C, then B times A equals C and the related division facts C over B equals A and C over A equals B.

In the case of A times B equals C, in that form, C is the subject of the equation.

When we rearrange it to C over B equals A, in that form, A is the subject of the equation.

B times A equals C.

You know what I'm gonna say next.

In this form, C is the subject of the equation.

How about C over A equals B? That's right.

In this form, B is the subject of the equation.

I'd like to check you've got that for multiplicative relationships.

In each arrangement of this formula, can you identify the variable which is the subject? It's the formula for every rectangle, four different arrangements of it.

Can you identify the subjects in each case? Pause, have a conversation with the person next to you, or a good think to yourself.

See you in a moment for the answers.

Welcome back.

I hope you said for the arrangement A equals B times H, that A was the subject.

In the form A over H equals B, B is the subject.

In the form A equals H times B, A is the subject.

That last arrangement A over B equals H, H is the subject.

Practise time now.

For question one, I'd like you to identify the subject in each of these formula/equations.

Eight examples there.

In each one, just put a circle around the subject.

Pause and do that now.

Question two.

Aisha is solving the equation 85 minus 3X equals 17.

The equation can be rearranged in these forms. Aisha says, "The form 85 minus 17 equals 3X is the best rearrangement because 3X is the subject." Could you write a sentence explaining what is wrong with Aisha's statement? Pause and do that now.

Feedback time now.

For question one, we were identifying the subject in each case.

For A, C was the subject.

For part B, we had Y as the subject.

For part C, it was D.

That was the subject For part D, we had A as the subject.

For part E, we add H as the subject.

For part F, we add A as the subject.

For G, we had subject D.

And for H, P was the subject.

For question two, I asked you to write a sentence explaining what was wrong with Aisha's statement.

You might have written something along the lines of, "Whilst 85 minus 17 equals 3X is the best form from which to solve, we cannot call 3X the subject because the coefficient is three and not one." Onto the second part of the lesson now where we're gonna be changing the subject of simple formula.

We can rearrange equations and formula using inverse operations.

If we came across the equation X plus 12 equals 37, you might ask yourself, what's the inverse of plus 12? The answer would be minus 12 or you might think of it as adding negative 12.

The expressions on each side remain equal because we perform the same operation to both sides.

It's an equation.

We need to maintain equality.

We do that by performing the same operation to both sides.

Once we simplify that, the left-hand side, we've got a positive 12 and a negative 12 where they cancel each other out so the left-hand side simplifies to X.

On the right-hand side, we've got 37 minus 12.

X is now the subject of the equation.

By making X the subject of the equation, we have the solution very quickly and efficiently.

What if the equation was X minus 2.

7 equals 8.

1? What's the inverse of minus 2.

7 or should I say negative 2.

7? It would be to add 2.

7 or to add positive 2.

7.

The expressions on each side remain equal because we perform the same operation to both sides.

When we simplify, left-hand side simplifies to X.

On the right-hand side, we've got 8.

1 plus 2.

7 X is now the subject to the equation and the joy of having X as the subject to the equation is we have the solution X equals 10.

8.

Sometimes we'll need to use multiple operations to rearrange.

In the case of 17 minus Y equals 4, we'll do the inverse of negative Y.

We'll add positive Y to both sides.

Y is now on the right-hand side of the equation, and importantly, it's positive.

From there, if we add negative 4 to both sides, we're going to isolate the Y.

We're left with 17 minus 4 equals Y.

You could say we've isolated the Y or we could say we've made Y the subject of the equation.

By making it the subject, we have our solution.

We can rearrange multiplicative relationships using the same principles.

If I said to you D equals S multiplied by T, I think you know that relationship.

We can isolate the S if we wanted to by doing the inverse of multiply by T.

What's the inverse of multiply by T? Divide by T.

Again, we perform the same operation to both sides to maintain equality.

And on the right-hand side, to multiply by T and to divide by T, well they will cancel and the right-hand side simplifies to just S.

And on the left-hand side, we've got D divided by T, which we'd write as D over T.

We've now made S the subject of the formula.

We started with D expressed in terms of S and T and ended up with a different perspective.

We've now got S expressed in terms of D and T.

When working with formula like this one, it's nice to know that you can change the perspective by rearranging and changing the subject.

The command word that you'll typically see in algebra is make S the subject of the formula, so we might see the formula D equals S times T and be asked to make S the subject.

D is currently the subject.

Inverse operation will isolate the S and we end up with D over T equals S.

By rearranging, we made S the subject of the formula.

So look out for that command.

Quick check you've got this.

I'm gonna do a couple of examples and then give you a couple of similar examples.

I'm asked to make X the subject.

I've got the equation X plus Y equals 10 and I want to make X the subject.

I want to isolate X and have it expressed in terms of the other terms and variables.

I'm gonna perform the inverse of add Y.

I'm gonna add negative Y to both sides.

That leaves me with X equals 10 minus Y.

I've made X the subject.

For my second question, I'm asked to make M the subject in the formula F equals M multiplied by A.

I need to isolate that M, so I'm gonna perform the inverse of multiply by A.

That is I'm gonna divide both sides by A.

That will lead me with M on the right-hand side, F divided by A on the left-hand side, which we would write as F over A, and there's my result.

I've made M the subject.

Your turn now.

Pause, have a go at these two questions.

Your method when you write it down should look just like mine.

See you in a moment for the answers.

Welcome back.

Let's see how we got along.

In the first equation, you're asked to make Y the subject.

You'd perform the inverse of add X.

That is to add negative X to both sides.

Then you'd be left with Y equals eight minus X.

You've made Y the subject.

Secondly, you're asked to make L the subject in the formula A equals L multiplied by W.

To isolate that L, we'll divide both sides by W and we'll end up with A over W equals L.

That's L expressed in terms of A and W.

That's L being the subject.

Practise time now.

Question one.

In each equation or formula, change the subject as instructed.

So for example, in question A, you have the equation X minus 2.

7 equals 8.

1 and the command make X the subject.

Isolate that X.

Make it the subject of that equation.

Do that for all six of those questions.

Pause and do that now.

Question two.

I'd like to rearrange this formula as instructed.

The formula is F minus E plus V equals 2.

I wonder if you've seen this one in your learning of geometry.

Hmm.

You might wanna look it up if you haven't.

It's a fascinating one.

Anyway, F minus E plus V equals 2.

For part A, I'd like you to rearrange to make F the subject.

For part B, I'd like you to make V the subject.

And for part C, I'd like you to make E the subject.

Pause and do that now.

Feedback time now.

Question one.

I was asking you to change the subject as instructed.

Part A, with the equation X minus 2.

7 equals 8.

1 and were told to make X the subject.

You should have added 2.

7 to both sides and that'll simplify to X equals 8.

1 plus 2.

7.

From this position having X as the subject, we've actually got the solution to the equation, X equals 10.

8.

For part B, you given the formula P equals 3L and told to make L the subject.

Currently, L is being multiplied by three.

We need to do the inverse.

We'll divide both sides by three.

That will isolate the L.

We end up with P over 3 equals L.

L is the subject Part C, we're asked to make Y the subject of that equation, 17 minus Y equals 4.

We'll start by adding Y to both sides.

That means Y is now positive on the right-hand side.

We can now add negative 4 to both sides and we end up with 17 minus 4 equals Y.

Having made Y the subject of that equation, we've reached a solution, Y equals 13.

For part D, I asked you to make W the subject of the formula A equals L times W.

To isolate that W, we'll divide both sides by L and we'll end up with A over L equals W.

After lots of practise, you'll be able to see instantly the family of rearrangements when you see an equation in this form.

If we have the equation 17 minus Y equals 4, you know that's born of 4 plus Y being 17.

Therefore, Y plus 4 equals 17 or 17 minus 4 equals Y.

If you can quickly see that family of rearrangements, you might be able to notice that the 4 and the Y have swapped positions.

If you get really familiar with this, it's good to think of it this way rather than having to write all those steps down because we do like to be efficient with our mathematics.

For part E, you were asked to make U the subject in the formula V equals U plus AT.

We need to do the inverse of add AT.

We'll subtract AT from both sides.

That'll give us V minus AT on the left-hand side.

U, on the right-hand side, we've made U the subject.

For part F, we're asked to make R the subject for C equals 2 pi R.

We need to do the opposite of multiplying by 2 pi.

That is we'll divide both sides by 2 pi.

On the left-hand side, we'll get C over 2 pi and that'll be equal to R.

We've made R the subject Question two.

Part A, I ask you to make F the subject of this formula.

In order to do so, I'm gonna do the inverse of negative E and the inverse of add V.

If I do the same to both sides, I'd be left with F isolated on its own on the left-hand side and 2 plus E minus V on the right-hand side.

I've made F the subject.

To make V the subject is gonna look awfully similar.

I need to do the inverse of negative E and the inverse of add F.

That'll leave V isolated on the left-hand side, so I'll do those operations to both sides of the equation and I'll be left with V equals 2 plus E minus F.

I've made V the subject.

For part C, a little bit trickier this one.

I ask you to make E the subject.

And on the left-hand side, we currently have a negative E term.

A good start, therefore, is add E to both sides.

That will rearrange the formula to F plus V equals 2 plus E.

In this form, we can easily isolate that positive E by subtracting two from both sides and we're left with F plus V minus 2 equals E.

E expressed in terms of the other terms and variables, E is the subject.

Well, here we are at the end of the lesson now.

In summary, we can apply inverse operations to simple formula in order to make a specific variable the subject.

For example, in this formula, D is the subject.

D equals S times T.

The inverse of multiply by S will make T the subject.

The inverse of that being to divide both sides by S.

There will be times where this will be really useful because when we're in the form D over S equals T, we have a different perspective on that formula.

Hope you enjoyed today's lesson and I look forward to seeing you again soon for more mathematics.

Goodbye for now.