Lesson video

Thank you for joining us in this lesson.

My name is Ms. Davis, and I'm gonna help you as you work your way through this lesson.

There's some really exciting algebra coming up so make sure that you've got everything you need and that you're really looking forward to getting stuck in.

Let's get started then.

Welcome to this lesson on combining equations.

Some of what we're doing in this lesson, you might find quite straightforward.

You might think some of it's a little bit easy.

If you're wondering why it is that we are doing this, what we're doing is we're exploring the structure behind some of the skills that we're gonna use for some trickier algebra problems. If we can work out how the structure works for simpler problems, then we can then apply that to trickier problems in the future.

So I want your focus today to be really thinking about laying out your working and what it is we're doing and why it works.

By the end of the lesson, you'll be able to additively combine two equations to create a third equation.

There's a couple of keywords that we're gonna use lots today, so pause the video and make sure you are happy with those.

Our new keyword for today is elimination.

Now in mathematics elimination is a technique to help solve equations simultaneously, and it's where one of the variables in a problem is removed, and we'll say then that that variable has been eliminated.

This lesson is split into 3 parts.

We're gonna start by adding and subtracting equations.

So here's a simple numerical equation.

3 plus 8 equals 11.

Notice I've given it a number and that's so we can use that number in the solutions, so we can see what equations we are manipulating.

What we can do is we can add the same value to both sides to maintain equality.

You'll have seen this before.

So we know that 3 plus 8 plus 5 is gonna equal 11 plus 5.

We've also seen that we can multiply both sides of the equation to maintain equality.

3 plus 8 all multiplied by 2 it's gonna be the same as 11 multiplied by 2.

And you can check that still works.

6 plus 16 equals 22.

Yes it does.

So this time we have two equations.

Instead of manipulating them separately, what would happen if we added them together? So we're gonna do equation number one, add equation number two, and we're gonna add the different elements of the equation.

So 3 add 7 is 10, 8 add 2 is 10, 11 add 9 is 20.

We have a new equation.

Is it still true? Yeah, of course it is.

10 plus 10 is 20.

So Laura says, "We haven't applied the same operation to both sides.

How do I know this always maintains equality." Pause the video, have a think what have we just done? Okay, well, we added to the left hand side 7 plus 2, we did 3 plus 8 and then we added 7 plus 2.

It doesn't matter that we added the 7 to the 3 and the 2 to the 8, in total we added 7 plus 2.

To the right hand side we added 9, but hang on a minute, 7 plus 2 equals 9.

So we have added the same value to both sides and therefore we have maintained equality.

That's gonna be an important point moving forward and we'll revisit it again when we've got some variables.

So let's try this one.

5 subtract 1 is 4, 3 plus 4 equals 7.

If I change that first one to 5 add negative 1, just to make the process a bit easier, and then add the elements.

Well we've got 8 plus 3 'cause negative 1 add 4 is 3, equals 11.

And again, just check that works.

Of course it does.

Right, quick check then.

These two equations have been combined but the new equation is not valid.

Can you spot the mistake? Well done if you spotted there was a mistake with the negative values, that should be negative 1 add 6 which is of course 5, not negative 7.

So that should read 8 add 5 is 13 and that is true.

Now this idea will work with subtraction as well.

So if I do 1 subtract 2, so 14 subtract 10, 6 subtract 3, and 20 subtract 13, I get 4 ad 3 equals 7.

Hopefully you are happy that that is still true.

We've made a third equation that is still valid.

Izzy says, "Subtraction is not commutative.

What if we do 2 subtract 1 instead? Well, let's have a look.

10 subtract 14 is negative 4.

3 subtract 6 is negative 3.

13 subtract 20 is negative 7.

Notice in my notation, I've written that I'm doing equation two subtract equation one, and it is still true.

It is a different equation, it's not the same equation but it is still valid.

It is still true.

What you need to be careful with when you're exploring these is the subtraction of negative values.

So we've got 1 subtract 2 this time, and let's just go really slowly, make sure that we're doing our subtraction correctly.

So 5 subtract 6 is negative 1.

4 subtract negative 10, 'cause that bottom equation could be written as 6 add negative 10.

So 4 subtract negative 10 is 14, it's the same as 4 add the additive inverse, so 4 add 10.

Then 9 subtract negative 4 is the same as 9 add 4 which is 13.

And we should know that we've done it right 'cause our new equation should be true, is negative 1 add 14 13? Yes.

So as long as we are careful with our negative numbers, we can add or subtract any two equations and maintain equality.

For numerical values, again, you might be wondering what's the point? What does that do? When we come to look at equations with variables, you'll start to see why this can be useful to us.

Okay, which of these equations has been subtracted from 6 plus 7 equals 13 to get 2 plus 5 equals 7? Read that again and then answer the question.

Of course it's 4 plus 2 equals 6.

All of those final equations were valid, but the only one that was a subtraction to get from 6 plus 7 equals 13 to 2 plus 5 equals 7 was B.

Which of these equations has been subtracted from 5 plus 3 equals 8 to get negative 2 plus 5 equals 3? Give this one a go.

Being careful that we're thinking about subtraction here, it is C.

5 subtract 7 is negative 2, 3 subtract negative 2 is 5, 8 subtract 5 is 3.

Fantastic.

Time for you to have a practise.

I would like you to add each term in the following equations to create a new equation.

Just check that your new equation is still true, otherwise you've made a mistake somewhere.

Have a go at those ones and then I will have a look at some subtraction.

So as promised, we're now gonna have a look at subtracting.

I would like you to subtract each term in the following equations to create a new valid equation.

Notice for each one I've written equation subtract equation two.

So make sure you are doing it that way round, and again, check your new equation is valid.

Off you go.

Well done.

Pause the video and check your answers, making sure that you've got a valid final equation.

I wanna draw your attention to question C.

Notice that the second terms when we added them together, the negative 3 and the 3 we got 0.

That's absolutely fine, is gonna be a useful point moving forwards.

For our subtractions, again, pause the video and check your answers.

Drawing your attention again to C where we noticed that that second term you subtracted them both in the original equations, got a 0 in the new equation, the same with F as well.

Bear that in mind whilst you're marking your answers.

Right, so now we're gonna use the same concept but we're gonna combine equations containing variables.

In these equations, x must represent the same value in both equations.

So we've got 2x plus 5 equals 11, and x plus 5 equals 8.

If we add these equations together, we can add them in any way, however it's gonna make most sense to add the x terms, and add the constant, and then add the right hand side of the equation.

So we've got 3x plus 10 equals 19.

If the value for the variable is a solution to one equation, then it should be a solution to all those equations.

Izzy reckons that x is 3.

I wonder if you agree.

Let's check it's valid for all our equations.

2 lots of 3 plus 5 is 11, and that does give us 11.

For two, 3 plus 5 does give us 8.

And for our new equation, 3 lots of 3 plus 10 is 19, and that is true.

Let's check that we've definitely maintained equality.

What have we added to the left hand side of equation one to get the new equation? Pause the video.

Right, well, we've added x plus 5, haven't we? We added x to the x term and 5 to the constant term, so in total we've added x plus 5.

What have we added to the right hand side of equation one to get the new equation? Well, we've added 8.

But remember, equation two tells us that x plus 5 is 8.

So the value we've added to our left hand side and the value we've added to the right hand side are the same.

Therefore we've maintained equality by adding the same value to both sides.

They were just indifferent forms. What that means is we can now do that for anything we know that will hold true.

The same works for subtraction.

Let's do 1 subtract 2.

So 2x subtract x is x, 5 subtract 5 is 0, 11 subtract 8 is 3.

And now look, we can now see that x is 3, which is what Izzy thought it was in the previous equation.

We can do exactly the same if the equations have two variables.

So we got 3x plus 5y is 11, and 2x plus 3y equals 7.

If I add them I could get 5x plus 8y equals 18, and I've maintained equality because I've added the same thing to both sides.

I've added 7 to both sides, just on the left hand side is in the form 2x plus 3y.

We can do the same as subtraction.

3x subtract 2x is x, 5y subtract 3y, it's 2y, don't forget to also do 11 subtract 7 is 4.

So we've maintained equality by adding or subtracting expressions of equal value to both sides.

There will be a solution for x and y which will work for all of those equations.

We just dunno what it is yet.

So Izzy wants a subtract these two equations.

Can you spot what mistake she has made? She has done some really dodgy algebra here.

She's subtracted coefficients of unlike terms, you cannot write 2ys subtract x as y.

That does not work.

You can't write as a single term.

She's given it another go.

Is this now correct? Take a moment to read over what she's written.

Yes, technically it is, she's done 2y subtract x and written it as 2y subtract x.

She's done 3x subtract negative 4y and written it as 3x plus 4y and she's done 27 subtract 2.

However, you might have said that she could have done this a lot simpler by adding like terms, so she could simplify now by adding like terms or she could have rearranged the equations first so that the order of the variables are the same in both equations.

Let's see what that looks like.

So Laura spotted that and said, "We could write this as 2y plus 3x equals 27 and negative 4y plus x equals 2.

Notice that second equation is the same as it was before, just with the terms the other way around.

Let's see what happens when Laura does this.

Well, that looks a lot easier.

We can now subtract the like terms and get a simple equation.

So there are two equations on the right hand side, which is the new equation when these are added together? Well, if you spotted it was 12y minus 5x equals negative 3.

1.

Just being careful with our negative number skills.

Again, these two equations are being added together.

Look carefully at the variables, which is the new equation? Hopefully I didn't catch you out here.

We need to write the variables the other way round as negative 3a plus 4b equals negative 3, and then it made it easier.

You should get a plus b equals 8.

Time for a practise then.

Be really careful, especially with your negative values, take your time with these.

For each of these you are adding equations one and two together.

Give those a go and then we'll have a look at the next step.

Well done.

Same again this time with subtraction.

Give that a go and come back when you're ready for the next bit.

And finally, here are three ways to combine the same two equations.

Look carefully at my working and finish off the working.

Can you then write me a sentence about what you notice from b and c? Come back when you're ready to look at the answers.

Well done.

Pause the video and check you've got the correct answers for each, particularly with e where the variables were written the wrong way round, and for f where you might have noticed that negative 2.

5q add 2.

5q got you 0q, which simplified our equation.

Once you check your answers, move on and we'll check the next bit.

Good.

So pay attention to our negative values, particularly with this set.

I'd like to draw your attention to a, where you notice that subtracting a this time gave us the equation b equals 4, and also with d we're subtracting negative 4h subtract negative 4h gives us 0h.

Mark those answers and then we'll have a look at the next bit.

And finally have a check of your answers for question three.

For question four, you might have noticed that b and c are very similar equations.

If you multiply b through by negative 1 you get the same equation as c.

What we've done is we've subtracted them the other way around.

So every term has got the opposite sign to what it did in the previous question.

Once you're happy with those, we'll move on to the final part of the lesson.

Right, I really hoping you're starting to see where this might be useful.

We touched on this idea of eliminating a variable in the last part of the lesson.

We're really gonna focus on it now.

So some choices for combining equations can create a new equation which is simpler than either of the starting equations.

You might have spotted that in the last task.

Let's look at an example.

What would happen if we subtracted these equations? Take a moment to give this a go.

So we should have x plus 0 equals 2.

If you subtracted them the other way, you would've got negative x plus 0 equals negative 2.

We now have a simple equation where we can see the solution for x.

We know straight away that x must be 2.

Have a think, can you explain why this has happened? What elements of that question has allowed this to happen? Well, there was the fact that we had a term of 5y in both equations, and 5y subtract 5y is 0.

So we got a new equation with no y term.

We call this eliminating the variable.

By subtracting these equations, we had eliminated y and got an equation with just x terms. Then we can find the solution for x.

We can see that x is 2.

Let's have a go at subtracting these equations.

2x subtract 2x, 0.

4y subtract y, it's 3y.

14 subtract 8, it's 6.

Again, because the x terms this time were the same, when we subtracted the equations, we eliminated the variable x.

We now have an equation for y, 3y equals 6, which will allow us to find out what y is.

We know that y must be 2.

Let's have a look at this one.

2x minus 4y equals 2.

Which of these equations could we subtract to eliminate a variable? What do you think? Well done if you spotted that it's 12x minus 4y equals 32.

All three of these equations were equivalent, you might have noticed that.

However, the one that's gonna allow us to eliminate the variable when subtracted from equation one is that final one.

The reason being it's the y terms are the same.

We've got a negative 4y and negative 4y.

So when they're subtracted they'll be eliminated.

Negative 4 subtract negative 4 is 0.

Izzy says, "This will be easier if I do 2 subtract 1.

Will that still work?" Of course it is.

We tried that with the previous questions.

You may wish to rewrite them the other way around, particularly if you're looking at trickier questions with a lot more steps, it may help to make sure you are writing your equations the correct way round.

Make sure you number them so you know which one is which.

So now we're gonna do 2 subtract 1, and that gives us 10x equals 30.

<v ->Of course, you could now tell me the solution for x</v> if we wanted it.

Have a look at this one.

Laura says, "This is slightly different.

Will I still eliminate the y variable if I subtract them?" Have a look, maybe give it a go.

What do you think? 2x subtract x doesn't eliminate x, that's fine.

Negative 3y subtract 3y, well hang on a minute.

Negative 3y subtract 3y is negative 6y.

So we didn't eliminate the ys either.

So no, the coefficients were not the same.

The new equation is valid, there's nothing wrong with it, but it doesn't eliminate a variable.

So it's not gonna help us solve the equation.

Let's have a look at what we can do instead.

So the coefficients of y are a 0 pair.

That means that they sum to 0.

You guessed it, we could add these equations instead to eliminate the y variable.

So 1 add 2, 3x, negative 3y add 3y is zero, 13 add 11, don't forget we're adding, is 24.

3x is 24 and you could now solve that.

Quick check then, true or false, subtracting these equations will eliminate a variable? You'll need to read the equations and then think about a justification.

Off you go.

Yeah, that's true.

The y terms have the same coefficient, so subtracting them will eliminate.

Have a look at these ones.

Again, true or false, subtracting will eliminate a variable? What do you think? Right, I've been a little bit mean here and written the terms in a different order.

Well done if you spotted it was false.

If we rewrite the top one, we'd get negative y plus 2x equals 5.

And then adding would actually eliminate the y variables because they are a zero pair, negative y add y gets you 0.

Okay, same again, subtracting these equations will eliminate a variable, decide whether it's true or false and then we'll think about the justifications? Well, if you spotted that it was true.

If you said false, just have a think about the justifications and we might clarify that in a moment.

What is the correct justification for y subtracting will eliminate a variable? Well done if you spotted that the x terms have the same coefficient, so 3x subtract 3x will give you 0x.

Now, the reason why this might have thrown a few of you is because adding would also eliminate a variable.

Subtracting eliminates the x variable, whereas adding eliminates the y variable.

It's really important that you get used to rearranging those equations so you can spot these things.

Right time for you to have a go.

I'd like you to add and subtract these pairs of equations and then tell me which option eliminated a variable and why? Off you go.

Well done.

Exactly the same again, give that a go.

And for two, a little bit of a puzzle.

I'd like you to pick two different equations from the options that satisfy each condition.

You can only use each equation once.

So if you've used it for question A, you can't also have it in question B.

Have a play around with that one and then we'll look at the answers.

Let's have a look at these answers then.

Pause the video and have a look at the equations and then I'll bring your attention to some of the answers.

So for A, subtracting eliminated the variable a as the coefficient of a in both equations was the same.

You add 2a and 2a.

In B, you'll notice that I decided to rewrite the equation, so the variables were the same way around in both equations.

And then adding eliminated the variable b as the coefficients of b in both equations are a zero pair.

For C, again, pause the video to check your equations, and then we'll run through our explanations.

So for C, neither option eliminated a variable as the coefficients of a were neither identical nor was it a 0 pair.

And the same applies for the coefficients of b.

For D, both options eliminated a variable because the coefficients of a were the same, so subtracting eliminated a, but the coefficients of b were a 0 pair, so adding eliminated b.

And finally for A, there were lots of options that eliminated a by adding, but because we needed the other ones in later questions had to be a plus 2.

5b equals 29, and 4.

2b minus a equals 38.

And then adding will eliminate a.

For B then, 3a plus 2b equals 32, and 2a minus 2b equals negative 12 will allow us to eliminate b by adding.

And C, 6a minus 0.

5b equals 19, and 5a minus 0.

5b equals 15.

B is eliminated by subtracting.

Doesn't matter that they're both negative terms, subtracting will eliminate them 'cause they're both the same term.

Fantastic.

So today we've looked at how two equations can be combined into a third equation by adding or subtracting.

It may not be something you thought about doing before.

What that is gonna allow us to do is it's gonna allow us to solve simultaneous equations in the future.

It'd be great to see you back to look at how we're going to do that.

Please join us again.

Thank you very much.