Lesson video

Thank you for joining us in this lesson.

My name is Ms. Davies 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 going to 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 three parts.

We're gonna start by adding and subtracting equations.

So here's a simple numerical equation, three plus eight 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 three plus eight plus five is gonna equal 11 plus five.

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

Three plus eight or multiplied by two, it's gonna be the same as 11 multiplied by two.

And you can check that still works.

Six 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 three add seven is 10, eight add two is 10, 11 add nine is 20.

We have created 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, seven plus two.

We did three plus eight and then we added seven plus two.

It doesn't matter that we added the seven to the three and the two to the eight.

In total, we added seven plus two.

To the right hand side we added nine, but hang on a minute, seven plus two equals nine.

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.

Five subtract one is four, three plus four equals seven.

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

Well we've got eight plus three 'cause negative one add four is three 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, if you spotted there was a mistake with the negative values, that should be negative one add six, which is of course five, not negative seven.

So that should read eight add five is 13 and that is true.

Now this idea will work with subtraction as well.

So if I do one subtract two, so 14 subtract 10, six subtract three and 20 subtract 13, I get four add three equals seven.

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 two subtract one instead? Well, let's have a look.

10 subtract 14 is negative four.

Three subtract six is negative three.

13 subtract 20 is negative seven.

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 one subtract two this time and let's just go really slowly make sure that we're doing our subtraction correctly.

So five subtract six is negative one.

Four subtract negative 10 'cause that bottom equation can be written as six add negative 10.

So four subtract negative 10 is 14.

It's the same as four add the additive inverse so four add 10.

Then nine subtract negative four is the same as nine add four which is 13.

And we should know that we've done it right 'cause our new equation should be true.

Is negative one 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 six plus seven equals 13 to get two plus five equals seven? Read that again and then answer the question.

Of course, it's four plus two equals six.

All of those final equations were valid, but the only one that was a subtraction to get from six plus seven equals 13 to two plus five equals seven was B.

Which of these equations has been subtracted from five plus three equals eight to get negative two plus five equals three? Give this one a go.

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

Five subtract seven is negative two, three subtract negative two is five, eight subtract five is three.

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 we'll have a look at some subtraction.

So as priced, 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 one subtract equation two.

So make sure you are doing it that way round.

And again, check your new equation is valid.

Off you go.

And finally, a little bit of a puzzle.

I'd like you to fill in the missing expression in each calculation.

Pay attention to my notation because A and B you'll notice we're doing one add two and C and D, you'll notice we're doing equation one subtract equation two.

Have a go at that one and then we'll have a look at the answers.

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 three and the three, we got zero.

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 notice that that second term when you subtracted them both in the original equations got a zero in the new equation.

The same with F as well.

There that in mind was you marking your answers and then we'll have a look at that puzzle.

And question three, you should have negative two add one equals negative one, which is also valid.

Three subtract nine equals negative six, which is true.

Eight subtract one equals seven and finally five subtract six equals negative one.

Well done for spotting that negative six subtract negative six is zero.

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 if we've got two x plus five equals 11 and x plus five equals eight, 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 three x 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 three.

I wonder if you agree.

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

Two of three plus five is 11 and that does give us 11.

For two, three plus five does give us eight and for our new equation, three lots of three plus 10 is 19 and that is true.

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

What if 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 five, haven't we? We added X to the X term and five to the constant term.

So in total we've added x plus five.

What have we added to the right hand side of equation one to get the new equation? Well we've added eight, but remember, equation two tells us that x plus five is eight.

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 in different 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 one subtract two.

So two x subtract X is x, five subtract five is zero, 11 subtract eight is three.

And now look, we can now see that X is three, 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've got three x plus five y is 11 and two x plus three y equals seven.

If I add them I could get five x plus eight y equals 18.

And I've maintained equality because I've added the same thing to both sides.

I've added seven to both sides.

Just on the left hand side is in the form two x plus three Y.

We can do the same as subtraction.

Three x subtract two x is x, five y subtract three y is two y.

Don't forget to also do 11 subtract seven is four.

So we've maintained the quality 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 don't know what it is yet.

So Izzy wants to 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 two y 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's is.

She's done two y subtract X and written it as two y subtract X.

She's done three x subtract negative four Y and written it as three x plus four Y and she's done 27 subtract two.

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 two y plus three x equals 27 and negative four y plus x equals two.

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, it was 12 y minus five x 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 around as negative three A plus four B equals negative three and then it made it easier.

You should get A plus B equals eight.

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 set.

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.

5 Q add 2.

5 Q got you zero Q, 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, but you notice that subtracting A this time gave us the equation B equals four and also with D, we're subtracting negative four H subtract negative four H gives us zero H.

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 one, 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 happen with those, we'll move on to the final part of the lesson.

Right, I'm 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 zero equals two.

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

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

We know straight away that X must be two.

Have a think.

Can you explain why this has happened? What elements of that question has allowed this to happen? Well, it was the fact that we had a term of five y in both equations.

And five y subtract five Y is zero.

So we got a new equation with no Y term.

We call this eliminating the variable.

By subtracting these equations, we have eliminated Y and got an equation with just X terms. Then, we can find the solution for X.

We can see the X is two.

Let's have a go at subtracting these equations.

Two x subtract two x, zero.

Four y subtract Y, it's three Y.

14 subtract eight is six.

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 three Y equals six, which will allow us to find out what Y is.

We know that Y must be two.

Let's have a look at this one.

Two x minus four y equals two.

Which of these equations could we subtract to eliminate a variable? What do you think? Whether if you spotted that it's 12 x minus four Y 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 the Y terms are the same.

We've got a negative four y and a negative four Y.

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

Negative four subtract negative four is zero.

Izzy says this will be easier if I do two subtract one.

Will that still work? Of course it's.

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 two subtract one and that gives us 10 x equals 30.

Of course, you could now tell me the solution for x 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? Two x subtract X doesn't eliminate X, that's fine.

Negative three y subtract three y.

Well hang on a minute, negative three y subtract three Y is negative six Y.

So we didn't eliminate the Y's either, so know 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.

The coefficients of Y are a zero pair.

That means that they sum to zero.

You guessed it.

We could add these equations instead to eliminate the Y variable.

So one add two, three x negative three y add three Y is zero, 13 add 11, don't forget we're adding is 24.

3 x 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, I don't know if you spotted it was false.

If we rewrite the top one, we'd get negative Y plus two x equals five.

And then adding would actually eliminate the Y variables because they are a zero pair.

Negative Y add Y gets you zero.

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, then 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, if you spotted that the X terms have the same coefficient, so three x subtract three x will give you zero x.

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 two A and two A.

In B, you'll notice that I decided to rewrite the equation so the variables were the same way round 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 a zero 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 zero 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.

5 B equals 29 and 4.

2 B minus A equals 38 and then adding will eliminate A.

For B then, three A plus two B equals 32 and two A minus two B equals negative 12 will allow us to eliminate B by adding.

And C, six A minus naught 0.

5 B equals 19, five A minus 0.

5 B 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!.