Lesson video

Thank you for joining us in this lesson.

My name is Ms. Davis and I'm going to 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.

In this lesson, we are going to be forming simultaneous equations.

Don't worry if you don't know what a simultaneous equation is yet.

We're going to start from things that we do know and then we're going to have a look at what this new phrase means.

By the end of the lesson, you'll be able to derive two different simultaneous equations from a context.

So equations which represent different relationships between the same variables are called simultaneous equations.

We're going to look at loads of examples so you can get your head around what these are.

So we're going to start by reminding ourselves of how to form and interpret equations in a context.

We're going to have a look at some really interesting problems and how we can use our algebra skills to solve those.

So letters can be used to represent unknowns and variables in context.

Sofia says, "An expression for the cost of visiting the local funfair is 2.

5 plus 1.

5r." To make sense of Sofia's expression, we need to know what the variable represents.

In this case, we don't know what R actually is.

We can make some guesses, but we don't know for sure.

Sofia has told us then, "R is the number of rides you go on." So if you look again at this expression, it's 2.

5 plus 1.

5 times the number of rides you go on.

So if you go on five rides, it's going to be 2.

5 plus 1.

5 times five.

Okay, we've got more of an understanding as to what this expression is telling us.

What do you think the values 2.

5 and 1.

5 mean? What else might we need to know to make this a valid expression? What do you think? So 2.

5 could be something like the cost to enter the fair.

It's not dependent on anything, it's just a fixed cost of 2.

5.

The 1.

5 is probably something like the price per ride.

'Cause we know we have to multiply it by r to get our overall cost.

What we could do with knowing is something like a unit of measure.

What are we measuring this in? It's likely to be pounds.

So it's likely to be two pound 50 and one pound 50.

We don't know for sure without that information.

The important thing to be aware of going forward is that we need to know what these variables mean in context to make sense of these problems. Where we have a statement of equality, we can form an equation.

So Sofia just had an expression.

We're now going to think about an equation.

So Andeep bought five apples with a five pound note and got two pound 40 change.

I wonder how much an apple costs? So we can write an equation for Andeep's purchase.

Remember we can use a letter to represent an unknown.

So let's use N.

We can use any letter we like and that's going to be the cost of an apple.

That's what we don't know.

That means the cost of five apples could be written as five n.

Then we can form an equation five n plus 2.

4 equals five or five n equals five minus 2.

4.

The key point here is that we've said that n is the cost of an apple.

We probably need to add that it's the cost of an apple in pounds 'cause the five and the 2.

4 are representing pounds.

If we were going to solve that equation then our answer would be in pounds.

And if you think about it, cost of an apple, it's probably going to be less than one pounds.

It's probably going to be a decimal between zero and one.

Where we have multiple unknowns, we can use different letters for each unknown.

So Sofia bought three cupcakes and two biscuits and it costs two pounds 70.

Again, to form an equation, we need to say what the unknowns are and what they're representing in context.

So how could we form an equation for Sofia's purchase? Pause the video.

What would you write? I went with three a plus two b equals 2.

7.

In my equation, what do the unknowns represent? She might have said something like a is the price of a cupcake.

You can't just write a is cupcakes.

A is specifically the price of a cupcake and it's in pounds as well, 'cause you can see that I've left 2.

7, that's in pounds.

B then is the price of a biscuit.

On Andeep's stall, the price of cupcakes is 70p and biscuits are 35p.

Lucas spent two pound 10.

See if you can do this one.

How would you form an equation for Lucas's purchase? Think about what your variables are representing.

Right, you got to be a little bit careful with units here.

You'd have 0.

7a, well you can use any letter, plus 0.

35b equals 2.

1 or 70a plus 35b equals 2.

10.

You need to make sure all your values are in the same units, but remember the letters can be any letters you like.

Right, what do our unknowns represent this time? Well, a is the number of cupcakes this time.

'Cause 0.

7 was the price.

So a is the number of cupcakes and b is the number of biscuits.

See how important it is to know whether the variables are representing a price or whether it's representing a frequency, what it actually is that it's representing.

Quick check then.

If Andeep doubles his age, he would be the same age as his uncle was five years ago.

Read that again.

Make sure you're happy with what's happening.

Then which of these could be an equation for Andeep's current age, A, and his uncle's current age, B? So which of those equations shows the proper relationship between Andeep and his uncle's current age? Take your time and then we'll check your answers.

So Andeep is doubling his age.

So that gives us two a and that'd be the same as his uncle five years ago.

So two a equals b minus five.

Of course you could rearrange that into other forms, but none of those are equivalent to the ones on the screen.

You could have had two a plus five equals b.

That's just not one of our options this time.

Okay, Sofia has some 10 pence coins and some 50 pence coins.

The value of her coins is four pounds 30.

So what would a represent in the equation 10a plus 50b equals 430? What do you reckon? Perfect, a is going to be the number of 10p coins if you do 10 multiplied by the number of 10 pence coins you have, that'll tell you what the value is of those coins.

Time to put this all into practise then.

So Izzy's current age is B and her friend's current age is B.

I would like you to write an equation to represent each statement below.

Give that a go.

If you really want to challenge, you can see if you can work out what Izzy and her friend's ages must be.

Don't worry if you can't at this stage we are just focusing on forming these equations.

Off you go.

Well done.

This time, I've written the equations and I want to know what the variables represent in context.

Make sure you're using full sentences and you're being really clear what they're representing.

Give those ones a go and then we'll look at it together.

So the first one, we've got a plus b equals 30.

Second one, a subtract b is two, or we can b subtract two equals two.

We don't know who is older, so we don't know which way round we're subtracting to get a value of two.

C, two and then we need brackets round a minus one and that's equal to b plus 10.

So a minus one is Izzy's age last year and we want two lots of that.

For D, a minus 10 is Izzy's age 10 years ago and then we were going to square that.

So a minus 10 all squared equals b.

And E, you've got ab over eight equals two a.

All of those equations can be manipulated into other forms, but at the moment we just need to write them in a form so we can use those in the future.

If you did fancy the challenge, you would've found that Izzy's age had to be 14 and her friend's age 16.

Don't worry about that at all, but you could use that now to go back and check your equations work.

Okay for A, a is the length of the rectangle in centimetres and b is the width of the rectangle in centimetres or the other way round.

For B, a is the cost of an adult ticket in pounds and b is the cost of a child ticket in pounds.

Make sure you have got that the right way round.

For C, a is the number of points for winning a match and b is the number of points for losing a match.

For D, a is the number of items of furniture bought and b is the number of clothing accessories bought.

Lovely, now we're going to investigate this idea of simultaneous equations.

So where we have multiple pieces of information about our variables, we can write multiple equations similar to what we just did with Izzy and her friends ages.

So if I know that two numbers have a sum of eight and a difference of two, I've been given two pieces of information.

The fact that the numbers add up to eight and when you subtract one from the other, you get two.

So that's two different pieces of information.

Let's have a go at forming some equations.

We'll let a be the one number and b the other number, we're going to use different letters here because it's quite possible that these are going to be different numbers.

In fact, for this case they have to be different numbers 'cause they have a difference of two.

The sum of eight, we could write as a b equals eight and a difference of two a minus b equals two.

Of course we could write b minus a equals two as well 'cause we don't know which way round these numbers are going to be.

In that second equation we're representing the same numbers as in the first equation, therefore we have to use the same letters in order to be able to use this to do anything useful.

So where we used a and b in the first equation, we're going to use a and b again in the second equation.

What we've got now is we've got two equations each about a different piece of information but both containing the same variables.

If we wanted to find a solution, if we wanted to find out what a and b were, we would need values that satisfy both equations.

They'd have to add to eight and have a difference of two.

Where we have two different equations connecting the same two variables, we can solve them simultaneously.

That means at the same time.

We call these simultaneous equations.

If we just look at one equation, a plus b equals eight, Aisha's spot on.

There's infinite solutions for this.

We can have one and seven, 2.

5 and 5.

5, five and three.

But when I add another equation, most of those answers that Aisha gave won't work anymore.

They work for the top equation but not for the one underneath.

We want values now that satisfy both.

Don't worry about how you'd get that at the moment if you're interested then five and three would work.

Five plus three is eight, five minus three is two.

What's important now is how we're forming these equations.

So let's say the perimeter of this rectangle is 32.

Andeep says you can calculate the perimeter by doing a plus b plus a plus b, and that equals 32.

So he's written an equation.

Sofia says you can sum the two different sides and double.

So two lots of a plus b equals 32.

Have Andeep and Sofia formed a pair equations that can be solved simultaneously? This might require a bit of thinking.

So pause the video.

What do you think? Right, the definition of simultaneous equations is going to help us here.

No, these are not simultaneous equations because they're not two different equations.

They're actually the same equation.

That's because they both use the same piece of information.

We've only got one piece of information.

The fact that the perimeter is 32, so both their equations are actually equivalent.

Now I've added an extra piece of information.

The length is four centimetres longer than the width.

Now we can form two equations.

We can use Andeep's or Sofia's or a simplified version, which is two a plus two b equals 32.

But now we can use that second piece of information.

The length is four centimetres longer than the width to write a equals b plus four.

These now are simultaneous equations.

In a rugby match, a team can score a try worth seven points or a penalty worth three points.

In this match we're going to assume that all tries were converted, which is why they're worth seven points rather than the usual five points.

Don't worry if you don't know anything about rugby, it's not essential to answer this question.

The score at the end of the match was 33-31.

Jacob says, "I can form these simultaneous equations where a is the number of tries and b is the number of penalties." So remember, a try is worth seven and a penalty is worth three.

So he's written seven a plus three b equals 33.

He's also written seven a plus three b equals 31.

We will not be able to solve these simultaneously.

Can you work out what the problem with Jacob's equations are? Take your time over this one.

Think about what he's done.

Okay, the problem here is a and b are not representing the same value in both equations.

If you think about it, the teams cannot have scored the same number of tries and penalties, otherwise they'd have the same number of points, wouldn't they? So a and b in the first equation will not be the same as a and b in the second equation and when we have simultaneous equations, the values of the variables have to be the same in both equations.

They have to represent the same numbers.

In fact, we cannot use simultaneous equations to work out the number of tries and penalties scored.

Let's try something different.

In a tournament, teams get five points for a win and one point for a loss.

No games are going to be left on a draw.

Izzy's team plays seven games and finish on 27 points.

Let's see if we can form some equations.

Now if it's five points for a win, then five times the number of wins plus one times the number of losses must equal the 27 points that her team got.

Can you think of a second equation we could form? What do you reckon? Right, well think about what piece of information we haven't used.

We've used for 27 points.

We haven't used the fact that they play seven games.

So if a is the number of wins and b is the number of losses, then a plus b must equal seven.

'Cause the number of games they win add the number of games they lose must equal seven games that they played.

These can be solved simultaneously as they're two different equations representing two different pieces of information that we were given and they connect the same two variables.

The more you practise these, the easier it gets to see where these equations are coming from.

So Jacob has some 10 pence coins and some 20 pence coins.

He has 38 coins in total with a value of six pound 10.

Which of these equations represent this information? What do you reckon? Yeah, similar to one of the ones we saw before, wasn't it? 10a plus 20b equals 610.

Be careful with your units here.

If 10 and 20 are being pence, then we need our answer to be in pence as well, so 610.

What do a and b represent in that equation? Can you write a sentence for what a and b are? So a is the number of 10 pence coins and b is the number of 20 pence coins.

Finally then, which of these would be a suitable second equation? So we came up with one equation.

Read the question again, which would be a suitable second equation.

Good, hopefully you spotted that we haven't yet used the piece of information that he has 38 coins in total.

That means that a plus b must be 38, where a is the number of 10 pence coins, b is the number of 20 pence coins.

Together there's 38 of them.

Perfect, time for you to have a go.

So for each scenario I'd like you to form a pair of equations which can be solved simultaneously.

You don't need to solve them at this point.

You just need to write two equations.

Underneath, make sure you state what the variables are representing in the context of your question.

Have a go at those three.

Lovely, there's three more for you to try this time.

Again, make sure you write down what the variables are representing.

Off you go.

Well done with those ones.

So for question two, I'd like you to explain each pair of equations cannot be solved simultaneously to work out the unknown values.

So read the context, look at the equations I have formed, what is wrong with them? Let's have a look at our answers then.

For each one you could use any letters that you like.

So just compare them to mine where I've used a and b.

So you've got two a plus two b equals 36 and a equals two b.

A therefore must be the length, and b is the width in centimetres.

Make sure you've got a equals two b, where a is the length and b is the width.

Get that the right way around.

For B, you've got two a plus three b is 25 and three a plus two b equals 27.

A is the cost of the adult tickets and b is the cost of the child tickets in pounds.

Well done if you added your units.

For C, we've got a plus b equals 12 and a equals b plus two or a minus two equals b would work as well.

A then has to be the number of goals scored by the winning team 'cause they have more goals.

And b, the number of goals scored by the losing team.

Make sure you get that the right way round.

Pause the video if you need to look over those again.

For D, we've got a multiplied by b, which we can write as ab equals 80 and two a plus two b equals 48.

There's other ways you can write the perimeter as we saw earlier with Andeep and Sofia, I think that's the simplest.

A here is the length and b is the width or they could be the other way round.

'Cause addition and multiplication are both commutative doesn't matter whether a or b is the length or the width.

Does matter that they are in centimetres.

E, we've got a equals b plus six, and then a plus one equals two lots of b minus one, making sure your b minus one is in brackets.

A is Andeep's current age and b is his cousin's current age.

And finally we've got a squared plus b squared equals 400, and then a minus b equals four or b minus a equals four.

Doesn't matter which way around we subtract those to get their difference.

And then a is the length of one side and b is the length of the other side in centimetres.

Let's have a look at question two.

Be really nice to see some good full sentences in your work here.

Let's have a look.

So c and s are representing different things in each equation.

In the first one, c is representing five coffees, but in the second one, c is representing four coffees.

That doesn't make sense to solve them simultaneously.

You might have said something like, we need to use the key information and write five c plus three s and then four c plus four s.

For B, both equations are identical, they're just in a different rearrangement.

We haven't used the piece of information that the total amount they have is 23 pounds.

So one of those is fine for the first equation, but the second equation should use that other piece of information.

Something like x plus y equals 23.

And finally, well done if you spotted that the variables got swapped in the second equation.

We need to make sure the variables are representing the same thing in both equations.

So if the first equation n represents the number of nights and p, the breakfasts, and then that should stay the same in the second equation.

Well done and hopefully you'll join us for a future video where we can look at solving those equations and actually finding out those values for n and p.

What you've shown today is that you are an expert now forming those equations so that you won't have difficulty doing that moving forward and you get really stuck in with the maths behind finding those solutions.

Simultaneous equations is something you can use often in life, either in the workplace or going about your daily lives where you've got information that you don't know that you need to figure out.

It might surprise you, you possibly actually use some of these skills already to solve problems. We've looked today at being clear about what each variable represents, and I want you to remember that as you move forward, and we've talked about what simultaneous equations are.

You may even have learned this new phrase of simultaneous equations and we know that they have to be two different equations, but representing the same two variables.

Well done today.

I hope you're really excited to come back and join us again for another lesson.

See you soon.