Lesson video

Hi there now, this is Ms. Bridgett.

In today's lesson, we're going to be looking at exploring systems of equations.

You'll need a pen, you'll need some paper, and just take a moment to make sure you remove any distraction.

Okay, off we go.

The shape on screen is called an arithmagon.

The numbers three, eight, 11, five, 13, and eight are in it.

Have a look at the arithmagon and see if you can work out a connection between the numbers in the circles and the numbers in the rectangles.

Pause the video and have a think about it.

Okay, hopefully you noticed that there was a connection between the numbers in the circles and the numbers in the rectangles.

The numbers in the rectangles are created by summing together the numbers in the circles that it is attached to.

For example, the number eight on the left is created by summing together the three and the five.

The number 11 is created by summing together the three and the eight in the circles that it's attached to.

And the number 13 is created by summing together the five and the eight, again, in the circles that it is attached to.

Okay, in this example, the way the arithmagon is presented has changed.

We've got some of the numbers, eight, 11, and six, but some of the numbers at the moment are unknown.

We don't know what P is, we don't know what Q is, and we don't know what R is.

Now, in the last lesson, we were looking at scenarios where we had more than one unknown.

The number of pens was unknown, or the number of rulers was unknown.

In this scenario, our unknowns are part of a system.

They're part of connected equations.

P plus Q is equal to eight, P plus six is equal to 11, and Q plus six is equal to R.

Now, those equations are connected.

P is appearing in the first equation and in the second equation.

Whatever the value of P is, it must satisfy that first equation and that second equation.

Q is also in the first equation and it appears in the third one.

Whatever the value of Q is, it must satisfy that first equation and that third equation simultaneously.

And that's why these sets of equations, these systems of equations, are called simultaneous equations.

What I would like you to do is see if you can work out the value of the unknowns.

Pause the video and see if you can work out the value of P, Q, and R.

Okay, I started with the second equation.

I started with P plus six is equal to 11, and the reason that I started with that is because there's only one unknown in it, so I can see, I can work out that the value of P has to be five.

Five plus six is equal to 11.

From there, I moved to the first equation, P plus Q Is equal to eight.

I now know that P is five, so Q must be equal to three.

Five plus three is equal to eight.

Finally, I'm going to move on to the third equation.

Here, I know that Q plus six is equal to R.

We've worked out that Q is equal to three.

Three plus six is equal to R.

R is equal to nine.

Here, we've got three arithmagons, each with three different unknowns in them.

What I would like you to do is to find the value of the unknowns in each.

Remember that the number in the rectangle is created by the sum of the values in the circle adjacent to it.

That eight in that first arithmagon is equal to the sum of the P and the five that it's connected to.

For the third one, I would also like you to see if you can come up with more than one way of solving it.

Okay, pause the video, off you go.

Okay, let's have a look at the answers.

Let's start off with that first arithmagon.

Looking at that arithmagon, I think P was probably the most straightforward one to find.

I know that P plus five is equal to eight, so I know that the value of P must be three.

And then I'm going to move on to Q.

From the bottom of that arithmagon, Q plus five is equal to five, which means the value of Q must have been zero, and then finally, I know that R is created from the sum of zero and three.

Moving on to that second arithmagon, I'm going to start with Q.

Now the reason I'm going to start with Q is because I know that is created from the three and the seven, so I've got an equation involving only one unknown.

I know that Q is the sum of three and seven.

It must be 10.

Moving on to the bottom row, again, I've got an equation just involving one unknown.

I know that R plus seven is equal to five.

That means the value of R must be equal to negative two.

Finally, I know that the sum of negative two and three is going to be equal to P, so P must be equal to one.

Now, the third arithmagon is quite tricky.

I think it's trickier than the other ones because all of those equations involve two unknowns.

P plus Q is equal to 12, Q plus R is equal to nine, P plus R is equal to 11.

There isn't a straightforward starting point like there was in the first two.

The answers to it are seven, four, and five.

Now, I wonder how you went about solving it.

Did you do it by trial and error? Did you just try out numbers 'til you found out some that worked? Maybe you drew a diagram.

Maybe you did something different.

What we're going to look at over the next few lessons is more formal ways of solving this.

For our final task in this lesson, we're going to do some more exploration within the arithmagons.

What I want you to think about is what would happen to the numbers in those rectangles if one of the numbers in a circle is multiplied by two.

Take one of those numbers in a circle, multiply it by two, what is the knock-on effect to the rest of the arithmagon? What would happen to the numbers in the rectangles if we multiplied all of the numbers in the circles by two? Take a moment to think about it and then see if your prediction is correct.

Pause the video and have a try.

Okay, let's have a look at that first question.

What happens to the numbers in the rectangles if one of the numbers was multiplied by two? If, for example, I took that number right at the very top of the arithmagon, three, and I multiply it by two, I'm going to get six.

Now, that's going to have an impact on the rectangles that it's connected to.

That number in that rectangle is now going to increase by another three to become 14.

The same with the eight on the other side.

The eight is connected to that circle that I've just meddled with, so this is now also going to be increased by three.

The 13 is going to be left completely alone there.

If I multiply one of the numbers by two, the numbers in the connecting rectangles are going to increase, and the amount that they're going to increase by is going to be equal to the value of the number in that circle.

The third rectangle isn't going to be altered.

Now, something different happens if I multiply all of the numbers by two.

If I multiply all of the numbers by two, all of the numbers in the circles by two, I'm going to get six, I'm going to get 16, I'm going to get 10.

Now, because I've changed all of those numbers in all of those circles, all of the rectangles are going to be impacted.

11 is now going to be become 22, the bottom rectangle is now going to become 26, and the rectangle on the right is now going to become 16.

Now, all of these numbers have doubled.

The number in the original rectangle that we looked at, the 11, has been increased by three and it's been increased by eight, which is going to have doubled it.

I wonder what would happen if we were to multiply all of the numbers by three.

I wonder what would happen if we were to multiply all of the numbers in the rectangles by two.

Unfortunately, we're not going to look at that today.

Thank you so much for your time today and I will see you in the next lesson when we go on to do a little bit more work on simultaneous equations.