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Hello, and welcome to this lesson about equating linear algebra expressions.

I'm Mr. Langton, and for today's lesson, all you'll need is a pen and paper.

Something to write with and something to write on.

Please take a moment now to clear away any distractions, including turning off any notifications on mobile devices.

And finally, if you can, try to find a quiet place where you won't be disturbed during the lesson.

Okay? So when you're ready, let's begin.

We'll start off, with a, try this activity.

The triangle and parallelogram below have equal perimeters.

What is the value of b? And what is their perimeter? I'm going to have to pause the video and have a go.

When you done all that you can, unpause it and we'll go through it together.

So pause it in three, two, one.

And get on.

I'm going to start to break this down.

We're told that the perimeters for each shape are the same, so that's where I'll start.

First, this triangle has three equal sides.

I'm just going to draw them like this.

The parallelogram has two equal pink sides and two equal blue sides.

And as we said, these two perimeters are equal.

Next thing that I'll label is lengths.

We're told the pink lengths are b centimetres.

We don't know what that is yet, but hopefully we'll find out if we keep going.

The green length is b centimetres plus six centimetres.

So we can label each green length b plus six.

And the blue length is 1.

5 centimetres longer than the green length so its b plus six plus 1.

5 and that's b plus 7.

5.

So let's start to simplify on the left hand side, the triangle, we have three bs and three sixes.

So together that's three b plus 18.

On right hand side.

There are four bs and two lots of 7.

5.

So that's four b plus 15.

And of course these two expressions they're equal.

If we know two expressions have equal value, we can equate them forming an equation.

We can solve these using written algebra or bar models.

I'm going to start off by drawing four b plus 15 as a bar model.

You see I've got my four bs and my 15, and then I'm going to draw three b plus 18 underneath that.

Now the really important thing to remember is that these bars must be equal length.

We're told that they're equal so the parts must be equal length.

I've drawn my three bs, and I've drawn my 18.

So now let's try and solve the equation, and see if we can find the value of b.

So, we can see, looking at the bars, trying to draw a line down here.

These three bs here are equal to these three bs here so I can take them both away.

And the top bar will still be equal to the bottom bar.

If I take away three b, mutual sides I'm left with b plus 15 there, and I'm just left with 18 there.

So I've got b and 15 Now these two are equal, but I can see if I split this up down here, that bit must also have a value of 15.

Mustn't it? So I can now subtract 15 from each side.

Now it's just going to leave you with b.

Yeah? And if I take 15 away from 18, you have three leftover.

So b has got a value of three.

So back to the original question.

We were first asked to find the value of b.

We've done that b equals three.

And now we want to find the perimeter.

So, we said that the pink lengths were b, each of those is three centimetres.

The green lengths were b plus six.

So three plus six is nine centimetres.

And finally the blue lengths were b plus six, plus 1.

5.

So three plus six is nine add the 1.

5 is 10.

5.

So let's check the perimeters.

over here, we've got nine centimetres plus nine centimetres plus nine centimetres, which gives us 27 centimetres.

Now with the parallelogram, we're told that the perimeters are equal.

So let's double check that we've got our three centimetres and our three centimetres is six centimetres.

We've got our 10.

5 and our 10.

5.

That's going to give me 21 centimetres.

And so in total together, six and 21 is 27 centimetres.

So in both cases, we've got the same perimeter.

We've worked it out 27 centimetres.

Now I'm going to move on to the independent task.

If you confident, then pause the video here and you can access the worksheet now.

If you want some support then leave the video playing, and I'll go through the first question with you.

If you want to get on it straight away pause the video in three, two, one, Okay? I'm going to go through this first question require a lot of detail.

Part a says, can you write an expression for the perimeter of each shape? So the triangle has got three sides and each side is x plus 10.

So I'm adding up all the sides, and if I simplify that I get three x plus 30.

And my rectangle has got four sides.

Two of them have a length of x and the other two, have got a length of X plus four I'm adding those together.

And if I simplify that I've got four x plus eight, now part b says can I write that as an equation? We're told that the two perimeters are the same, which means that the four x plus eight is equal to three x plus 30.

Part C says, can you represent this as a bar model? So I'm going to draw a bar model, now, can you see, that I've made sure the bars are the same length each time I've made sure that my xs, the box rejects is the same and it all matches up.

So finally, want to calculate the perimeter, I'm going to do that by solving the equation and then use a bar model to help me four x plus eight equals three x plus 30.

So first thing, I'm going to look at how many xs I've got that match up.

So I've got x, x, and x there, I can take those three xs away from each side.

And the length of the bars will still be the same.

They'll still be equal.

So that leaves me with x plus eight equals 30.

Next step.

I can split that up down there.

We know that these two bits are equal so that must be eight.

I can then take that eight away from each one that would leave me with an x there.

And if I take away eight from this side, I'm going to be left with 22.

So x equals 22.

It doesn't really matter that I've made the bar for my eight bigger than the bath of the 22.

There was no way I could've known in advance that it would be bigger, but I've been able to use the bar model to help me get to my answer, that's perfectly fine.

If I know x equals 22, I can pop back over here.

This side is 22, that's 22, that's 26 and that's 26.

So now I can calculate the perimeter.

You said that x is 22.

So over here, that's 32, that's 32 and that's 32.

And if I add them up in either case, I'm going to get 96 centimetres for the perimeter.

Okay? Okay? Should we just go over those last two questions? The triangle and the rectangle both have the same perimeter.

Can you calculate the perimeter? So we know one side is x and one side is three let's label those other sides, x and three.

So the perimeter of the rectangle is two x plus six.

And that's going to be equal to the triangle, which has got three x plus four as its perimeter.

Now know, if you solve this and you look at what I've got and then subtract two x from each side.

Which means that I've got here six and over here, I've got x plus four.

So if six equals x plus four I subtract that four from each side, I'm going to get x equals two.

So now I know x equals two.

I can jump back up here.

That side is two, that side is four, that side is four, which gives me a perimeter of 10 centimetres.

Double check that here.

If that side is two, that side is two, two, and three is five, six, seven, eight, nine, 10.

Yep.

I'm definitely right.

So look at the other side, these two triangles, they've also got the same perimeter.

Let's see if we can calculate it.

So, first up, we've got add these three sides together there are five xs plus two and four, six, add three is nine.

And the triangle over here, we've got three, four, five, six, seven xs plus three.

So I'm going to subtract five x from each side, which just leaves me with nine.

Yeah? And that equals two x plus three.

If I subtract my three from each side, six equals two x.

And that means that one x I need to divide by two, one x is three.

x equals three.

Pop back up here, two three is six, and two is eight, three plus three is six and two three is six and four is 10.

So that perimeter is 10 add eight add six, which is 24 centimetres.

It never hurts to check if each of the x is three, then over here, two, three is six, add two is eight.

Three times three is nine and two times three is six plus one is seven, nine add eight is 17.

17 add seven is 24 centimetres.

And finally, just before I go, here's an explore task for you.

So pause this video and just have a go and see what you can find out.

Good luck.