Lesson video

Hello, Mr. Robson here.

Rearranging to Solve Linear Equations.

Great choice by you today to go for this lesson.

I'm excited, so let's get started.

Our learning outcome is that we'll be able to recognise that equations with unknowns on both sides of the equation can be manipulated so that the unknowns are on one side and then solve the equation.

Some keywords we're gonna need throughout the lesson, equation and equality.

An equation is used to show two expressions that are equal to each other.

Two expressions form an equality if, when substitution takes place, the expressions have the same value.

For example, substituting x = 5 into 3x - 1 = x + 9 would give us three lots of 5 - 1, which is equal to 5 + 9.

We have equality.

Two parts of the lesson, and we're gonna start by representing equations with unknowns on both sides.

Some pupils are given an equation to solve.

Aisha says, "I've got 3x = 9.

That's easy!" Lucas says, "I've got 3x + 1 = 10.

That's okay." Whereas Sam says, "I've got 5x + 1 = 2x + 10.

I've got the hardest one by far!" Sofia says, "Hold on, Sam.

I think your equation is almost the same as Lucas and Aisha's!" Can you see what Sofia has spotted? Pause this video and have a look at the three pupil's equations.

Let's look at how the equations are connected.

What is it that Sofia spotted? She says, "It's easier to see the link if you write them like this." Add +1 to both sides of Aisha's equation and we get Lucas's.

Oh, look, 3x and +1, 9 and +1 turns Aisha's equation into Lucas's equation.

Add 2x to both sides of Lucas' and we get Sam's.

If we add +2x to both sides of the equation, we end up at 5x + 1 = 2x + 10.

You can see this building up on our balance scales.

I'll start again with 3x being equal to 9.

Add 1 to both sides.

That's how the algebra looks.

And we end up at 3x + 1 on the left-hand side, 10 on the right-hand side.

Add 2x to both sides of the balance scales, and that's how the algebra looks.

We've created the equation 5x + 1 = 2x + 10 by adding 1 to both sides and then adding 2x to both sides of Aisha's equation.

We can undo this building up in order to solve equations with unknowns on both sides.

This is Sam's equation, 5x + 1 = 2x + 10.

We'll start by adding -2x to both sides.

That looks like that, and then we remove the 0 pairs, a +2x and a -2x will cancel each other out and disappear from our scales.

That leaves us with 3x + 1 being equal to 10.

And I think you can see the step that we want to take next.

We wanna remove that 1 from the left-hand side.

We do that by adding -1 to both sides.

That's how the algebra looks.

And then the +1 and the -1 make a 0 pair, they disappear, and that leaves us with 3x is equal to 9, therefore 1x is equal to 3.

That's a solution.

If I remove the visual representation of the balance scales and just show that algebra, we started a 5x + 1 = 2x + 10, we started with unknowns on both sides.

The unknown term 5x on the left-hand side, the unknown term 2x on the right-hand side.

We manipulated the equation by adding -2x so that we have the unknowns on only one side.

That's an important step, gets us much closer to our solution.

We then isolated the unknown by adding -1 to both sides, leaving just the unknown term on the left-hand side.

They're important steps when we're solving equations with unknowns on both sides.

We can represent the solving of this very same equation using what we call a bar model.

That is the equation 5x + 1 being equal to 2x + 10.

My bar in my top row has five x's and 1 the bar in my second row has two x's and 10.

They're the same length, and it's important that they're the same length because that demonstrates the equality between the two expressions.

What then happens when I draw a line there? Enables me to focus on that moment.

Effectively I've added -2x or removed 2x from both sides of the equation.

We now have 3x + 1 is equal to 10.

Just our visual representation looks different to the balance scale.

The next question is what is that length? Well, it's 10 - 1, it's 9.

Oh look, we now have three x's being equal to 9, therefore x must be 3.

Bar models are a really powerful way to solve equations with unknowns on both sides.

Quick check that you've got everything I've said so far.

What equation is represented by this balance model? Is it 5x = 9, 4x + x = 3 + 6, or 4x + 3 = x + 6? Pause and tell the person next to you.

I hope you said option C, 4x + 3 on the left-hand side, x + 6 on the right-hand side.

Which useful first step starts to solve this equation, 4x + 3 = x + 6? What first step are we gonna take? We're going to add x to both sides, are we going to add -x to both sides, or will we add -4x to both sides? What do you think? Pause, tell the person next to you.

Adding -x would be a really useful first step.

If we add -x to both sides, it creates a 0 pair, which we can remove, leaving us with 3x + 3 on the left-hand side and just 6 on the right-hand side.

That enabled us to turn the equation 4x + 3 = x + 6 into the equation 3x + 3 = 6.

What step might we take next? Would it be to add -3 to both sides, to add +3 to both sides, or to add -6 to both sides? What's the most efficient thing we could do? Pause, and tell the person next to you.

It was option A, adding -3 to both sides.

That will look like that, <v ->3 and +3 become a 0 pair</v> and cancel each other out.

We'll remove them from both sides of the equation.

We've now got 3x is equal to 3, therefore a solution x = 1.

A bar now.

Which equation is represented by this one? Is it 10x = 2x + 13, 5x + 8 = 7x + 5, or 8x + 5 = 5x + 7? Which one is it? Pause, tell the person next to you.

It's option B, 5x + 8 = 7x + 5.

Same equation, what does this line show us? So this it shows that 5x = 5x, that 8 = 5, or that 8 = 2x + 5? Pause and have a think about those three options.

I hope you said it does show us that 5x = 5x.

It doesn't show us that 8 = 5 because the 8 and the 5 are not the same length and 8 and 5 are not equal.

And it also shows us that 8 is equal to 2x + 5.

To the left-hand side of the line, we can see five x's and five x's being equal in length.

This is really useful.

Because they're equal, we retain equality when we remove them from both sides of our equation.

A manipulation to have the unknowns on one side only is what we read from the right-hand side of the line.

The length of 8 being equal to the length of 2x + 5 means we've removed five x's from both sides.

We've now got unknowns on only one side.

That means we're much closer to a solution.

What is this length? What length? This length.

Is it 2, 3, or 5? Pause, tell the person next to you.

I hope you said 3 because it's 8 - 5.

So what is our solution? The last step.

Is it x = 1.

5, x = 3, or x = 3/2? Pause and tell the person next to you.

x = 3/2.

We have two x's being equal to 3, therefore x is equal to 3 divided by 2, which we write as 3/2.

You've noticed something haven't you? 3 divided by 2 can also be expressed as 1.

5, but it's not 3.

The solution is not 3.

Most often you'll see solutions written as a fraction, but sometimes in context we might want to call it x = 1.

5 instead of x = 3/2.

Aisha and Sam are solving the equation.

x + 13 = 6x + 3.

Aisha says, "You always have to start with the variable terms so I'm going to add -x to both sides." Sam says, "You don't have to start with the variable terms. You can manipulate the constants first.

I'm going to add -3 to both sides." Who's right? Pause and have a discussion with the person next to you.

Both of our pupil's methods work.

There are no rules about what you manipulate first.

In Aisha's case, she wants to add -x to both sides first.

When we do that, we have a 0 pair, or a +x and the -x disappear, leaving us with 30 in the left-hand side, 5x + 3 on the right-hand side.

We can then add -3 to both sides to turn our equation into 10 being equal to 5x, giving us a solution x = 2.

In Sam's case, Sam wanted to add -3 to both sides of the equation, manipulating the constants, not the unknown terms. By adding -3 to both sides, we get a +3 and a -3, a 0 pair.

They cancel each other out and disappear from our model.

We're left with x + 10 = 6x.

The next step, add -x to both sides, which leaves us with 10 being equal to 5x, a solution x = 2.

Let's compare the two methods.

That was Aisha's method, adding an unknown to both sides to start.

And there's Sam's method, manipulating the constants first.

Compare the two, what do you notice? Pause, have a good look and a good think.

Did you notice same steps, different order? Aisha went for add -x and then had to add -3, whereas Sam started with add -3 and then later had to add -x.

Exactly the same steps just done in a different order.

You also notice the exact same result.

They're both accurate methods so we both led to the same solution.

It's fine to do it either way.

Quick true or false, when solving the equations you have to manipulate the variable terms first? Is that true? Is it false? Could you justify your answer with one of these two statements? We need the unknowns on one side in order to solve, or you can start by manipulating either the variable terms or the constant terms? Pause, and have a think about this question.

I hope you said false because you can start by manipulating either the variable terms or the constant terms. Practise time now.

Question 1, build an equation with unknowns on both sides.

Start with this balance which shows 2x being equal to 8.

We have unknowns on only one side.

And share what you need to add to both sides to make the equation 7x + 2 = 5x + 10.

You should draw some extra things on your balance scale model and write down the algebraic notation to reflect what you're adding to the balance model.

Pause and do that now.

For question 2, I'd like you to undo this balance model to solve this equation.

Write your algebraic notation at the same time alongside your model.

The equation is 6x + 4 = 3x + 10.

What are you gonna do to your balance model to solve? Make sure you write the algebraic notation that matches whatever manipulations you make to the model.

Question 3, use this bar model to solve this equation.

That's 2x + 161 being equal to 8x + 29.

Whilst using the bar model, be sure to write your algebraic notation alongside.

Pause and do that now.

Let's see how we did.

We are building an equation with unknowns on both sides.

Transforming 2x = 8 into 7x + 2 = 5x + 10.

What did we need to do? A good start would be to add 5x to both sides of your balance scale.

The algebraic notation of that would look like so.

We now have 7x on the left-hand side, 5x + 8 on the right-hand side.

What else do we need? We need to add 2 to both sides.

Adding 2 to both sides algebraically looks like that.

We created the equation.

7x + 2 = 5x + 10.

For question 2, undo this balance model to solve the equation.

First useful step to undoing.

There are multiple steps.

I think a good one is to add -3x to both sides.

The 0 pairs disappear, leaving you with 3x + 1 on the left-hand side and 10 on the right-hand side.

I'd like to see that 4 disappear.

I'm gonna do that by adding -4 to both sides of my balance model.

When that 0 pairs disappear, we're left with 3x is equal to 6, therefore x is equal to 2.

If you manipulated the constants first and still arrived at the same result, that's absolutely fine.

For question 3, using this bar model to solve the equation.

We'll start with a line there because we're gonna remove 2x from both sides of this equation.

We're then focusing on that moment, the length of 161 being equal to 6x + 29.

That's 161 being equal to 6x + 29.

There's your algebra notation.

We're then interested in that length.

Well, that length's 161 - 29, it's 132.

What does it tell us? It tells us that 132 is equivalent to 6x, therefore we have the solution x = 22.

Second half of the lesson now.

We're gonna solve equations with unknowns on both sides really efficiently.

Andeep and Laura are looking at a problem.

They're trying to solve x + 3 = 2x - 4 and Andeep makes a plea for help.

"I'm not sure how to deal with negatives," he says.

Luckily Laura is here to help.

"Here.

Let me show you!" Laura draws a balance model, x + 3 on the left-hand side and 2x and -4 on the right-hand side.

"Now you add positives to undo the negatives." Ah, we've added four +1s to both sides.

It's the same as before.

We've got +4, -4, a 0 pair.

That can disappear from our model.

That means we're left with x + 7 being equal to 2x.

Now look, all the terms are positive.

Now we end up with a solution x = 7.

Quick check you've got that.

Which useful first step starts to solve this equation, x + 3 = 4x - 6? Is it add +3, add -6, or add + 6? Pause, tell the person next to you.

I hope you said option C, add +6.

When we add +6 to both sides of the equation, we'll transform it into x + 9 = 4x.

From there, we can add a -x to both sides and we'll get 3x being equal to 9, therefore x = 3 Andeep says, "Can you do negatives with bar models?" Laura is a bit of an expert, "Absolutely! It's more difficult to set them up, but if you master it, they can be easier to solve!" That's x + 3.

That's nice and straightforward.

2x - 4.

Hmm, there's two x's or two positive x's, two x's moving in the positive direction towards the right.

The negative goes in the negative direction.

Can you see that is a move of 4 backwards or in the negative direction? That's 2x - 4, those bottom bars.

From there, x = 7 as a solution is really rather obvious.

It's just a little trickier to draw this bar model.

Check you've got that.

Which equation does this bar model represent? Hmm, is it x + 13 = 3x - 5, x + 18 = 3x, or 2x = 18? Which equation is represented by that model? Pause and have a think about those three.

I hope you said option A, x + 13 = 3x - 5.

Why? Because that's x + 13 and that's 3x is moving in the positive direction and 5 moving in the opposite direction or the negative direction, hence that's 3x - 5.

Did you also say, "I can see x + 18 = 3x"? I hope so because you can.

There is our 3x - 5.

When we apply the x + 13 there, can you see x + 18 on the top bar and three x's on the bottom bar? Did you also say, "I can see 2x is equal to 18"? Can you see it? You can when you remove x from both bars.

The answers were the steps to solving this equation.

If I write those three answers there, the step from A to B was to add +5 to both sides of the equation.

The step from B to C was to add -x to both sides of the equation.

Our solution x = 9.

Andeep and Laura are solving another equation.

"Whoops! I've started to solve 8x + 4 = 3x + 19, but I've made a mistake.

I'll have to start again." What mistake has Laura made? Oh, she's added -8x to both sides.

4 = -5x + 19.

She thinks she's made a mistake and she wants to start again.

Andeep says, "No.

Keep going!" Add -19 to both sides." Do you agree with Andeep's advice or do you think Laura should start again? Pause and have a think.

It is okay to say yes, it's also okay to say no.

We need to be really fluent when we're solving equations.

Laura could start again and maybe add -3x to both sides of the equation, but there's no need to start again.

Andeep's right, if we add -19 to both sides, we'll get there.

Add -19 to both sides, <v ->15 = -5x,</v> divide through by -5 and we get the solution x = 3.

You don't have to manipulate the smallest x first.

When we look at that equation, 8x on the left-hand side, 3x on the right-hand side, the temptation is to add -3x as a first step.

That's fine, it's efficient.

We don't have to.

Adding -8x works perfectly as valid.

Quick check you've got that.

Which of these are valid starts to solving this equation, 8x + 7 = 29 - 3x? Is it add 3x, add -7, add -3x, add -8x? There may be more than one correct answer.

Pause and have a think Option A was a good one, add 3x to both sides.

Option B was a good one as well, add -7 to both sides.

Option C, not good.

Option D, absolutely, add -8x to both sides, good start.

So adding 3x to both sides will transform the equation into 11x + 7 = 29.

We're closer to a solution.

Adding -7 to both sides of the equation will transform it into 8x = 22 - 3x.

We're closer to a solution.

Adding -3x to both sides.

No, no good.

We'll be left with 5x + 7 = 29 - 6x.

We've still got unknowns on both sides of the equation, we're no closer to our solution.

Adding -8x gives us 7 being equal to 29 - 11x, another valid start.

Sometimes a model is inappropriate.

In this case we just use our fluency in manipulation.

If I said draw a balance model or a bar model for 1/5th of x - 7 = 13 - 4x, it's a really tough diagram so we just need to be fluent in our algebraic manipulation.

A sensible start here is to multiply both sides of our equation by 5 because multiplying by 5 is the inverse of multiplying by 1/5th.

Multiplying both sides by +5 means we no longer have a fraction in our equation.

We multiply out those brackets, we get x - 35 = 65 - 20x.

Then we can add 20x to both sides, meaning we now have a positive unknown term on one side.

From there, add 35 to the equation and we'll get x = 100/21.

Quick check you've got that.

Which of these is a useful first step to solving this equation, 7 - x = 13 - 1/4x? Would it be to multiply by +4, multiply by +1/4, or multiply by -4? Pause and have a think.

I hope you said option A is a good one, multiplying by +4, I hope you said option B is not a good one, and I hope you said option C is a wonderful one.

Not efficient to multiply by +1/4, why? Because we'll end up with that.

Multiplying by +4 was a really efficient start.

Multiply all those terms by 4 and we're left with 28 - 12x = 52 - x.

We can solve that.

Arguably even more efficient was to multiply by -4 because you're left with 12 x- 28 = x - 52.

Our unknown terms are positive, a little easier to work with.

Practise time now.

For question 1, I'd like you to solve these equations in two different ways.

Pause and give this a go now.

Question 1 part B, I'd like to solve these equations in two different ways.

Pause and give this a go.

For question 2, find the perimeter of this rectangle.

Pause and think about this one.

For question 3, solve 1/8x - 7 = 3 - 7x.

Pause and give it a go.

Some feedback now.

Solving this equation in two different ways.

The method I've shown you there is to start by manipulating the unknown terms. I've added -3x to both sides of the equation first to turn it into 2 = 4x - 10, and I'll add + 10 to both sides to leave us with 4x being equal to 12, therefore x = 3.

My other different way was to manipulate the constants by adding +10 to both sides and then adding -3x to both sides, arriving again at 4x = 12, therefore x = 3.

There were multiple other methods you might have used which should have led to the same solution.

For part B, again I started by manipulating the unknown terms first.

Adding a +5x to both sides is quite an efficient way to start.

Turns the equation into 7x + 2 = 23.

Add -2 to both sides, 7x = 21, so x = 3.

What if we manipulated the constant terms first? Adding -2 to both sides, add the +5x to both sides, therefore x = 3.

Question 2, find the perimeter of the rectangle.

If we know that the top length, 17 - 2x, is the same as the bottom length, 3x + 2, we can make an equation out of that.

Those two lengths are equal.

If we make the equation 17 - 2x = 3x + 2, you can solve like so.

We find that x has a value of 3.

We'll substitute that into the length.

Three lots of x + 2, three lots of 3 + 2, that length must be 11.

If the base length is 11 and the height is 7, our perimeter must be two lots of brackets 11 + 7, our perimeter must be 36.

Solving this beautiful equation, what's the opposite of 1/8? Hmm, multiplying by 8.

So multiplying both sides by 8 will look like that.

And then when you multiply out those brackets, you'll get x - 56 = 24 - 56x.

From there, you can add 56x to both sides of the equation and then add 56 to both sides of the equation, 57x = 80.

Hmm, that's a tricky one, or is it? Just divide through by 57.

Our solution x = 80 divided by 57, which we write as 80/57.

What a beautiful equation.

In summary, equations with unknowns on both sides can be manipulated in many ways so that the unknowns are on one side and we can therefore solve.

I hope you've enjoyed today's lesson, I have, and I look forward to seeing you again soon for more mathematics.

Goodbye for now.