Lesson video

Hello, thank you for joining me today.

My name is Miss Davies, and I'm gonna help you as we work through this lesson.

I'm really looking forward to working alongside you.

There's lots of exciting things that we're going to have a look at.

I really hope there's bits that you enjoy and bits that you find a little bit interesting.

Maybe there'll be some things you haven't seen before as well.

Let's get started then.

Welcome to our lesson on solving simple linear equations with a multiplicative step.

By the end of the lesson, you'll be able to solve a linear equation requiring a single multiplicative step.

Take your time to read today's keywords.

I'm gonna focus on this idea of a reciprocal.

The reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal is equal to one, and we're gonna use that today to solve our equations.

We're gonna start by looking at multiplicative steps using representations and in the second part of the lesson, we're gonna take away the representations and work on solving equations algebraically.

An equation with a multiplicative step can be represented with a bar model.

What equations can you write from this bar model? Here's just a few that you might have come up with.

a plus a equals 14, 2a equals 14, a equals 14 minus a, or 14 minus a equals a.

There are others you might have also come up with.

Do any of these equations show us the solution? What do you think? At the moment, none of these equations show us the solution.

2a equals 14 probably gives us the most information, but none of these tell us exactly how to calculate the solution yet.

To find the solution, we need to work out the value of one positive a.

How could we isolate one a from this bar model? Pause the video.

What do you think? So what we could do is we could split the bar in two.

Each half of the bar would then be worth seven.

The solution then must be when a is seven.

What operation have we used to isolate one a? Pause the video.

What do you think? You might have said, we've divided by 2, we've divided 14 by 2 to get 7.

You might have said we've halved or multiplied by half.

Dividing by 2 and multiplying by half are the same operation.

In this bar model, either one of those options will isolate one positive a.

So the solution is when a is seven.

We can check our answer by substituting into any of our equations.

We're gonna do them all.

So 7 plus 7 equals 14.

So 14 equals 14.

2 multiply by 7 is 14, so 14 equals 14.

That's true.

That's balanced.

7 equals 14 minus 7, so 7 equals 7 and 14 minus 7 equals 7, so 7 equals 7.

So again, that one is balanced when a is seven.

This bar model represents the equation 5s equals 30 or you could write that as 30 equals 5s.

It doesn't matter which bar the unknown is part of and in an equation, it doesn't matter which side of the equation the unknown is on.

This is a multiplicative relationship.

What we're saying is 5 lots of s have the value of 30 or 5 times s equals 30.

To find the value of one positive s, we need to split the bar into five sections.

What do we need to do to the value of 30? We need to divide 30 by 5 or find a fifth of 30.

Those two things are equivalent.

If we find a fifth of 30, we get a solution when x is 6.

Unknowns can have values which are non-integer as well, and they often do.

Let's think about the value of a this time.

To get the value of a, we need to divide by 2 or find a half, which is the same as multiplying by a half.

7 divided by 2 is 7/2.

Equally, 7 multiplied by a half is 7/2.

You could write that as 3.

5, but often, fractions are easier to work with.

Izzy says, "I want to solve the equation 7x equals 2.

Will this have the same solution as the previous question?" The previous question was 2x equals 7.

What do you think? Let's draw a bar model.

We've got 7x equal to 2.

So what we need to do is find a seventh of two.

We need to split that two bar into seven pieces.

2 times a seventh is 2/7.

So, the solution is when x is 2/7.

That doesn't convert nicely into a decimal, but that's not a problem.

Answers as fractions are really nice to use.

This is different from the previous equation 'cause that had a solution of 7/2.

2/7 and 7/2 are different values.

We're gonna have a go at doing one together and then you'll go try one on your own.

So we're gonna find the value of x for this bar model.

What we want to do is split the bar into four pieces, so 24 multiplied by a quarter.

24 multiplied by a quarter is 6.

So, each of those sections is six.

The solution then is when x is six.

We're gonna check our answer.

So 6 plus 6 plus 6 plus 6 or 4 lots of 6, those are equivalent, they both equal 24.

So, the top bar and the bottom bar are balanced.

Time for you to have a go.

Find the value of x for this bar model and then check your answer by substitution.

Off you go.

Well done.

So, we want to split our bar into three sections.

27 times a third is 9.

So there's a solution when x is nine.

Let's check that 9 plus 9 plus 9 is 27 and 3 times 9 is 27.

The top bar and the bottom bar are balanced when x is nine.

What equation is represented in this bar model? See if you can write it down.

This bar model represents the equation a half x equals 6 or x/2 equals 6.

Remember that second one means x divided by 2, so x divided by 2 equals 6.

Those two equations are equivalent.

I'd like you to think about this one.

What could we do to work out the value of one positive x this time? Good thinking.

If we know what a half x is, we want to work out what 1x is, we need to double or multiply by 2.

So, a half x times 2 is equivalent to x.

We also need to multiply the bottom bar by 2 so that they are equal.

So 6 times 2 is 12, so 1x must be 12.

We can check our answers by substituting, and I think it's really important this time to make sure we have got the right value.

So we've got a half of 12 equals 6, so 6 equals 6.

If we use our second equation, 12 divided by 2 is 6.

6 is 6, so this does work when x is 12.

What could we do to work out the value of one positive x this time? See if you can find a solution.

This time, we need to multiply by 3.

We have a solution of 21.

Let's check it with our substitution.

A third of 21 is 7 and that works, doesn't it? A quick check, I'd like you to complete these calculations so they are equivalent to x, so a half x multiplied by what would be equivalent to x.

Off you go.

Let's check our answers.

A half x times 2 is equivalent to x times the third times the seventh times 4.

Time to put that into practise.

I would like you to match up the bar models with the calculations, which will find the solution.

There may be more than one calculation for each diagram.

That's fine.

Come back when you're ready for the next bit.

Well done.

For this second question, I'd like you to use the bar models to find the value of x.

I'll see you for the answers when you're ready.

Good work.

So that first one should match with 4 multiply by half or 4 divided by 2.

Those operations are equivalent.

For b, 8 times a quarter or 8 divided by 4.

For c, 4 times 2.

Well done if you also spotted that 4 divided by a half is the same thing.

For d, 4 times 4 and again, well done if you spotted that 4 divided by a quarter is equivalent.

And finally, we've got 8 divided by a quarter or 8 times 4.

We're gonna use those in the second part of our lesson, so make sure you're happy with them before you move on.

For the second one, you should have a solution of x is three for a.

For b, x is five.

For c, x is three.

For d, x is 5/3 or 1 2/3.

X is 9/2 or 4 1/2 and f, x is 50.

Well done.

Let's look at the next part of our lesson.

Right, now, we're gonna have a go at doing exactly the same thing but with our equations.

We've got a lot of focus now on writing solutions algebraically and how we're presenting our work.

So that's gonna be our challenge as we work through is to make sure that we're keeping our working neat and logical.

So, we can use the same method for equations without representations.

Let's look at 7x equals 42.

To isolate one positive x, we need to multiply by the reciprocal of seven.

That's because 7x multiplied by a seventh gives us 1x.

So, I'm gonna put that on both sides of my equation to maintain equality.

So 7x multiplied by a seventh equals 42 multiplied by a seventh.

That gives me x equals 6.

We have written our working out down the page and we've shown how we've maintained equality by multiplying both sides of the equation by the same value.

Sofia says, "7x divided by 7 is equivalent to x, so I could divide both sides of the equation by 7 to solve." What do you think? Yes, Sofia is correct.

To solve this equation, we could multiply by seventh like we did previously or we can divide by 7.

They're the same thing.

Let's see with Sofia's way.

7x equals 42.

7x divided by 7 equals 42 divided by 7, so x equals 6.

We can check our answer by substituting the solution into the original equation.

So, 7 lots of 6 equals 42.

42 is 42.

Both sides of the equation balance when x is six.

So Laura says, "I want to solve the equation 5x equals 7," and Jacob has the equation 7 equals 5x.

Should they get the same answer? What do you think? Yeah, they're the exact same equation.

It does not matter what side of the equal sign the unknown is on.

What do we need to do to isolate one positive x? What do you think? We need to multiply by the reciprocal of five, which is multiply by fifth or similar to what Sofia did with the previous question, you could divide by 5.

5x divided by 5 will get you 1x, so you can divide both sides by 5.

Let's have a look.

7 equals 5x.

That was the equation Jacob had.

Multiply both sides by a fifth.

That gives me 7/5 equals x.

The solution is when x is 7/5.

I could write that as a decimal if I wanted to, but I don't need to.

I'm happy with it in that form for the moment.

Let's check by substituting.

So we've got 7 equals 5x, so 7 equals 5 lots of 7/5, 5 lots of 7/5 is 5 times 7/5.

5 times 7 is 35/5.

35/5 is 7.

So, both sides of that equation are balanced.

Right, we're gonna have a go at one together and then you are going to try one.

Pay particular attention to how we're laying out our working out.

15x equals -45.

I can times by 1/15, or I can divide by 15.

15x times 1/15 leaves me with x.

<v ->45 times 1/15 is -3.

</v> Let's check our answer.

Particularly because we had a negative value, we want to make sure we didn't make any mistakes.

15x equals -45.

That means 15 multiplied by -3 should equal -45 and that is true.

Remember, the solution is the value for x that makes this equation balanced, so the solution is when x equals -3.

I'd like you now to do the same for this equation and check your answer by substituting.

Off you go.

So 4x equals -28.

You could have chosen to multiply by a quarter or divide by 4.

That will give you x equals -7.

Let's check it.

4 lots of x equals -28.

So, 4 lots of -7 should equal -28.

4/7 to 28, so 4 times -7 is -28.

Our solution is x equals -7.

We can solve equations where the coefficient of the variable is a fraction.

Let's look at the equation a third x equals 6.

The reciprocal of a third is 3/1 or 3.

To isolate one positive x then, we can multiply the equation by 3.

If you want to draw a bar model and split your x bar into three, that'll give you a third x.

Then, if you multiply it by 3, you'll see what x is.

So, third x equals 6, we need to multiply by 3.

A third x times 3 is equivalent to x, and 6 times 3 is equivalent to 18.

How could we the equation 3/7x equals 9? I'm gonna let you think about this one.

So, did you think about what we need to multiply 3/7 by to get 1? We need to multiply by the reciprocal, which is 7/3.

So let's try it.

3/7 x equals 9, multiply by 7/3.

That gives me x on one side, and then 9 multiplied by 7/3 is 63/3.

That gives us an answer as x equals 21.

I'm definitely gonna wanna check this one 'cause there was quite a few steps there and some fraction skills as well.

So 3/7 of 21 equals 9.

Let's see if that's true.

So you can find 1/7 of 21 and then times by 3 if you wish.

So 1/7 of 21 is 3 times by 3 is 9.

That does work.

Let's have a look at nought.

3x equals 6.

Sofia says, "I'm not sure what the reciprocal of nought.

3 is, so I'm not sure what to do." What could Sofia do? I wonder if you found a solution to this that you think would work.

I think there's two obvious ways of doing this.

Firstly, she could convert nought.

3 into a fraction or nought.

3x divided by nought.

3 is equivalent to x.

So she could divide both sides of the equation by nought.

3.

If you're not sure about that, pause the video and just have another read.

All right, let's look at both of these methods.

So nought.

3 is a fraction is 3/10.

So, I can multiply by the reciprocal.

So 3/10x multiplied by 10/3 equals 6 multiplied by 10/3.

3/10x multiplied by 10/3 is just x.

That's why we've chosen to multiply by that value.

6 times 10/3 is 60/3.

That does simplify to x equals 20.

All right, so let's have a look at this other method.

So nought.

3x equals 6.

We could divide nought.

3x by nought.

3, which means we have to divide 6 by nought.

3.

Nought.

3x divided by nought.

3 gives us x.

That's why we chose that operation.

6 divided by nought.

3 is the same as 60 divided by 3.

If you want to, you can write that as a fraction 6 over nought.

3 and then equivalent fraction would be 60/3.

That means x is 20.

We got the same answer doing both methods.

I wonder which one you preferred.

All right, we're gonna try some together.

I'm gonna do the first one, and then you're gonna try the second one.

So, 1/10x equals -20.

I'm gonna multiply by the reciprocal, which is 10.

So I have x equals -200.

As always, I'm gonna check my answer.

1/10 of -200, that's -200 divided by 10 is -20.

Our solution is x equals -200.

I'd like you to try the one on the right and then check your answer by substituting.

Off you go.

A fifth x equals -7 times in both sides by 5 gives us x equals -35.

Well done if you got that.

Let's show it's correct by substituting a fifth of -35, which is the same as -35/5 does give us -7.

So, we know that x equals -35 is our solution.

Lucas and Jacob are solving the equation x/3 equals 12.

Lucas says, "Here is my work.

I think x equals 36." Take a second to read his working.

Jacob says, "I don't need to show my working for this one.

It's easy.

3 times 4 is 12, so x equals 4." Who do you think is correct? Well, you know how important working out is, so well done if you said it was Lucas.

Jacob has made a really common mistake in thinking he needs to divide 12 by 3.

He's seen the division, and he's just divided the numbers he can see.

However, if we substitute that in, we can see it doesn't work.

4 divided by 3 is not 12.

Now, Lucas, we can follow clearly his working out.

We'll have a look at this in more detail on the next page.

So, when solving equations like this, you can read it as x divided by 3 and then remember that the inverse operation will be multiplied by 3, or you can read it as a third of x.

So multiply by the reciprocal of a third, which is 3/1 or just 3.

That's essentially what Lucas did.

Both those things are exactly the same.

Let's have a go at one together.

We're gonna solve the equation h/4 equals 20.

I'm gonna multiply both sides of my equation by 4.

That gives me h on the left-hand side and 80 on the right-hand side.

A quick check would show that 80 divided by 4 is 20, so that works.

Try this one on the right-hand side.

Let's have a look.

Y divided by 6 equals 12, so I can multiply both sides by 6 and this gets me as y equals 72.

And just check that makes sense, is 72 divided by six 12? Yes, it is.

Well done if you got that one.

We're now gonna look at some equations with -x.

We can solve these with a multiplicative step.

So here are our algebra tiles.

We've got the equation -x equals -3.

What can you multiply -x by to make positive x? If you've got an answer for this one, well done if you worked out that you can multiply by -1 or divide by -1.

They're equivalent.

If you're using algebra tiles, you can fit that algebra tile over.

So we've now gone from -x to positive x by multiplying by -1.

What do I need to do to the right-hand side to maintain equality? Of course, I'm gonna need to do exactly the same to the right-hand side.

So I'm gonna need to multiply by -1 again.

When using physical algebra tiles, we can flip them over to multiply by -1.

Our equation -x equals -3 when multiplied by -1 has become x equals 3.

Three pupils are solving the equation -5x equals 35.

We're gonna have a look at their methods, and I want to know which one do you think is best.

Andeep has multiplied by -1 then multiplied by a fifth.

Jun has divided by -1, then divided by 5.

And lastly, Laura has multiplied by negative a fifth.

Read all three methods.

Which one do you think is the best? All three methods work perfectly, and all three methods would be fantastic for you to use when solving this equation.

It's down to personal preference which one you prefer.

Andeep and Jun, you can see, have solved in two steps, and Laura has solved in one step.

Sometimes, doing two steps stops us from making a mistake, but it all depends on what you feel most comfortable with.

You might have also thought that both pupils could have divided both sides by -5, that would've worked as well.

So Izzy has tried to solve the equation -8x equals -56.

Have a look at her working.

She has said that solution is when x equals -7.

Firstly, I want you to think about what she has done well, see if you can work out where she's made a mistake and can you tell me the correct solution.

Off you go.

I think Izzy has done a great job at laying out her working out.

She showed that she's maintained equality by dividing both sides by 8.

In fact, all of her working out is correct.

What she hasn't done is she hasn't found the solution.

She needs to find the value of one positive x.

Her equation still says -x equals -7.

We want a solution for one positive x.

The correct solution, if you multiply both sides by -1, shows you that x equals 7.

Well done.

Time to put all of that into practise.

For each question I would like you to solve the equation, I want you to pay particular attention into your working out.

There's plenty of space on the page to show that working out.

Off you go.

Well done.

We're gonna look at our working out carefully for each question.

For a, you could divide both sides of the equation by 2 or multiply by half.

You get x equals 9.

For b, you could divide both sides by 6 or multiply by 6, so x equals 5.

C, you can multiply both sides by a fifth or divide them both by 5 and that gives you 9/5.

You could change that into a mixed number or a decimal.

I'm happy with 9/5.

D, want to divide by 21, so we get an answer of x equals -1.

For e, if you wanted to do it in one step, you can divide by -2, or you could divide by 2 and then multiply by -1.

So, divide in both sides by -2 gives you an answer of x equals 6.

You might wanna check that one.

<v ->2 times 6 does give you -12.

</v> Well done if you got that one.

For f, dividing by -8, 18 divided by -8 is the same as -18/8.

That does simplify to -9/4.

There's quite a few steps to that one.

And also, being careful with your fraction skills, so well done if you've got that answer of -9/4.

G, we multiply both sides by 4, we get x is 52.

H, if we multiply both sides by the reciprocal, which is 3/2, we get x equals -15.

And I, I'm gonna do this one by dividing by 1.

4 and then multiplying by -1, so dividing by 1.

4, 4.

2 divided by 1.

4 is equivalent to 42 divided by 14.

There are three 14s in 42, so -x must be equal to 3.

Don't forget, we want one positive x a few times from both by -1, get x equals -3.

Well done.

You're now confident at solving equations with a single multiplicative step.

We have learned today that equations with a multiplicative step can be represented in a bar mode and we use those bar models to isolate a single positive unknown to find the solution.

Equations with a multiplicative step can be solved by multiplying both sides of the equation by the reciprocal of the coefficient of the unknown.

When coefficients are negative, it's important to multiply by a negative value to isolate the single positive unknown.

And those are all those things we talked about in that final task.

Fantastic work today.

Please join us again for another lesson.