Lesson video

Thank you for choosing to learn using this video today.

My name is Ms. Davis, and I'm gonna help you as you work your way through this lesson.

Feel free to pause the video and work through things at your own pace.

Lots of exciting things to look at, so let's get started.

Welcome to our lesson on preparing to solve a two-step linear equation.

Our lesson outcome for today, "I can understand that an equation needs to be in a format to be ready to be solved, through collecting like terms on each side of the equation." The additive inverse of a number is a number that, when added to the original number, gives the sum of zero.

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

To simplify an expression is to write it in a more efficient and compact form, without affecting the value of the original expression.

And like terms are terms that have the same set of variables and corresponding exponents.

Pause the video if you feel you need to write any of those down or review any of those.

We're gonna start by using representations to solve a two step equation.

Some equations can be solved with one calculation, you may have seen these before.

These are often referred to as one step equations.

Let's look at some examples.

X plus 17 equals 53.

If I put that in a bar model, this equation has a single additive step.

By adding the additive inverse, which is -17, to both sides of the equation, we can find a solution for x.

That was one step.

7x equals 42, as a bar model, would look like this.

This one has a single multiplicative step.

Multiplying both sides by the reciprocal, which is 1/7, will find the solution.

Some equations can be solved with two steps.

This equation is a two step equation.

Pause the video and think about how you'd represent it with a bar model.

Fantastic, I wonder if your bar model looks like mine.

Let's think about 3x minus 1 equals 20.

How could you represent this equation with a bar model? Sometimes drawing bar models with a subtraction is harder to do.

So we've got 3x, we need to subtract 1, and that would give us 20.

Well done if yours looks like mine.

These are not one step equations, because one positive x cannot be isolated by either adding or multiplying a single value.

These are two step equations, as they have a multiplicative and an additive step.

They are written in the form axe + b equals c.

Let's have a look at that form now.

So you can see that we've got 3x + 2 = 20, and that's in the form axe + b = c.

The same with the next one.

We've got axe + b = c.

a, b and c represent constants and x the unknown.

So just be aware, there's lots of letters there.

a, b and c are just standing for the constants, whereas x is the unknown.

We know that other letters can be used as unknowns in equations, but here we just chose to use x to represent the unknown.

We can find the solutions to these equations by considering the bar models.

So if we start by adding -2, we can draw this bar model now as 3x is the same as 18 and now we can see that x must be 6.

The solution is when x is 6.

Let's do the same with our other bar model.

Right, well let's start by doing the 20 add the 1 to find the length of the whole bottom bar.

So 3x must be equal to 21.

So each x has a value of 7.

The solution is when x is 7.

I'm gonna have a go at one now, I'd like you to follow the steps carefully and then you're gonna try a similar one.

So I'm gonna find the solution to the equation 4x + 5 = 49 using a bar model.

So we need to start by drawing our bar model.

So I should have 4x + 5 = 49.

Now I know that if I add -5 or subtract 5, that leaves me with 4x = 44.

I want to know what 1x is.

So times by 1/4 or divide by 4, you get x as 11.

I'd like you to use the same method to find the solution to the equation 3x + 8 = 35.

We'll go through the answers when you're ready.

Well done.

Hopefully your bar model looks similar to mine.

Then we were adding -8.

So we know that three x's must be worth 27.

Each x must be 9.

So well done if you've got a solution when x is 9.

So in a bar model we can group the variables and the constants together so they represent equations in this form axe + b = c.

So if we have a look at this bar model.

At the moment it seems to represent the equation 4x + 5 + 2x + 9 = 38.

If I group the x's and group the constants together, I can rewrite it like this.

If we want to, we could add the two constants to simplify, 9 add 5.

This equation could be simplified to 6x + 14 = 38.

Let's look at another one.

How could we represent the equation 2(3x + 4) = 25 in a bar model? Well, let's have a look.

Bracketing the expression means we have two lots of the expression 3x + 4.

So there's 3x + 4 and another 3x + 4.

We know that that has to equal 25.

Let's rearrange to group the variables again.

And I've added those two constants to make it look simpler, 4 add 4 is 8.

We can now see that our equation can be written as 6x + 8 = 25.

We now have an equation which only requires two steps to solve.

It's in that form, axe + b = c.

Time for a check.

I'd like you to match up the bar models on the left to the grouped versions on the right.

Off you go.

Well done.

The top one on the left matches with the second one on the right.

We've got 6x + 10 = 60.

The second one down matches with the top one on the right.

So we've got 6x + 9 = 60.

The third one down matches with the bottom one on the right, which gives us 3x + 18 = 60.

And the last one matches with the third one down on the right, which gives us 8x + 16 = 60.

Let's put that all into practise.

I'd like you to use the bar models to find a solution for x, in each question.

Focus particularly on how you're laying these out with bar models.

I don't just want to see a solution, I want to see your working to go with it.

Give it a go and then come back for the next bit.

Well done.

This second set of questions, I would like you to match the bar model with the equation.

You'll notice the equations have all been written in the form axe + b = c, whereas the bar models have not been rearranged to group the variables and the constants together.

Off you go, come back for the answers.

Superb work.

For this first one, have a look at my working out.

You should have got a solution when x is 8.

For B, a solution when x is 25.

For C, a solution when x is 1.

And for D, a solution when x is, you could have written it as 7.

6 or 38/5.

You could have changed it into a mixed number as well if you wished.

Do take some time to have a look at the working outs and make sure your working out matches.

For two, the first bar model could be written as 4x + 17 = 50.

That's now in the form axe + b = c and ready to be solved.

B, 5x + 5 = 30.

C, 4x + 16 = 50.

D, 6x + 15 = 30.

E, 6x + 8 = 50.

And F, 6x + 8 = 30.

Well done on all of those.

For the second part of the lesson, we're gonna have a go at writing linear equations in this form, axe + b = c.

Any linear equation can be written in the form axe + b = c by collecting like terms. So we're gonna start by making sure we're really confident about what like terms are.

Like terms are terms that have the same set of variables and corresponding exponents.

We'll see what that means with some examples.

Group these into like terms. Pause the video and see what you come up with.

Well done.

Let's have a look at what we've got.

4x, 3x and -2x are all like terms, as they have a variable of x.

4xy and xy are like terms, their variable is xy.

0.

2, 4 and 4/5 are all constant terms. They are like terms, they're all constant.

They have no variable.

The other two did not match up with any.

So Lucas said, "But I thought 4x and 4y would be like terms as they both have a 4." Can you explain to Lucas why he's incorrect? Both terms have the same coefficient of the variable but the variables are different.

What makes something a like term is the fact that it has the same variables.

Jun says, "So why are -2x squared and -2x not like terms?" Can you explain this one? Like terms have the same set of variables and corresponding exponents.

So for example, if it has an x squared in one term, a like term would also have to have an x squared.

The algebra tiles may help you see why x and x squared are not the same thing.

So -2x squared and -2x are not like terms. That x would have to have the same exponent for them to be like terms. Lovely, now we're confident with what like terms are, we can simplify an expression by adding like terms together.

So if I had 5x - 3x + 4x, this is equivalent to (5 - 3 + 4) lots of x.

Because they're all like terms, this gives me an equivalent expression of 6x.

Where an expression contains both like terms and unlike terms, we can simplify each type of term separately.

So in 8x - 3 - 10x + 7, I've got two x terms and two constant terms. I'm gonna rewrite them the other way round so that I've grouped the terms together.

So 8x - 10x - 3 + 7.

And then I can add the x terms together, add 8, add -10, and then -3 add 7.

We can now see that's equivalent to -2x + 4.

Time for you to have a go yourself.

Can you simplify each expression by collecting like terms? Let's have a look then.

The first one we get 8a + 6b.

For B, -5x + 8y.

You can write that as 8y - 5x.

5x negative, the 8y is positive.

For C, 7p - 5, or -5 add 7p.

So let's put that together with our equation skills.

How could we write this equation in the form axe + b = c? Give this one a go.

Let's have a look.

I'm gonna group the like terms together.

Then I've got 4 + -7 and 3 + 5.

4 + -7 gives me -3, so my coefficient of x is -3 and my constant is 8.

<v ->3x + 8 = 10.

</v> If expressions contain brackets, it's possible to expand the brackets before collecting like terms. Let's try this one.

Have a go using your expanding bracket skills and then we'll check it together.

We need to multiply the whole of that first expression by two.

Two lots of x is 2x.

Two lots of -3 is -6, so 2x - 6.

The second bracket four lots of 3x is 12x.

Four lots of 5 is 20.

Got plus 12x plus 20 and that equals -10.

I can now collect like terms by grouping the x terms together and the constants together.

So I've got 2 plus 12, lots of x plus -6 plus 20.

So that gives me 14x plus 14 because -6 plus 20 is positive 14.

Equals -10.

That's now an equation ready to be solved.

Jacob and Lucas need to solve this equation.

Jacob says, "We need to expand both brackets and collect like terms." Lucas says, "I think there's a quicker away." Will Jacob's method work? What do you think Lucas is talking about? Pause the video.

Yes, Jacob's method will work.

This will give us an equivalent expression on the left hand side in the required form.

Lucas, though ,has noticed that the expression in the brackets is the same.

We can then write the left hand side as a multiple of a single bracket.

Don't worry, we're gonna have a look at that.

So let's start with Jacob's method, which is expanding both brackets.

We get 4x plus 12, subtract 2x subtract 6 equals 22.

Make sure that you've multiplied that -2 by x and the -2 by 3.

Then collecting like terms, you have 2x + 6 = 22.

The right hand side of the equation has not changed because we haven't begun to solve the equation.

We haven't performed any operations.

We're just writing equivalent expressions.

Let's have a look at what Lucas was talking about.

Because these brackets are the same thing, I've got four lots of x plus 3, minus two lots of x plus 3.

If I have four lots of something and I take away two lots of that thing, I'm left with two lots of that thing.

So 4 subtract 2 is 2.

So I'm left with two lots of x plus 3.

I can now expand my bracket to get that as 2x + 6 = 22.

Notice it's the same thing we had in the other method.

Either method works fine.

Knowing two methods does mean you can check your answers with an alternative method.

It's really easy to make mistakes where you've got quite long equations.

So being able to check it by doing it another way can stop you from making those mistakes.

Now Jacob wants to solve this equation.

He says Lucas's method will not work this time.

Is he correct? Yes, he is correct.

The expressions in the brackets are different.

Got eight lots of x plus one, subtract five lots of x plus three.

So we can't unitize the brackets and just write them as a single multiple of one bracket 'cause they're different things.

If we wanted to write it in the form axe + b = c, then Jacob will need to expand both brackets and simplify.

Time for you to have a go.

Which equation is equivalent to 5 bracket x minus 1 plus 3 bracket x minus 1 equals 22.

What do you think? Well then if you spotted it's the bottom one.

'Cause the brackets are the same, I could write that as eight lots of x minus one, which gives me 8x minus 8.

You could have expanded both brackets and then collected like terms. That would've worked as well.

This skill could be very useful when forming equations to solve perimeter problems. This shape is made up of a regular pentagon on top of a rectangle.

How could we form an expression for its perimeter? What do you think, first of all, and then we'll look at it together.

Well done, if you thought about how you were gonna start this problem.

Now this is a regular Pentagon.

That means that all four lengths that I've highlighted now at the top is the same as the bottom length of the rectangle and the top left of the rectangle, which we can't see, which is x plus 1.

So all four of those lengths are part of the perimeter and they're all x plus 1.

There's also a length of x plus 1 at the bottom, so that's five lengths of x plus 1.

I'm gonna write that as five bracket x plus 1.

Now we've got the two sides on the left and the right of the rectangle, which both have a length of 2x plus 3.

So I've got two lots of 2x plus 3.

If you want to, pause the video now and check you agree.

I'm now gonna tell you that the perimeter of the shape is 83 centimetres.

Let's look at how we could form an equation for x in the form axe + b = c.

So there was the form we had before.

I've now put it equal to 83.

And I'm gonna expand my brackets.

I've got 5x plus 5 plus 4x plus 6 equals 83.

I could collect like terms to get 9x plus 11 equals 83.

This is now in a format ready to be solved for x.

A quick check.

The perimeter of this rhombus is 92 centimetres.

A rhombus, remember, has all four sides the same length.

Which of these is an equation to find the value of x? What do you think? Well, I don't know if you spotted, there are three correct answers to that one.

A, B, and D.

They are all equivalent.

D is already in the form, axe + b = c.

So we'll be ready to solve for x, if we wanted to.

Time for you to have a go.

For each equation, I would like you to write it in the form axe + b = c.

You do not need to solve the equation.

Off you go, come back for the next bit.

Well done.

For this second question, I'd like you to match the shapes below to one of these expressions for its perimeter.

You've got 10x minus 4, 10x minus 2 or 9x plus 11.

The first shape is made out of an equilateral triangle on top of a rectangle.

B is made out of a parallelogram on top of a rectangle and C, a regular hexagon on top of a rectangle.

You'll need to think about your shape properties to help you.

Superb work.

Let's have a look, then, at our answers.

We're gonna be really careful to look at our positive and our negative values and check we're collecting like terms properly.

So the first one, x just means 1x so if that (indistinct), where we don't see a coefficient, it means we have a coefficient of 1.

So 7 plus 1 is 8, so we've got 8x plus 3 plus 1.

So 8x + 4 = 12.

For B, we've got -3x add 5x.

That gives me 2x.

5 add -8 gives me -3.

2x - 3 = 1.

For C, -x, remember needs-- means -1x so I've got -1 minus 4 gives me -5x and I've got 10 plus 7, which gives me 17.

<v ->5x + 17 = 2.

</v> Let's have a look at the ones with brackets.

Because these brackets are the same, I can write that as eight lots of x minus 2, which is 8x - 16 = 16.

For E, these brackets were different.

I'm gonna need to expand them.

So 4x minus 20 plus 15x plus 10 equals 85.

So 19x - 10 = 85.

Now this bottom one, I wonder if you found this one a little bit trickier.

However, it makes it easier if you remember that when the brackets are the same we can actually unitize them and collect like terms a bit quicker.

So six lots of 4 minus 2x, subtract one lot of 4 minus 2x gives me five lots of 4 minus 2x.

End up with 20 - 10x = -40.

Equally, you could have expanded both brackets.

If you expanded the second bracket, be careful that you should have -4 add 2x.

<v ->1 times -2 is positive 2x.

</v> Well done.

Take your time to read back through them, if there's anything you want to check.

And finally, the first one, you should have 9x plus 11.

You've got three lots of x plus 5 and two lots of 3x minus 2.

For B, you should have 10x minus 2.

This one was a little bit trickier.

You've got two lots of x minus 2, two lots of x, and two lots of 3x plus 1.

And finally, we've got 10x minus 4.

That gives us two lots of 2x plus one.

Then we've got six lots of x minus 1.

Well done.

So we have seen that an equation with both a multiplicative and an additive step could be represented and solved using a bar model.

Any linear equation can be written in the form axe + b = c, by collecting like terms and we've done lots of that today.

When it's in that form, it's ready to be solved.

Thank you very much for joining us and I look forward to seeing you again.