Lesson video

Hello, thank you for joining me today.

My name is Ms. Davis 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 are 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 different types of equations.

In this lesson, you'll be able to recognise that there are many different types of equations, of which linear is just one type.

Quite a few keywords we're gonna use today.

Particularly, I want to draw your attention to this word linear.

The relationship between two variables is linear if, when plotted on a pair of axes, a straight line is formed.

The gradient is a measure of how steep a line is.

It is calculated by finding the rate of change in the y-direction with respect to the positive x-direction.

We're gonna start by reviewing our skills on plotting graphs.

Equations which show a relationship between two variables can be plotted on a graph.

The relationship between two variables is linear if it forms a straight line when plotted.

If the graph of a relationship is not a straight line, then the relationship is non-linear.

How can we plot the relationship Y equals X squared minus two? This may be something you've done before.

It may not be.

Pause the video.

How would you do it? Right, we need to pick some coordinates that follow the rule the y-coordinate is the x-coordinate squared minus two.

Sofia says, I think the coordinate two, two is on the graph.

How could we check if she is correct? We need to check if it satisfies this relationship.

So let's substitute X equals two into the right-hand side of our equation.

So two squared minus two is two.

So when X is two, this gives us an answer for Y as two also.

That means two, two is on the graph.

Let's plot it.

Aisha says, let's plot the coordinate when X equals four as well.

We'll do this together.

If X equals four, we've got four squared minus two, which is 16 minus two, which is 14.

So when X equals four, Y equals 14.

Let's plot that one as well.

So now Sofia's thinking, we can plot these two coordinates and draw the line between them.

Aisha says, this must be a linear graph.

Ooh, Jun says, but I don't think Y equals X squared minus two is a linear relationship, so how can its graph be a straight line? Pause the video.

Think about those three statements.

What do you think? Right, there's lots to pick apart in that one.

The coordinates they have plotted are correct and will be on the graph.

However, they could only join two points with a straight line if they know the graph should be linear.

So that was a method, if you know that a graph is linear, you can plot two points and draw the straight line between them.

However, they haven't plotted enough points or got enough information to know whether they form a straight line or not.

In fact, most relationships in maths and in real life are not linear relationships.

We delve into linear relationships because they have a really key structure.

Jun is actually correct.

Y equals X squared minus two is not a linear relationship, and it's only the linear relationships that form straight lines.

Y equals X squared minus two will not form a straight line.

So we can't just draw the line in from two coordinates.

What we can do if we don't know whether a relationship is linear or not, or if we know it's definitely not linear, is we can use a table of values to plot some more points.

So Sofia has now drawn a table of values.

We're gonna try and plot some more points.

I always like to start with zero and then the positive values of X.

So zero squared minus two is negative two.

So 0, -2 will be on our graph.

One squared minus two equals negative one.

So we've got 1, -1.

We already know we have 2, 2, 3, 7, 4, 14 we had already, and 5, 23.

Using our negative values, we've got -1, -1, and -2, 2.

It is useful to plot a couple of negative X values as well to give us an idea of what this graph is going to look like.

There we go.

We can see clearly now that it is not a straight line.

The graph of this relationship is actually a curve.

Has Sofia now plotted the graph of all points that fit this relationship? Pause the video.

What do you think? No, she could have used non-integer coordinates as well.

She just picked integer values.

There are an infinite number of coordinates that follow this relationship.

We have only plotted a few just so that we could get an idea of what the relationship looked like.

If we wanted to show the graph of all points, we need to draw a curve.

If we just want to explore what the graph looks like, then it's okay just to plot a few of the coordinates.

If we know a relationship between two variables is linear, then we only need to plot two points.

It's always a good idea, however, to plot a third point just to help you spot any mistakes and to check it definitely is a linear relationship.

If we don't know whether a relationship is linear, then plotting from a table of values helps us to see the shape of the graph.

Let's try some more.

We can graph expressions by looking at how the value of the expression changes the different values of the variable.

So how could we graph the expression three A plus four? Well, we could label the horizontal axis A, instead of X, and then on the y-axis, we could plot the value of three A plus four as A varies.

So let's explore this expression as A varies.

Using a table of values.

Just like before, I filled in my positive values first because I find them easier and then filled in my negative values.

I have plotted them and we can clearly see a pattern.

If I want to draw the exact graph of three A plus four, I would need to draw the line in.

It is linear so I can draw a straight line.

Let's investigate some of the following expressions.

Before we start, I would like you to have a look at the axes I have drawn.

What do you notice? This was quite subtle.

Don't know if you spotted it.

They all have different scales.

The reason I've done that is because some of these graphs are quite narrow, quite difficult to plot when they're drawn on an axes with a step of one.

So when you are plotting these coordinates, make sure you read the scales on the axes carefully.

If you are choosing to draw your own axes, you should fill in your table of values first so you can then decide what your scales should go up to and decide on appropriate scale.

Let's try drawing B squared minus five.

We're gonna fill in a table of values.

Pause if you want to check my values.

And then we've plotted it.

We've only plotted coordinates between negative three and five on the x-axis, but this does give us an idea about how the expression changes as the variable changes.

It's not a complete graph, but we can see some of its features.

Be aware that changing the scales does change the shape of the graphs in some ways.

So you can see that on the one I've drawn at the moment, the y-axis has a step of two.

If I actually draw it with a step of one, you'll see that the graph now looks a bit narrower.

So just be aware that changing the scale does change the way your graph looks in some ways.

Let's have a look at three minus two T.

Filling in our table of values.

And plotting our coordinates.

Again, if you want to check them, just pause the video.

Let's have a go at a quarter H plus two.

Now this time, you'll see that I've picked integer coordinates and I've also picked multiples of two, because I know that I'm gonna have to find a quarter, and finding a quarter of 2, 4, 6, 8, it gives me nicer answers than trying to find a quarter of sort of 1, 3, 5.

I filled those in, and then I can plot those.

So we have explored lots of different relationships so far.

You are gonna get a chance of doing the same in a moment.

Andeep is plotting the graph of the expression 60 over X.

I'd like you to have a look at his table of values and his graph.

Give an example of something he has done well, and then can you spot any mistakes? Off you go.

Andeep has actually done mostly a really good job.

He has drawn his table of values.

He has chosen X values that give integer Y values.

So you think about 60 divided by X.

He's chosen values for X, which give him integer answers.

You'll notice he didn't use X is seven or X is eight.

He used the y-axis scale correctly.

So notice the y-axis is actually going up in steps of five and he did use that correctly.

Did you spot his mistake though? He did make a mistake with doing 60 divided by five.

That should have been 12.

You can probably see where that is on the graph because when he's plotted it, it doesn't actually fit with the relationship that the rest of the points seem to have.

Again, this is a different type of relationship, not a linear one, but one that you may have come across before.

You may see again in the future.

Time for you to have a practise.

I would like you to fill in the table of values and plot the points for the graphs of each expression.

You don't have to draw in the full graph.

You can just plot the points that are given to you in the table of the values.

A couple more for you to try on this page as well.

Come back when you're ready for the answers.

Well done.

So you're gonna want to have a look at the screen to see whether you've plotted your points correctly.

So four A minus five does give us a linear graph with gradient four and y-intercept negative five.

B, it does not give us a linear graph.

You can see we have a curve forming.

C, we have a linear graph again, with gradient negative three and y-intercept five.

B over five.

It's the same as a fifth of B.

So yes, we do end up with a straight line again.

Just check your coordinates.

24 over X.

This is similar to the one Andeep was plotting before, which was 60 over X.

Do check your values carefully.

Try and be as accurate as you can with those.

Right, well done! Lots of plotting-graph skills there that we can see that we're still really confident with.

Let's have a look then at identifying linear equations from graphs.

We've talked a little bit about this already.

Let's do it in a bit more detail.

Equations of straight lines can have many forms. You may have already looked at this before and seen them written in the form Y equals MX plus C.

Here are some examples of equations of straight lines in the form Y equals MX plus C.

Pause and read through them.

I want you to have a think.

Are there any values of M and C which would not form a linear graph? You could use graphing software to help you explore.

You could type in some different equations using different values of M and C and seeing if there's any that don't form a linear graph.

Try it out.

All constant values for M and C will form a linear graph.

So you probably tried lots of different values if you were using graphing software, or you just thought about the equation of a straight line if you weren't, and you probably thought, yeah, it's always gonna be a linear graph with a gradient M and y-intercept zero, C.

A slight exception.

If M or C are variables, then it's possible to make many relationships which are non-linear.

Here is a table of values for Y equals two X plus three.

How do we know that this will form a linear graph when plotted? Pause the video.

Have a think about your answer.

Okay, I've drawn some arrows on to help us.

As X increases by one, Y increases by two, and that is constant between every two coordinates.

This means the gradient of this line will be two.

Equations that form a straight line when plotted are linear equations.

They will have the same gradient anywhere on the line.

It's easier to see if an equation forms a linear graph when it's written in the form Y equals.

I'd like you to look at these three.

Which of them do you think are linear equations? I've added some table of values to see if this can help us.

Y equals X plus one is a linear equation.

It's in the form Y equals MX plus C, isn't it? You can see it has a constant gradient of one.

Y equals X squared plus one is not a linear equation.

You can see that as X increases by one, the increase in Y changes.

So that is not linear.

It's not in the form Y equals MX plus C because we've got an exponent of X squared there.

Y equals X cubed plus one.

Again, as X increases by one, the value that Y changes by is different each time.

They do not have constant gradients.

Therefore, those two are not linear.

Which of these are linear equations, do you think? Again, I've added some tables in to help us.

The first one, as X increases by one, you can see that the amount that Y changes by is different each time.

So definitely not linear.

Y equals X over 12 has a constant gradient of a twelfth.

Again, that's in the form Y equals MX plus C with a gradient of one-twelfth and a y-intercept of zero.

Here is a table of values for the graph of equation Y equals a quarter X plus two.

Izzy says, this is not a linear equation because the Y values do not increase by the same amount each time.

Is Izzy correct? Y equals a quarter X plus two is in the form Y equals MX plus C.

We're expecting it to be linear.

So what's gone wrong with her table? Well, the X values are not increasing by the same amount each time.

That's absolutely fine.

When you're plotting a graph, you can pick any X values that you like.

However, if we're trying to spot a pattern, then it's not easy to then spot that there's a constant gradient.

So linear equations can take many forms and some are harder to identify than others.

Here are some more equations with two variables.

Okay, just for a little bit of fun, pause the video.

Which of these do you think are going to be linear? Don't worry, we're gonna investigate them all in a moment.

Hold onto your answers.

It is not easy to see if they're linear at the moment.

What we are gonna do is draw their graphs to help.

At this point, you may feel like you want to draw these on graphing software because some of them are quite difficult to generate coordinates for.

I'm gonna show you how to do it manually.

Feel free to use graphing software if you wish.

Two Y minus X equals five.

Let's pick some coordinates that follow this rule.

One, three would.

Three, four would.

Five, five would.

Seven, six would.

Feel free to check that you agree.

I have plotted those and I have used graphing software, and I can see that that is a linear equation.

So it's a different form for an equation of a straight line.

Let's try Y equals X squared minus four.

Let's try so when X is negative two, we get a Y value of zero.

When X is negative one, we get a Y value of negative three.

When X is zero, we get a Y value of negative four.

And when X is one, we get a Y value of negative three.

2, 0, 3, 5 also on that line.

If I plotted those and drew the full graph, that is what I get.

So this time, not linear.

Right, Y equals one over X.

So when X is negative two, one divided by negative two, negative a half.

Check it on a calculator if you want to.

When X is one, one divided by one is one.

When X is negative one, one divided by negative one is negative one.

And when X is two, one divided by two is a half.

Again, if you're using a calculator, you could input more values in for X and find some values for Y, or you can draw using graphing software.

This is a really cool graph, possibly harder to see if you just plotted a few coordinates.

You might not have known where the rest of the graph was gonna go.

But clearly, not linear.

YX equals 24.

This is one we probably can do in our heads, because I want coordinates where X times Y equals 24.

Can you think of any coordinates where X times Y is 24? These are the ones I went with.

Two, 12.

Three, eight.

Four, six.

Six, four.

When I plot them, I get another graph which is not linear.

Right, we're getting onto the ones with even more terms. So coordinates that follow this rule, let's have X is negative one.

So negative one plus two multiplied by negative one plus one.

One multiplied by zero, which is zero.

Let's try when X is zero.

Zero plus two multiplied by zero plus one.

Two multiplied by one is two.

When X is one, I've got one plus two multiplied by one plus one, three times two is six.

And you can do this with even more values for X, especially if you've got the use of a calculator.

Plus in that, you can see again another non-linear graph.

Final one.

This one is gonna be really tricky to do without using graphing software.

So let's say X was negative two.

I've got Y plus negative two equals two lots of negative two minus three lots of negative two minus two squared.

I can do those multiplications and then I can simplify by collecting like terms on the right.

Y minus two equals negative two.

So what number subtract two gives you negative two? Zero does.

I've got negative two, zero.

That was a lot of effort to get one coordinate, but well done if you managed to follow along with that.

There are some others.

If you'd like to pause and try them out, that would be fantastic.

Otherwise, let's look at the graph.

There we go.

So that crazy-looking equation does give us a linear graph.

This equation is linear.

A bit of a summary for what we've just looked at.

There are many different types of equations and linear is just one type.

Some equations may not look like typical linear equations, but they could be rearranged into a linear form.

For now, plotting the graphs or using graphing software can help you check if an equation is linear.

There's a couple of examples underneath of ones that might have thrown us.

Okay, time for a check.

Which of these are linear equations? I am gonna show you the graphs to help you.

Before I show you, have a think about which ones you think might be linear.

Okay, there are the graphs.

So make your final decision and then we'll see which are linear.

I wonder if you were right first time or whether you changed any of your answers when you saw these graphs.

It is B and C which are linear equations.

Fantastic.

Time for a practise.

So you need to fill in the missing values in the tables.

Then, using those tables, match the equations to the graphs.

Finally, state whether each equation is linear or not.

It's up to you what level you want to challenge yourself at.

You may want to check your answers with a calculator or you may want to use a calculator for generating some of these coordinates.

If you're not sure when matching your graphs, again, it's your choice whether you want to use graphing software to help you or not.

Off you go, and then we'll look at our answers together.

So this first one, you should have -12, -2, 0, 4.

Might have been quite hard to see which one that matched up with from just those four coordinates.

Well done if you spotted it was F, and it's definitely non-linear.

I'm also going to show you C.

So C, you've got -2, -2.

5, 2.

5, and 2.

Again, this has been a tricky one to match up with the graph with just the four coordinates.

So well done if you spotted it was H, and again, non-linear.

B and D then were our linear equations.

So for B, you should have 7, 5, 1, -1, and that matches with G.

And for D, -2, -1, 1, and 2.

That was a crazy equation to substitute values in for.

So well done if you've got those either in your head or used the calculator.

And that matches with E, which is again linear.

Let's bring all those things we've talked about together.

So we've talked lots about this word linear today.

We know a relationship between two variables is linear if it forms a straight line when plotted on a graph.

We've looked at how we can graph an expression by looking at how the value of the expression changes as the variable changes.

And we can identify linear and non-linear expressions from their graphical form.

We know that sometimes equations that look like they're not going to be linear equations can be rearranged into a linear form.

And we know that using graphing software can help us spot whether things are linear.

Right, well done for that, guys, and I really hope to see you again soon.