Lesson video

Well done for logging on and choosing to learn with this video today.

My name is Ms. Davis and I'm gonna help you as we move through this lesson.

There's lots of interesting, exciting bits that we're gonna talk about.

So please feel free to pause things to move at your own pace and I will help you by adding in any hints or any suggestions that will help you answer similar questions in the future.

Let's get started.

Welcome to this lesson on the equation of a straight line.

By the end of this lesson, you'll be able to appreciate that writing linear equations in the form y equals mx plus C helps to reveal the structure.

So a couple of keywords that you might have come across before, but that we're gonna be using a lot today.

So first one is this idea of a numerical coefficient.

So a numerical coefficient is a constant multiplier of the variables in a term.

A constant is a term that does not change, so it contains no variables.

So let's see what that looks like in an example.

So say we had the equation y equals 3x minus five.

Three is the coefficient of x.

It is the constant multiplier of the variable x.

The constant term is negative five, and I'm gonna be referring to the coefficient of x and the constant term in this lesson.

Our new keyword is the equation of a line.

So an equation of a line is any equation whose graph forms a straight line.

We're gonna be picking apart different types of equations and their structure.

We're gonna start by looking at different forms of linear relationships and then we're gonna move on to a more specific form.

We can plot linear relationships to investigate their features and that's where we're gonna start.

We're gonna start with the relationship y equals x plus two, and think about what features this graph has.

In order to plot this relationship, we need at least two coordinates that follow the rule.

So can you think of two coordinates that would follow the rule y equals x plus two? Lovely, I wonder if you thought about any of the same ones as me.

So I went with 0, 2, 3, 5, 10, 12.

We can have negative numbers, negative two, zero, negative four, negative two.

You could have come up with non-integer coordinates as well.

Even though I said that we only need two coordinates, it's always helpful to plot at least three in case one of your coordinates is wrong and to get a bit of an idea of how the line's going to look.

So I've plotted three and I've joined them up with a ruler.

This is now the graph of the relationship y equals x plus two.

We call y equals x plus two the equation of the line.

It's an equation, and when we plot its coordinates, it forms that line.

So how can we describe this line using keywords and features? Pause the video and have a think about the key features of that graph.

Think about what you already know about how to describe graphs and then we'll see what we've said together.

So some things you might have said: as x increases, y increases.

You might have used this language of positive gradient.

It has a positive gradient.

You might have been more exact and said that it has a gradient of one.

You might have looked at the y-intercept and say it has a y-intercept of zero, two.

And then this relationship, y is always two more than x.

I wonder if you said any of those things.

We can plot any linear equation on a graph.

So let's look at this one.

Y minus x equals two.

Pause the video.

Can you come up with any coordinates where the y value subtract the x value is two? Some choices here, 8, 10, 2, 4, 0, 2, 5, 7.

But there's an infinite number that you could have come up with.

I've plotted these again and drawn a line.

So this is the line with equation y minus x equals two.

What do you notice about this line? Lots of things to notice.

I wonder if you spotted that it has the same features as y equals x plus two.

It's the same line.

We're gonna try this one.

X equals y minus two.

Do you think this is gonna be the same as the others? Let's have a look.

So this time, I'm not giving you free reign.

I would like you to come up with the x coordinates for these y coordinates.

So remember, x equals y subtract two.

See if you can fill them in.

Lovely, so we have 1, 3, 7, 9, negative two, zero, negative five, negative three.

And we've plotted them and drawn a line.

And I wonder if you said this to start with, it is the same line as y equals x plus two or y minus x equals two.

The equation of a line can be written in different forms. Those three graphs that we drew before are all the same because they represent the same linear relationship.

I've just written them a different way round.

You might have noticed when you were coming up with the coordinates that you were doing the same sort of process to get your sets of coordinates each time.

We're gonna look at why this is the case with a bar model.

So y equals x plus two.

Let's look at the same bar model.

So y minus x equals two and x equals y minus two.

Some forms of equations of lines are more common than others and we are gonna look at some different forms later in the lesson.

Different forms have different uses at different times.

Let's have a go at plotting the equation x plus y equals seven.

This is a nice form of this relationship because it's really easy to come up with coordinates that work.

Can you think of any coordinates where x plus y equals seven? So I went with 0, 7, 7, 0, 4, 3, 6, 1, 2, 5.

There's plenty more, especially if you are using negative values or non-integer values.

What do you think they'll look like when plotted? Specifically, what is the gradient and y-intercept? Have a think yourself and then we'll look by drawing the graph.

Lovely, there's the line x plus y equals seven.

It has a gradient of negative one and a y-intercept of zero, seven.

Okay, let's try x plus y equals 10.

While do you think about what it will look like, think about coordinates if you want to have a go at plotting.

So we'll work through it together and we'll see whether you were correct as to what it would look like.

So this time, we've got this graph here.

Some coordinates might be 1, 9, 2, 8, 0, 10.

The gradient is negative one, same as the previous one, and the y-intercept this time is zero, 10.

We're gonna look at slightly trickier linear relationships.

So again, we're gonna just explore the features by having a go at plotting it.

So 2x plus y equals 10.

Pause the video.

What do you think it's going to look like? Can you think of any values for x and y that satisfy this relationship? And it's not as easy.

See what you come up with.

Lovely, I wonder if you thought about the previous one and what similarities and differences there's gonna be.

Let's have a look.

So coordinates that might work.

Zero, 10 'cause two lots of zero plus 10 is 10.

1, 8, two lots of one plus eight is 10, and so on.

If you plot that this time, it looks like that.

So some similarities, it goes through zero, 10 like the previous one does and it is steeper this time though.

The gradient is negative, two and the y-intercept is zero, 10.

So each of those four graphs has a gradient and a y-intercept and I'd like you to match them up.

Off you go.

So A has a gradient of negative three and a y-intercept of four.

B, a gradient of three and a y-intercept of four.

C, a gradient of 1/3 and a y-intercept of four.

And D, a gradient of negative 1/3 and y-intercept of four.

Let's have a go at this set.

Off you go.

Well done.

So A, we've got a gradient of negative two and a y-intercept of negative five.

B, a gradient of negative two, but a y-intercept of five.

C, a gradient of positive two and a y-intercept of negative five.

And D, a gradient of two and a y-intercept of five.

Well done.

You've shown that you know all about your gradient and your y-intercepts and you can link that to the appropriate graphs.

Time for you to have a practise.

So I would like you to plot each equation and then write down the gradient and the y-intercept.

What you might want to do is write down some of the coordinates that satisfy that relationship.

So for example, in A, can you think of some coordinates where x plus y equals five? I recommend you plot at least three so that you have an idea of what the graph will look like and you can spot any mistakes.

And then once you join your graph, you can get the gradient and the y-intercept.

D is possibly the trickiest one here.

So again, see if you can come up with some coordinates that fit that rule.

So you need to be able to double the x, double the y and it equal 10.

And then you can get your gradient and y-intercept.

Part two, which equations represent the same relationships? So were there any that actually you think are the same relationship, just written in a different way? Off you go.

Well done.

So same idea, just a different set of equations this time.

So plot each equation, give me the gradient and the y-intercept and then match any that represent the same relationship.

If you're struggling to plot the equations, remember, you're looking for coordinates that follow that rule.

Off you go.

Superb.

Let's have a look at these then.

So A has a gradient of negative one and a y-intercept of zero, five.

Look at my graph to see if it looks like yours.

B, a gradient of two and a y-intercept of zero, negative five.

C, a gradient of negative two, a y-intercept of zero, five.

And D, a gradient of negative one and a y-intercept of zero, five.

So any of those equations then that represent the same relationship seems to be A and D.

They have the same features, they are the same graph, they must be the same relationship.

Let's look at the second set.

So x plus y equals four has a gradient of negative one and a y-intercept of zero, four.

B, y equals four minus x has a gradient of negative one and a y-intercept of zero, four.

C has a gradient of 1/2 and a y-intercept of zero, two.

And D has a gradient of negative 1/2 and a y-intercept of zero, two.

So although those last two have similarities, they are not the same relationship.

However, A and B are the exact same relationship.

You could draw yourself a bar model if you wish to show that those two relationships are the same thing.

So now we've looked at lots of different types of relationships, we're now gonna look at a specific form for an equation of a straight line.

So we're gonna look at equations of a straight line in the form why equals mx plus c.

And we're gonna see if we can identify gradients and y-intercepts.

So some forms of equations of lines make it easier to see the key features than others.

One of the more common forms to write an equation of a line is y equals mx plus c.

Here are some examples of equations of lines that are written in that form.

Y equals 2x plus four, y equals 3x plus one, y equals x plus eight, y equals x.

Y equals 2x minus three.

Y equals negative 5x.

Y equals six minus 4x.

Y equals seven plus 2/3x.

You might wanna pause the video and just see if you agree that they've been written in the form y equals mx plus c.

Are there any that you're less sure about? Things that are not in this form: x equals three.

X plus y equals six.

Similar to some that we looked at in our first part of the lesson.

2y equals x plus eight.

Y plus x plus one equals zero.

So the general equation y equals mx plus c is similar to how you might write the nth term of a linear sequence.

So all those ones we looked at in our examples, they had a coefficient of x and a constant, didn't they? So here is the sequence 3n plus two and I've represented it on a graph.

What does the three in the nth term represent? Take a look at the graph.

Can you see where that three is in the diagram? Okay, what does the two in the nth term represent? Again, pause the video.

Where is that two in the diagram? Okay, so 3n plus two.

So the three, each term is adding an extra three each time.

So can you see the pink blocks of groups in three? So you've got one three, two threes, three threes, four threes.

So that three is how much our sequence is increasing by each type.

The two is the constant term.

Can you see that there are two blue blocks at the bottom of every term? And that remains constant.

So that's where the three and the two can be found when we're writing a linear sequence.

But that has similarities to when we're drawing a linear equation on a graph.

So here's the graph of the equation y equals 3x plus two.

Notice, we can draw a line through it this time because it is the equation of a line and it's defined for any possible value of x.

What does the three in the equation of the line represent and what does the two in the equation of the line represent? Can you see it anywhere in the graph? So the three is actually the gradient of the line.

Can you see if you take one step right, you have to go three up to get back to the line.

So the three is the gradient and the two is the y-intercept.

So zero, two is where that line crosses the y-axis.

So let's see if we can explore that a bit further.

So here are the graphs of y equals 2x plus one, y equals 2x plus five, and y equals 2x minus one.

Which graph do you think belongs to which equation? So think about the similarities in the graphs.

Think about the differences.

Which is which do you reckon? Awesome, right, so A, that blue one is y equals 2x plus five.

B, the green one, is y equals 2x plus one.

And C, that pink one, is y equals 2x minus one.

They all have the same gradient, they're all parallel.

And notice in the equation, they all said 2x, 2x, 2x.

However, the y-intercepts are different.

So for A, the y-intercept is zero, five.

For B, the y-intercept is zero, one.

For C, the y-intercept is zero, negative one.

So think about the elements of the equation, dictating where the y-intercept is.

When the equation of a line is written in the form y equals mx plus c, the gradient and the y-intercept are easy to identify.

So you might want to make some notes on this.

So where we have y equals mx plus c, the coefficient of x, so that letter m is just the letter we've picked to mean the coefficient of x.

So the coefficient of x is the gradient of the line.

The constant, whether it's positive or negative, is the y-intercept.

It is not actually known for sure where the m and the c come from.

However, it's been suggested that m comes from the French word monter, which means to climb or to ascend.

Okay? And obviously if that means the gradient, the gradient is how steep a line is.

You can think of it as how much the the line is rising by, and monter meaning to climb or to ascent.

C could come from commencer, which means to start.

So if we're talking about the y-axis, that is where the line is crossing the y-axis, which could be where it comes from.

So some pupils are trying to work out the gradient and the line set by looking at the equations of lines.

Each person has made a mistake.

Let's see if we can help them out.

So Aisha says the y-intercept is zero, five and the gradient is four.

Jacob says the gradient is two and the y-intercept is zero, one.

Lucas says the gradient is zero and the y-intercept is zero, negative two.

Can you spot their mistakes to help them out? Let's start with Aisha's.

So Aisha's looked at the wrong values and she's got 'em the wrong way around.

The coefficient of x is the gradient.

So the gradient is five.

The constant is the y-intercept.

There you go.

She's got it right now.

The y-intercept is zero, four.

So Jacob has got the gradient correct this time, but his y-intercept should be negative.

Can you see that the constant is negative one? That's the same as saying 2x plus negative one.

So the y-intercept should be zero, negative one.

Fantastic.

So he's learnt now and he's corrected himself.

And Lucas, Lucas got a little bit confused with the x.

Because there was no number before the x, I don't think he was that sure what to do.

But remember, where there is no number written, it means one of that variable.

So we could actually write that as y equals 1x subtract two.

There we go, he's corrected himself now.

The gradient is one and the y-intercept is zero, negative two.

These are all easy mistakes to make when you're starting out.

Hopefully by looking at these, you've got a bit more of an understanding of what value represent what in an equation.

Do be careful 'cause sometimes the x term and the constant term are written the other way around.

That's not a problem.

So in the equation y equals five minus 2x, this still counts as being in the form y equals mx plus c.

What do you think the gradient and the y-intercept's gonna be? Give it a go.

Doesn't matter if you're wrong first time, and then we'll talk about it together.

So the coefficient of x is negative two.

So the gradient this time is going to be negative two.

The constant is a positive five.

So the y-intercept is zero, five.

So Laura said in the equation 2y equals 10 minus 4x, the gradient is negative four and the y-intercept is zero, 10.

Can you explain why Laura is still incorrect? Hopefully you saw why she came up with those values.

She's seen the negative four is the coefficient of x.

She's seen that the 10 is the constant.

Did you spot where she went wrong though? Her equation isn't in the form y equals mx plus c.

Her equation says 2y equals 10 minus 4x.

Y has to have a coefficient of one.

It needs to be a single y equals for us to be able to identify the gradient and the y-intercept.

Laura's gonna try a different one.

She says, what can I do to work out the gradient and the y-intercept for the equation y equals two lots of 3x minus four? Well, what we can do is we can use our expanding bracket skills.

We'll have a quick review as we go through.

So don't worry if you've forgotten.

Two lots of 3x minus four is equivalent to 6x minus eight 'cause we've got two lots of 3x, which gives us 6x and then we're adding two lots of negative four, which gives us negative eight.

That sign that we've got in the middle is the identity symbol 'cause these two things are identical.

They're just different ways of writing the same expression.

So that means we can write our equation as y equals 6x minus eight.

It's really easy now to see that the gradient is six and the y-intercept is zero, negative eight.

So true or false? The coefficient of x in any linear equation represents the gradient.

What do you think? Well, I don't if you noticed that was false.

Let's look at our justifications.

Right, hopefully reading these justifications helps us see why it has to be false.

This is only true when the equation is in the form y equals mx plus c.

So keep your eye open for that.

It is the coefficient of x that represents the gradient when the equation of a line is in the form y equal mx plus c.

Try this one.

When in the form y equals mx plus c, the constant represents the y-intercept.

True or false? Well done.

We've clarified what form our equation has to be in this time.

So that is now true.

What is the correct justification? Good, it's the fact that the y-intercept represents the value of y when x equals zero.

Remember, on the y-axis, x is always zero.

If you times zero by anything, it's still zero.

So that constant represents the value when x is zero.

It doesn't have to be the last term in the expression, remember? Let's have a practise.

So for the first section on the left-hand side, you need to write the gradient and the y-intercept for each equation.

Please can you write the y-intercept as a coordinate.

For the second half, you need to fill in the missing pieces of information for each question.

So sometimes there'll be a missing coefficient of x or there might be a missing constant or you might need to work out the gradient or the y-intercept.

See if you can get those filled in and then we'll look at it together.

Well done, guys.

We're really starting to master this idea of finding the gradient and the y-intercept from an equation.

I wonder if there was any that you thought were a bit interesting or any that caught you out.

Let's look through each one.

So the first one, gradient of eight.

That's the coefficient of x.

Y-intercept of zero, four.

B, gradient of three, y-intercept of zero, negative seven.

Check you've got the negative.

C, gradient, negative five, y-intercept of zero, one.

D, gradient of negative three, y-intercept of zero, 10.

I wonder if this one caught you out.

The gradient is one 'cause you've got one X, there's a coefficient of one before the x.

We just don't always write it.

There's no constant.

That is the same as adding zero.

So well done if you got the y-intercept of zero, zero.

If you're not sure about that one, see if you can write down another equation that you think might have a y-intercept of zero, zero.

Something like y equals 2x or y equals 3x.

If it doesn't have a constant in the equation, then it must be equivalent to having a constant of zero.

It comes up again in question F, doesn't it? So this time, our gradient is negative one.

And again, our y-intercept is zero, zero.

Y equals five is an interesting one.

Y equals five is a horizontal line, so it has a gradient of zero.

The y-intercept then is zero, five.

So there's no coefficient of x.

There is a constant of five.

And the last one really making sure that you've got your head around which part of the equation represent what feature.

The gradient is q plus three.

The y-intercept then is 0p.

Fantastic work on that one.

Let's look at this bit on the right-hand side.

So the first one, our equation should now read y equals 9x plus four.

Gradient of nine, y-intercept of zero, four.

B, we've got our gradient of negative, six and our y-intercept of zero, 1/2.

We can have negative values and we can have fractional values as y-intercepts or gradients.

C, this time, we need to expand our bracket to make it easier to find the gradient and the y-intercept.

So our gradient is 12 and our y-intercept is three times two, which is six.

So zero, six.

D, our equation is y equals 10 minus x with a gradient of negative one and a y-intercept of zero, 10.

E, so our equation is five lots of one minus 2x equals y.

To get our gradient and our y-intercept, we're gonna want to expand our brackets.

So our gradient is gonna be five multiplied by negative two.

That's negative 10.

And our constant is five multiplied by one, which is five.

So our y-intercept is zero, five.

F, our equation should now read eight minus 5x equals y 'cause we have a gradient of negative five and a y-intercept of zero, eight.

And finally, our equation should look like y equals u plus tx.

We need a coefficient of x as t and we need our constant to be positive u.

Fantastic work on that one.

There's lots and lots of different equations.

Hopefully when you come across 'em now in the future, you'll be able to find the gradients and the y-intercepts.

So this is what we've looked at.

Equations of straight lines can be written in different forms. We looked at how writing in the form y equals mx plus C allows us to see the gradient and the y-intercept easily.

In the general equation y equals mx plus c, the coefficient of x, that m, is the gradient and the constant, that c, is the y-intercept.

Fantastic work, well done.