Lesson video

Thank you for joining us.

Well done for choosing to learn using this video.

My name is Ms. Davis and I'm gonna be helping you as you progress through this lesson.

There's lots of new bits that you might come across as well as maybe things that you've seen before that you want to recap.

So make sure you're moving through the video at your own pace.

There's plenty of opportunities to pause things and have a think before I then go on to explain how things work.

Really look forward to working with you.

Let's get started.

This lesson is about finding the equation of the line in the form y = mx + c.

Our lesson outcome is that you'll be able to find the equation of the line in this form by the end of our lesson.

Here are some key words which you may have come across before, but are gonna be vital to today's lesson.

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

We're gonna be looking at specific forms of equations of straight lines today.

The gradient is a measure of how steep a line is, and an intercept is the coordinate where a line or curve meets a given axis.

So we might talk about the y intercept or the x intercept.

That's where the line would meet that axis.

So this lesson is split into two parts.

First, we're gonna write the equation of a line from key features and then secondly, we're going to find the equation of a line.

So equations of lines can be written in a variety of ways.

Any equation representing a linear relationship between x and y, will form a linear graph when plotted.

When an equation of a line is written in the form y = mx + c, the gradient and the y intercept are easy to identify.

If you haven't got notes on this, you may wish to get some notes down.

So when something is written in the form y = mx + c, the coefficient of x, which is an m, is the gradient of the line and the constant represented by c here, is the y intercept.

And that's a really important point that we're gonna be using.

What this means is if we know the gradient and the y intercept, we can write an equation for the line in the form y = mx + c.

Let's look at an example.

This line has a gradient of four and a y intercept of (0, 7).

You might want to pause the video to check that you agree.

The equation for this line then is y = 4x + 7.

Has a gradient of four, so that's the coefficient of x and it has a y intercept of (0, 7) so that's our constant.

Could we draw a different line with a gradient four and a y intercept seven? Pause the video.

What do you think? No, it's impossible.

This graph is uniquely defined by its gradient and y intercept.

Any line with a gradient of four and a y intercept of seven is this graph.

Lucas has drawn a line with y intercept (0, 5) and gradient negative two.

Some of his classmates are trying to guess his equation.

Who is correct? Well done if you spotted that it's both Sam and Izzy.

The y intercept is the constant.

So they both have a positive five term in their equation and the coefficient of x is the gradient.

So you'll see that they've both got coefficients of x as negative two.

Let's have a go yourself.

So write the equation for the line which has these features.

Give it a go, and then we'll look at our answers.

So the first one, you should have y = 5x -4 or you could have written that as y = -4 + 5x.

B, you should have y = -x + 3.

If you wrote -1x + 3, that's the same thing.

Or y = 3 - x or if you wrote 3 - 1x, that's the same thing.

C, y = -4x + 0 but actually we don't need to write the plus zero.

So you could have written that as y = -4x.

So let's see if we can work out the equation of a line from this graph.

We need to calculate the gradient and find the y intercept.

So good practise for our gradient calculating skills.

So we're going for a movement of positive one in the x direction.

Then we need to look at how we get back to our graph.

So we need to subtract two in the y direction and that means our gradient is negative two.

The y intercept is (0, 3).

We can see it on our graph this time.

So the equation of the line is y = -2x +3 or if you want to write it the other way around, y = 3 - 2x.

So if the purple graph is y = -2x + 3 what do you think the equation of the green graph is? See if you can work this one out.

Hopefully, you wrote this as y = -2x + 1 or y = 1 - 2x if you want to write it the other way around.

The reason being, it has the same gradient as A 'cause they're parallel.

So it's always gonna have that negative two x term.

The y intercept is (0, 1) so this time, we should have a plus one.

The green graph is a translation of the purple graph.

It's just been translated down two, which is why our plus three has now become a plus one.

We need to be careful to check the scales when reading graphs.

You may have come across this before.

Let's look at the equation of this line.

So I've picked the coordinate (0, -12) and then I've moved two squares in the positive x direction.

But if you look at my axes, two squares actually represent 10 in the x direction.

And then to get back up to my line, I need to go two squares up.

But looking at my y axis, two squares actually represents four in the y direction.

So my change in x is plus 10.

My change in y is plus four.

To get a gradient, I need my change in x to be one.

If I divide them both by 10, I get 0.

4 or four tenths or two fifths.

My y intercept is easy to see.

It is negative 12.

So the equation of the line is y = 2/5x - 12.

I like to write gradients on y intercepts as fractions 'cause I think they're easier to use.

I also like to write my fractions in their simplest form, which is why I've chosen that form.

True or false then? This line has the equation y = 3x + 2? Well done if you spotted that was false.

What is the justification for why that cannot be true? Well done.

We shouldn't have had to do any working for that one.

We should have been able to look at it and go, it can't be correct 'cause the gradient should be negative.

I should have a negative coefficient of x.

What about this one, y= 4x +2.

See if you can do the minimum working out to work out if that's true or false.

That one is true.

We can see that the y intercept is two, and it has a gradient of four.

Every one in the x direction, we have to go up four.

It's two squares but it's representing four on the y axis.

How about this one? y = 12 x + 2.

True or false? Well done if you went with false.

Which justification is correct? This was a bit of a tricky one.

It looks like the gradient is less than 12.

It isn't actually.

If you find some coordinates to work out the gradient, which isn't easy to do on this graph, the gradient is 12.

We've just stretched the scales.

The thing that should have stood out to you is that the y intercept is negative.

It crosses the y axis below the x axis line.

So it should have a negative y intercept.

Time to put all that into practise.

So for each graph, can you work out the equation of the line by writing the gradient and the y intercept and then putting it together in an equation? Please keep your lines in the form y = mx + c for now.

Superb, same idea for this second set.

Can you work out the equation of the line, looking at the gradient and the y intercept? The third one you might find a little bit trickier.

I have plotted two coordinates for you to help you draw in your triangle to get your gradient.

Off you go.

Well done.

So the first one, a gradient of two, a y intercept of (0, 3), should have y = 2x +3.

Give yourself a tick for each of those elements.

If you made any mistakes, see if you can work out what it was.

B, gradient of one, y intercept of (0, -4).

The equation y = x - 4.

If you wrote it as y = 1x - 4, nothing wrong with that.

Again, give yourself a mark out of three for that one.

And C, gradient of negative three, y intercept of (0, 0), equation y = -3x.

If you put -3x + 0, it's not wrong.

However, in mathematics, we like to be as efficient as possible.

So y= -3x is a better way of writing it.

All right, looking at the second set.

So this gradient was a third.

The y intercept negative 3.

So our equation, y = 1/3x - 3.

E, gradient of negative 6.

It's a little bit trickier to work out.

You have to look at your scales.

Y intercept of negative eight.

So our equation y = -6x - 8.

And the final one.

Now the good thing about this question is that the steps in the x and the y direction are actually the same.

So although they're going up in zero point twos, the x and the y direction are the same.

What that does mean is you can just count the squares if you want because they're in the same proportion as they would be if each square was one.

So for the gradient, we've got negative four over three and the y intercept (0, 0.

2).

So you could change that to two tenths to keep your constant and your coefficient of x in the same format.

But because it's written on the graph as 0.

2, there's nothing wrong with leaving that one as 0.

2.

So final answer of y = -4/3x + 0.

2.

Equally, you could have written that the other way around as y = 0.

2 -4/3x.

In this second part of the lesson, we are gonna look at finding the equation of a line.

We can calculate the equation of a line from its table of values.

We're bringing together lots of our skills.

So here's a table of values for an equation.

We can see that when x increases by one, y increases by four.

That means the gradient of the line formed by these points is four.

We can also see that our y intercept is (0, -4).

That means we can write the equation of the line as y = 4x - 4.

Now if we don't know the y intercept, we can calculate it from the gradient.

So here I've got a table of values, but I haven't included the coordinate where x is zero, but we do know our gradient is negative two.

So I can work backwards and add two to find my y intercept of (0, 14).

You might just want to check that your x is increasing by one each time and that your y is decreasing by two each time.

The y intercept then is (0, 14).

So the equation of our line is y = -2x + 14.

We can calculate the equation of the line if we know the y intercept and another coordinate.

So we've got the y intercept element.

All we have to do is work out the gradient.

So looking at our coordinates, we've got a change in x of four, a change in y of eight, divide them both by four, we get our gradient of two.

Now we can put those elements together.

So the equation of the line, it could be y = 2x +4.

So so far we've looked at calculating the equation of a line from a table of values and then from two coordinates where one of them is the y intercept.

I'm gonna have a go at an example.

I would like you to watch carefully what I do and then you're gonna give this a go yourself.

So we're gonna calculate the equation of this line.

I've been given the y intercept and another coordinate.

I'm gonna use the coordinates to help me find the gradient.

So from zero to three, the change in x is three.

From eight to negative four, the change in y is negative 12.

To find my gradient, I need to divide both values by three.

So my gradient is negative four.

The y intercept is on my graph as (0, 8).

My equation then could be y = -4x + 8.

We know there's other forms that could be written in, but we're focusing on this form y = mx + c for the moment.

Right, have a go at doing the same with this one.

So calculate the equation of the line that goes through (0, -4) and (2, -10).

Off you go.

Well done.

So we've got the y intercept.

We can see it, but we need to calculate the gradient.

So from zero to two, we've got an increase of two.

From negative four to negative 10, we've got a decrease of six.

So that means our gradient must be negative three.

The y intercept, we can see is (0, -4).

Putting those elements together, equation of a line could be y = -3x - 4.

Make sure that you've got your y intercept and your gradient correct, and then you've got them written the right way round in your equation.

We can calculate the equation of a line if we know any two coordinates on the line.

So before we had the y intercept and one other, but actually any two coordinates on the line is enough to find the equation.

We're gonna look at the equation which passes through (3, 5) and (12, 14).

Before we look at some methods for this, remember that the equation of a line shows a relationship between the x and y values as a coordinate pair.

You might wanna just see can you spot a relationship between (3, 5) and (12, 14)? What's the relationship between x and y in both of those? Right, you might have said this, y seems to be x + 2.

This was a nice example.

However, with a lot of coordinates, it's just not gonna be easy to spot it straight away, which is why we're gonna need a method using our gradient and our y intercept.

But it's good to realise that there has to be some kind of relationship that works for both coordinates if they're on the same line.

So there's a few others there for you to try.

Can you spot the relationship between x and y? Well done if you spotted we've got y = 2x, y = x, and y= x- 3.

Like I said before, most of the time, it's just gonna be too tricky to spot that relationship just from the two coordinates.

We can find the gradient in the y intercept like we did previously and then work out the equation from them.

Where this is useful though is to check that relationship is correct.

We can see if our coordinates follow that relationship.

So let's start by calculating the equation of the line joining (3, 4) and (5, 8).

We're gonna do it by looking at the gradient.

So from three to five we have an increase of two.

From four to eight, we have an increase of four.

My gradient then, must be two.

So now I'm gonna use the gradient to work out the y intercept.

So I know that I have the coordinate (3, 4) and I have a gradient of two.

I know that I now need to count back through the gradient until I get to the coordinate zero something.

So if my x coordinate is currently three and the gradient is two, two times three is six.

So I need to take off six to get back to my coordinate with an x value of 0.

4 subtract six is negative two.

Because our coordinate (3, 4) is quite close to the origin, you might wanna just count through and check that that works.

So from zero to three, we've gone up 1, 2, 3 in the x direction.

So from negative two to four, we should have added two, then two, then two again, and that does work.

So the equation of the line is gonna be y = 2x - 2.

Remember, that when we're just given coordinates, just sketching the points can help you check whether the gradient is positive or negative.

So it is worth spending some time just sketching a quick axes, plotting roughly where the points would be in relation to each other to help support you.

Now we can check that this relationship holds true for our coordinates.

So is two lots of the x coordinate.

Subtract two, the Y coordinate.

So we're gonna do two lots of three.

Subtract two which is six, subtract two which gives us four, which is the y coordinate.

So (3, 4) is on that line.

Let's try it with this one.

Two multiplied by five, subtract two which is 10.

Subtract two or eight.

So (5, 8) is on that line.

So well done.

We seem to have found the equation of the line that goes through both of those coordinates.

So there's quite a few steps we looked at there.

So we're gonna break it down slowly.

I'm gonna show you how to calculate the equation of the line through these two points, and then you're gonna have a go at one yourself.

So first, I need the gradient.

You may wish to roughly sketch these points so you've got a bit of an idea of what is happening.

So from five to nine is plus four.

From four to 16 is minus 20.

Our gradient then is negative five.

Putting this into a broken table to help us.

So I know my gradient is negative five.

I want to work backwards to when x was zero.

So five lots of five is 25.

So I need to add 25 onto the four to get back to 29.

And again, you could use a calculator or you could count through to check that that is the case.

The y intercept is (0, 29).

So the equation of my line must be y = -5x + 29.

We're gonna look at checking that in a moment.

So have a go at calculating the equation of the line, which goes through (4, -3) and (6, 17).

Let's do this step by step together.

So from four to six is add two, negative three to 17 is add 20.

Looks like we're gonna have a positive gradient.

So our gradient is 10.

Looking at our table then, so we're adding 10 each time.

So we need to do four lots of 10.

That's 40.

We're gonna need to subtract 40 to work back to our y intercept.

Our equation of a line then is y = 10x - 43.

Well done if you've got some or all of those elements correct.

So now I'm gonna check that our answers work for the coordinates that we had.

So our equation was y = -5x + 29.

So let's try it with (5, 4).

So negative five times five plus 29 is negative 25 plus 29, which is four, which is what we were hoping for.

Let's try it with the other coordinate.

So negative five times nine this time.

So x coordinate is nine, plus 29, which is negative 45 plus 29, which is negative 16, which again, is what we were hoping for.

So both of those coordinates lie on that line.

So we found the correct line for those coordinates.

I would like you to do the same now for your two coordinates and your equation of the line.

Off you go.

Fantastic, so you should have shown that substituting four for x gives us negative three for Y and substituting six for x gives us 17 for y.

So yes, both those coordinates lie on that line.

Right, time to bring that all together for a practise.

So I'd like you to have a go at calculating the equation of the line for each of those tables of values.

Where you don't have the y intercept, obviously, you might need to do a couple of calculations just to work backwards or work forwards to what the y intercept would be.

If you're struggling, sketch yourself out a graph, put on the coordinates that you know, see if you can work out the y intercept and the gradient from that.

Give it a go, and then we'll look at the next set.

Fantastic, this time, you've got the y intercept and one other coordinate.

Can you work out the equation of each line? The main skill here is you're gonna need to find the gradient.

Give it a go.

Lovely, so now we're gonna have a go at calculating the equation of the line just from the coordinates.

So I've given you some steps to follow.

We want to find the gradient.

Use your ratio table to help you.

Then, we need to find the y intercept by counting backwards or forwards in our table of values.

Once you've got those two elements, you can write the equation.

Give it a go.

Fantastic progress you've shown there, guys.

So first one y = 4x - 7.

B, y = -11x + 3.

C, y = -2x and the y intercept is plus seven.

Well done if you counted back and got that y intercept.

And D, y = x - 22.

And again, well done if you worked all the way back to get that correct y intercept.

For the second section, so our challenge here was getting our gradient, so you should have a gradient of four for the first one.

That y = 4x + 12.

Gradient of negative 12 for the second one.

y = -12x + 12.

You should have a gradient of negative four for the third one, y = -4x - 12.

And finally, y = -12x - 12 for that last one.

Superb, let's see how many of the equations and lines you got just from the coordinates.

There's minimal information you've been given here.

You had to do most of the work so well done.

So A, our gradient is four and our y intercept is negative seven.

So y = 4x - 7.

B, our gradient is negative five, our y intercept is 20.

So we've got y = -5x + 20.

C, we've got a gradient of negative one and our y intercept of 8.

We've got y = -x + 8.

And the last one, we've got a gradient of a third.

Well done if you got that correct.

And a y intercept of 4.

So y = 1/3x + 4.

Fantastic work guys.

Please don't worry if there was bits of that that you didn't get completely correct.

I do advise that you pause the video though, look back through, follow your working, and see where it is that you made any mistakes.

Well done.

Well done, guys.

That was a fantastic amount of effort you put in today.

Let's have a look at what we learned.

We learned that we can write an equation of a line in the form y = mx + c when we know what the gradient is and the y intercept.

We're getting really good now at that link between the form of the equation and those key features.

We can write the equation of any line given the graph of the line.

We can pick the key features out and write the equation.

We can write the equation of the line given a table of values by finding out both the gradient and the y intercept and we can do the same given any two coordinates on the line.

We talked about how even though we're only given two coordinates, we can help ourselves out by drawing, sketching a graph, drawing tables, okay, and providing ourselves with a method to getting to that equation.

Well done for joining us on that journey today, and I look forward to seeing you in future lessons.