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Today we're going to be talking about equations of straight lines.

Now the title is y is equal to mx plus c, and you'll see why this is during the course of this lesson.

Remember, I am Mr Maseko.

Please make sure you have a pen, a pencil, a ruler, and something to write on before you start this lesson.

Okay, now that you have those things, let's get on with today's lesson.

Try this activity.

Pause the video here and give this a go.

Okay, let's see what you have come up with.

Binh says, I'm thinking of a linear graph with a gradient of 2 that goes through the point zero three.

What are the coordinates are of Binh's line? Well, we have the point zero three.

So where is zero three on this graph? It's not the point that's labelled because that is zero two.

So zero three is, there.

We know this line had a gradient of two, What does that mean again? We did this in the previous lessons.

What does having a gradient of two mean? Good, it means that when we go up one in the x-axis, you go up two in the y.

So we know that the coordinate one five will also be on the line, and we can do the same thing up one of the x up two on the y, and then with the coordinates two seven.

And then we can go up one on the x, up two on the y, and we can have the coordinates three nine.

And so on and so forth.

Now let's connect this.

If we look at this, this line that blue line, that is Binh's line.

Because you see it goes through the point one five.

It's got the point zero three, it has a gradient of two.

So what other equations of each of these lines? Let's start with Binh's line.

What is the equation of that line? Remember, we discussed this last lesson.

When we refer to the equation of the line, we are talking about the name of the line.

So, if we look at our blue line, how do we find our y ordinates? Well we know the gradient it two, so we know about the coefficient of x will be two.

So the y is equal to two x, or is it just two x? Well, let's see.

If x is one, what's two times one? Well two times one would be two.

Well then when x is one, y is five.

So how do you get from two to five? Well you add three.

Now that add three, is interesting.

And we'll see why it's interesting.

Look at the graph and look at the equation of the line and let's see if you can figure out why it's interesting.

Now that red line, what's the equation of that red line? Well let's pick some points on this line.

We have the point one two.

We have the point two four.

We have the point negative one negative two.

Well on that line, all the y ordinates are double the x ordinates.

So that red line is just y is equal to two x.

That's just y is equal to two x.

Now let's look at the green line.

If you look at all these lines they all have the same gradient, because they're all parallel.

So we know the green line will also be y equals two x.

But is it just y equals two x? No it's not because y equals two x is that red line.

So something is happened to two x.

So let's pick a point on this line.

We'll pick the point here, that is two zero.

So two times two gives you four.

To get from four to zero, well that's take away four.

Again that take away four is interesting.

Can you spot why it's interesting? Well, let's explore this further.

When we write equations of lines, they have a general form.

This y equals mx plus c is the general form.

Now m, the coefficient of x, m.

So what does m represent? We've talked about this already.

M, the coefficient of x, represents the? Good, it represents the gradient.

Now what do you think c represents? What do you think c represents? Look at our graphs.

The first graph is y is equal to two x plus three.

That's the blue graph.

That's y equals two x add three.

The green graph is y is equal to two x take away four.

Well that add three and that take away four, those are the values of c, but what do the represent.

You got it? If you look at the graph look, for the y equals two x plus three, that graph crosses the y axis at the point zero three.

For the line y equals two x take away four that graph crosses the y axis at zero negative four.

The line y equals two x well that just crosses at zero zero.

Are you spotting it? So what is c? C is C that is the y intercept.

So it's the y intercept.

So where the line crosses the y axis.

If you look at the green line it crosses the y axis when y is negative four.

Hence two x takeaway four.

Look at the blue line it crosses the y axis when y is three.

Hence two x add three.

The red line is just y equals two x because two x I could write add zero but we're mathematicians we know we never write add zero.

Cause it crosses the y axis when y is zero.

So here's an independent task that I want you to try.

Pause the video here and give this a go.

Okay let's see what you've come up with.

For the line y equals two x plus one what is the gradient? Well we know the gradient is the coefficient of x.

So m is two.

Remember y equals mx plus c.

So gradient is two and y intercept coordinates? Well that would be zero one.

Because it crosses the y axis when y is one.

For the second one m would be three.

And the y intercept coordinate would be zero negative one.

This m is five and it crosses the y axis when zero add zero four.

So when y is four.

Now for that last line, what is the coefficient of x? Cause that's written a bit differently to y equals mx plus c Remember the gradient is always the coefficient of x so the gradient here is negative two.

And the y intercept coordinates well that would be zero five.

Another way that line could have been written, would have been y is equal to negative two x add five.

Now we have a line with a gradient of four that goes through the point zero four.

So the gradient is four so we know that that is y is equal to four x.

Well it goes through the point of zero four.

Well the coordinate of zero four.

That's a coordinate of the Good.

That's a coordinate of the y intercept.

So what is the y intercept? Well that would be four x add four.

So the equation would be y is equal to four x add four.

Because this graph would cross the y axis when y is equal to four.

Now really really well done if you got this.

So for this explore task fill in the gaps using these cards to make an accurate statement.

If you want to do this without a clue, pause the video in three, two, one.

So your clue.

Think about where this a graph would cross the y axis first So think about the y intercept first.

So let's say we have negative one.

So the graph is crossing at zero negative one.

Let's pick any of these let's say we have two x take away one.

So this graph has a gradient of two and a y intercept of negative one.

So what coordinates lie on the line two x take away one.

Well let's see what if x was equal to three? If x was three, what would the y coordinate be? Well two times three, that gives us six.

And six take away one that would be five.

So when x is three the y ordinate would be five.

So two x minus one goes through the point three five.

Okay so pause the video here and give this a go.

Okay let's see what statements you have come up with.

Well what else could we have done? Let's say that we have three x take away two.

Where does this line cross the y axis? At zero negative two.

So it crosses the y axis when the y ordinate is negative two.

So we want a coordinate with an x y value.

So let's say the x ordinate was four.

What would the y ordinate be? Well three times four gives you twelve.

Then twelve take away two well that gives us ten.

Well that doesn't quite work we don't have ten there.

So we can't have an x ordinate of four.

Let's try an x ordinate of, let's say what? What can we have? Five? An x ordinate of five.

Will that work? Well three times five.

Three times five is fifteen take away two that's thirteen.

Hm that still doesn't work we don't have thirteen.

Well let's try an x ordinate instead of saying an x ordinate of five let's say an x ordinate of one.

What would this be? Well three times one that is three takeaway two.

That gives us our y ordinate of one.

So we could have one one, but we have used one twice.

And this is the process that you would've had to go through to use these squares to fill in those empty boxes there.

Now I look forward to seeing whatever answers you came up with.

So if you want to share your work, ask your parent or carer to share work on twitter tagging @oaknational and #learnwithoak.

Thank you very much for participating in today's lesson.

Bye for now.