Lesson video

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

My name is Miss 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 our lesson on finding the equation of the line in the form AY plus BX plus C equals zero.

In this lesson, we are gonna be looking at writing linear equations in the form AY plus BX plus C equals zero.

And by the end of the lesson, you'll appreciate that writing in this format sometimes may be more appropriate than other forms of the equation of a straight line.

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

There's some challenging elements to this lesson, so if you haven't already looked at the equations of straight lines in other forms, then you might want to do that first before continuing with this.

So we're gonna split this into two parts.

We're gonna look at equations in the form Y equals MX plus C, which you might have investigated a little bit already.

And then we're gonna look at equations in the form AY plus BX plus C equals zero, which should contain some exciting new bits and pieces that you haven't looked at before.

So this format of Y equals MX plus C for an equation of a straight line is useful to identify the gradient and the Y-intercept.

The coefficient of X is the gradient, as this tells us how much Y will increase for every increase of one in the X direction.

If we're multiplying X by a constant, then every time X increases by one, Y will increase by that constant.

The constant term in Y equals MX plus C represents the Y-intercept, as this is the value of Y when X is zero.

This is a point we're going to explore in this lesson.

So all those coordinates there on the Y-axis, the X-coordinate is zero.

We can call the Y-axis X equals zero.

So what we can do is we can find the Y-intercept from an equation by thinking about when X is zero.

So we can substitute X equals zero into the equation of the line.

So this would give us Y equals two lots of zero plus five, Y equals zero plus five, so Y equals five.

So we've shown why the Y-intercept is going to be zero, five by substituting zero in for X.

This works for every value because multiplying any value by zero gives an answer of zero.

So in this form, Y equals two X plus five, if X is zero, the whole X term will be zero.

That just leaves the constant, so it'll leave you with Y equals the constant.

It's really important that you understand why it is that the constant represents the Y-intercept moving forward through this lesson.

So if you're not sure, pause, maybe rewind and follow that through again.

See if you are happy with why that constant represents the Y-intercept.

So knowing these key features can help us draw the graphs from their equations.

So how can we draw the graph Y equals four minus three X? Well we could use a table of values, so we can think of some coordinates for X, substitute into our equation to get our coordinates for Y, and then plot our relationship.

Hopefully, if we haven't made any mistakes, they will form a straight line because that's a linear equation.

Don't forget that if we're drawing the graph, we need to draw a line through them, going across our page with a ruler.

Or we could use the key features of the graph.

This method has its advantages and disadvantages, which we will explore in a moment.

So we know the Y-intercept must be zero, four.

So before we do anything else, we can actually plot the Y-intercept.

And we know that the gradient is negative three, so that we can plot our next integer coordinate along because for every one increase in X, Y decreases by three.

So we go right one, down three, plot another coordinate, right one, down three, plot another coordinate, and so on, making sure we draw our line all the way across our page.

So now we're gonna have a look at the pros and cons of this method.

So Aisha has tried to draw the line with equation Y equals two X plus four using this method.

Is she correct? What do you think? Well done if you said no, she's not correct.

So the pros of this method is it's really easy to plot the Y-intercept, and then you don't have to substitute into your equation anymore.

You can just draw your line using your gradient.

Where you might have to be careful is looking at the scale on your axes.

So you'll notice that the Y-axis actually goes up in twos.

So her gradient is actually four.

She's gone one right, up four, one right, up four.

So it is really important to check that scale on the axes before we draw in the lines using that method.

What advice could you give Aisha, do you think, so that she now gets it right? So you might have said that you can actually check the coordinates work.

So do two times two is four, plus four is eight, so two, eight should be on that line.

Oh, but it's not, we've got two, 12.

So that is one way of checking that it doesn't fit our rule.

The other thing is to make sure you read the axes, don't just count squares.

So Aisha has reflected on that method and has said, "As long as I'm careful with the scales on the axes, I can plot any linear equation using this method." Do you agree or disagree with her statement now? Most of the elements in that statement is absolutely fine, but just be aware that this only works for equations where we can identify the gradient and the Y-intercept.

So if they're in the form Y equal MX plus C, this is a great method to use, as long as you're careful with your scales.

However, lots of formats of equations of straight lines might not be in the form Y equals MX plus C.

It might not be as easy to see the gradient and the Y-intercept, so there might be better ways of plotting our graph, using a table of values, for example.

Can you think of any equations of lines which cannot be written in the form Y equals MX plus C? I'll give you a hint.

Can you think of any lines which will not have a Y-intercept? Well done if you said vertical lines.

They are an exception to the rule that straight lines have a Y-intercept.

Because they are vertical, they do not cross the Y-axis.

What coordinates lie on that line? Do you think you could tell me what the equation of that line there is? That line is X equals three.

There's no relationship between the X and the Y coordinate.

The only rules for the coordinates on that line is that the X value has to be three.

So equations for vertical lines can be written in the form X equals A, where A, zero is the X-intercept.

We'll do some more work with X-intercepts in the next part of the lesson.

They cannot be written in the form Y equals MX plus C.

They do not have a Y-intercept.

And they are perfectly vertical, so the gradient is infinite.

It cannot be written as a numerical value.

So have a go at this one.

If you know the Y-intercept and the gradient of a line, you can draw the line on a set of axes.

What do you think, true or false? And then justify your answer.

Well done if you spotted that is true.

You do not need to draw a table of values.

You can plot using the gradient and the Y-intercept, as long as you are careful by looking at your scales to get your gradient correct.

True or false, this time all straight lines can be written in the form Y equal MX plus C? Use the justifications to help you if you're not sure.

That one is false.

Vertical lines cannot be written in the form Y equals MX plus C, as they do not have a Y-intercept and the gradient is infinite.

So time for a practise.

What I would like you to do is match each graph with the equations underneath.

Think about the key features that we have talked about.

Off you go.

Well done on that first set of questions.

Let's look at the second set.

So for each question, I'd like you to work out the equation of the line in the form Y equals MX plus C.

I've given you the key features, and then I'd like you to write down the equation.

Once you've done that, can you draw the line on the axes? See if you can use those key features to draw that line.

Off you go.

Superb, the final set we're going to have a look at, exactly the same as before, but I've been a little bit more creative with some of the gradients I have chosen.

Okay, so take your time when using the gradient, and make sure that you've written your equation as well as drawing your line, off you go.

Fantastic effort.

Let's see if we've got most of these matched up correctly.

So A is Y equals X.

B, Y equals two X minus two.

C, Y equals two minus X.

D, X equals two, it's a vertical line.

E, Y equals negative X.

F, Y equals two.

G, Y equals two minus a half X.

Y wipes a half X minus two, make sure you've got those two the right way round.

I, Y equals X plus two.

And J, Y equals two minus two X.

Not as easy as it looked, that one, because lot of the graphs were similar, and you had to pay real attention to your fractions and your negative numbers, so well done.

For question two, the equation should be Y equals three X minus two.

And then just check that your graph looks similar to mine.

B, you've got Y equals negative X plus five.

And again just check you've got the correct graph.

Sometimes it's easy to check the Y-intercept and the X-intercept, if they're both visible, to check that the graph is in the right place.

C, we've got Y equals negative four X minus three.

And again you might wanna check some key points on that.

You should cross the Y-axis at zero, negative three.

And then you need to make sure you're also going through the coordinate negative two, five.

And that should hopefully make sure you've got your line in the right place.

This second set, we've got Y equals a half X plus four, so it should go through the coordinate zero, four, then two, five, then four, six.

So E, the equation of line is Y equals negative a third X minus one.

So make sure you're going through the coordinate zero, negative one, and then three, negative two, six, negative three, and you've got that same line as me.

Last one, Y equals two thirds X.

So it should have a line going through the origin, and then you need to go three right, two up to get to the coordinate three, two, and then three right, two up again to get to the coordinate six, four.

It was helpful this time that our scales on our axes both had steps of one so we could use the squares to help us.

Well done, in our second part of our lesson, we are gonna look at equations in the form AY plus BX plus C equals zero.

Writing equations of lines in other forms can help us identify other features.

Here is a bar model to show the equation Y equals two X minus three.

Y equals two Xs minus three.

What other equations can we write for this bar model? You could have Y plus three equals two X because the top bar and bottom bar are the same.

Two X minus Y equals three.

Two X equals three plus Y.

Let's think about this one.

What value would you get if you did two X, subtracted Y, and subtracted three.

Pause and have a think, and then we'll look at it together.

Okay, so there's our two X, and we're gonna subtract the Y, and subtract the three, zero.

Because those two bars are equal, if we do the whole of one, subtract the whole of the other, we'll have nothing left.

So we could write the equation, two X minus Y minus three equals zero.

All these relationships are the same, so all the graphs of these equations would be the same as Y equals two X minus three.

So another form of equations of straight lines is AY plus BX plus C equals zero.

Here are some examples.

Have a look at the features of these.

Here are some equations not written in that form.

Notice, for our examples, we had them all written equal to zero, we had a coefficient of X, a coefficient of Y, potentially a constant, but it was equal to zero.

We can use bar models to help us write the equations in this form.

So if we want to write in the form Y equals MX plus C from this bar model, we can see that Y is equal to four X plus five.

If we want to write in the form AY plus BX plus C equals zero, remember we're trying to get everything equal to zero, to nothing.

We can do four X plus five, and if we take away that whole other bar, we'll be left with nothing.

So four X plus five minus Y equals zero.

And remember, it doesn't matter which way round we write our X terms and our Y terms, it's still in the same form of AY plus BX plus C equals zero.

If you wanna rewrite that, so we've got our Xs, then our Ys, then our constants, we could have four X minus Y plus five equals zero, or we could have negative Y plus four X plus five equals zero.

Let's have a look at this relationship.

So in the form Y equals MX plus C, let's find our Y.

So this Y here is equal to three Xs subtract one, so three X minus one.

If we want to write in the form AY plus BX plus C equals zero, we've got our three Xs, subtract Y, subtract one, and we'll be left with zero.

So setting the equation of the line equal to zero can help us work out the intercepts of the graph.

So where would with the graph Y plus X minus five equals zero intercept the Y-axis? So we can substitute X equals zero into our equation.

So we've got Y plus zero minus five equals zero.

That means Y minus five equals zero.

What value of Y would make that true? What value, subtract five, gives you zero? Did you find it? Y has to be equal to five.

So the Y-intercept would be zero, five because when X is zero, Y is five.

Let's try the Y-intercept for this equation, Y plus X minus 10 equals zero.

Again, let's substitute X equals zero, 'cause it's the Y-intercept when X is zero.

So Y plus zero minus 10 equals zero.

Y minus 10 equals zero.

What value of Y would make this statement true? See if you can get it.

Well done, it's when Y is 10.

So when X is zero, Y has to be 10, so the Y-intercept is zero, 10.

Andeep reckons he's noticed a pattern, "I think the Y-intercept for Y plus X minus eight equals zero will be zero, eight." Do you agree? So the Y-intercept has an X value of zero.

Substituting it, Y minus eight equals zero.

That means Y has to be eight.

Perfect, Andeep is correct.

So Lucas says, "The Y-intercept for Y plus X plus eight equals zero will also be zero, eight." Do you agree? Let's try it out.

So substitute X is zero, so Y plus zero plus eight equals zero.

Y plus eight equals zero.

Or hang on, what value of Y makes this true? What number plus eight equals zero? It's negative eight.

So actually the Y-intercept is zero, negative eight for this example.

So when the equation of a line is in the form Y plus X minus eight equals zero, notice we've only got one Y and one X at the moment, we've got no coefficients larger than one, the Y-intercept is gonna be zero, A.

When the equation of the line is in the form Y plus X plus A equals zero, the Y-intercept is zero, negative A.

I don't suggest that this is something you memorise, but I do recommend you know how to test this out.

So we're substituting that X equals zero to work out the Y-intercept, and that's the skill that's really important.

What do you think the Y-intercept for Y minus X minus four equals zero will be? Why don't you just have a guess, then see if you can work it out, then we'll look at it together.

Let's try it, so Y minus zero minus four equals zero.

Zero minus four is minus four, so we've got Y minus four equals zero.

In order to make this true, Y has to be four.

So the Y-intercept is zero, four.

I would like you to have go at working out the Y-intercept for each graph.

As we said before, don't just guess, use your substitution skills.

Can you find the value of Y when X is zero? Off you go.

Let's have a look.

We should have zero, one for the first one, zero, negative 12 for the second one, zero, negative five for the third one, and zero, three for that final one.

Good substitution skills, guys, let's look at the next bit.

So we can use a similar idea to find the X-intercepts.

We've talked loads about Y-intercepts, but what about the X-intercepts? The X-intercept is where the graph crosses the X-axis.

So the coordinates of the X-intercepts on this graph are given there for you.

What do you notice? Pause the video.

What do you notice about the X-intercepts? All the X-intercepts have a Y value of zero.

So we can use a similar method for finding the Y-intercepts to find the X-intercepts.

The line crosses the X-axis when Y equals zero.

So let's substitute Y is zero into this equation.

So zero plus X minus five equals zero.

So X minus five equals zero.

What value of X will make this statement true? What does X need to be? X needs to be five.

Five minus five is zero.

So when X is five, Y is zero.

And you can just check that in your equation if you wish.

Make sure that when X is five, Y is zero.

So where would this graph, Y plus X plus five equals zero, cross the X-axis? Let's do it together.

Crosses the X-axis when Y equals zero.

So zero plus X plus five equals zero.

X plus five equals zero.

What number, when you add five, gets you zero? Well done, it is negative five.

The X-intercept this time would be negative five, zero.

Okay, so which of these could be an X-intercept for a graph? Think about what makes something an X-intercept.

Off you go.

An X-intercept when Y is zero, so three, zero, zero, zero, or negative 12, zero.

What do you think the X-intercepts are for these graphs? You don't need to guess, you can use your substitution skills.

Remember it crosses the X-axis when Y is zero.

Well done, we have one, zero for the first one, negative 12, zero for the second one, three, zero for that third one.

So why is this important? Why is this helpful to us? Well, we can use the X and Y-intercepts to help us easily sketch graphs when we've got equations in this form.

So let's try drawing the graph of the equation Y plus X minus seven equals zero.

The Y-intercept is going to be seven, 'cause the Y-intercept when X is zero.

The X-intercept is also gonna be seven.

We can see really clearly now where our line is going to go.

So we're gonna have a go at a couple of these together.

So I'm gonna do the first one.

I'd like you to watch what I've done, and then we'll work through the second one together.

So draw the graph of the equation, Y plus X minus four equals zero.

Well, we need some coordinates, so let's try the X-intercept and the Y-intercept.

So the Y-intercept, to start with, is when X is zero, so that gives me a Y-intercept of four.

If X is zero, Y has to be four.

I can plot that then as zero, four.

The X-intercept then is when Y is zero.

So I've got zero plus X minus four equals zero.

So X has to be four to make that true.

The X-intercept is four, zero.

Now that I've got those two coordinates, I can draw my line in.

Right, use mine as a support and have a go at drawing the graph of the equation Y plus X minus two equals zero.

Okay, our Y-intercept when X is zero, Y has to be two, so we can plot zero, two.

X-intercept, when Y equals zero, X has to be two, so we've got two, zero.

Drawing our line in, it should look like that.

Well done.

Let's try this one, Y plus X plus three equals zero.

So our Y-intercept, when X is zero, Y is negative three, so I've got zero, negative three.

Our X-intercept, when Y is zero, X is negative three, so we've got negative three, zero.

Drawing our line in, it looks like that.

We've had a few examples now.

So see if you can give this one a go on your own.

Off you go.

Outstanding, let's look at our working.

So our Y-intercept, when X is zero, Y has to be negative one, 'cause negative one plus one is zero.

When Y is zero, X has to be negative one, 'cause negative one plus one is zero.

So we've got our coordinates of zero, negative one and negative one, zero.

This might not be as easy to draw our line in this time, so you might want to plot a couple of coordinates on your grid that's following the same gradient before you try and draw your line in, but it should look like that.

Our final element, we're gonna have a look at some examples where A and B are not one.

Together, we are gonna draw the graph with equation two Y plus three X minus six equals zero.

Let's go.

First, let's find the Y-intercept.

We know that lines cross the Y-axis when X is zero.

So our equation looks like this, two Y plus three lots of zero minus six equals zero.

Now three lots of zero is zero, so this now looks like two Y minus six equals zero, and that means that two Y must be six, 'cause six minus six is zero.

Now we're gonna use our bar model to help us with this.

If two Y equals six, then that might mean each Y must be equal to three.

So this equation is true when Y equals three.

That means the Y-intercept would be zero, three.

Let's look at our next step.

So lines cross the X-axis when Y is zero, so let's try that.

So we've got two lots of zero plus three lots of X minus six equals zero.

Two lots of zero is zero, so we end up with three x minus six equals zero.

Now let's use a little bit of logic.

Six minus six is zero, so that three X must be equal to six.

I'll draw a bar model to help me.

So three Xs must be the same as six.

If we split that six up into six ones, then we can see that each X must be the same as two, so X must be two.

The X-intercept then is two, zero.

We've got our Y-intercept and our X-intercept, so we can draw our graph.

Look at that.

Well done.

There were so many skills that we were using there.

Do not panic if you found some of those elements a little bit tricky.

We are just building all those skills in to look at some fun and interesting equations.

So key bit of information then, select the correct option to complete this sentence.

The X-intercept is what? The X-intercept is where the Y coordinate is zero.

So the X-intercept is where the Y coordinate is zero.

The Y-intercept is when the X coordinate is zero.

It's important to get those the right way round.

True or false, when a graph is in the form AY plus BX plus C equals zero, the X and Y intercepts are just the value of C, but negative, true or false? And think about your justification.

That one is false.

That was the case when A and B are one.

So when it was just X plus Y plus something equals zero, that was fine.

But we looked to that example at the end, didn't we, where we had twos and threes as our coefficients, and then that wasn't the case.

So I'd like you to work out the Y-intercept and the X-intercept, and then have a grow at drawing your graphs.

Give it a go.

Well done, so for question two, I would like you to use the bar model to write the relationship in the form Y equals MX plus C, and in the form AXE plus BY plus C equals zero.

There's two questions to do, both formats for both questions.

And finally for three, I would like you to fill in the missing working where I've tried to draw the graph two Y plus five X minus 20 equals zero.

So you're gonna have to pay attention to what I've done, see if you can fill in the missing steps, and then draw the graph to finish off.

Give it a go.

Well done, so we've got Y-intercept of zero, four and X-intercept of four, zero, check your graph.

For B, a Y-intercept of zero, two, and X-intercept of two, zero.

For C, you've got a Y-intercept of zero, negative two, and an X-intercept of negative two, zero.

Question two, Y equal MX plus C should be Y equals three X minus two, and the other format three X minus Y minus two equals zero.

For B, Y equals 10 minus two X, or two X plus Y minus 10 equals zero.

Final question, so the missing line of working, two Y has to be equal to 20, 'cause 20 minus 20 is zero.

Our coordinate then is zero, 10, and you can plot that.

For our X-intercept, that first line of working should read two lots of zero, 'cause we're substituting Y is zero.

In our second line of working, we've got the missing value of 20.

We should be able to say now that five X minus 20 is zero, that means that our five X must be 20, 'cause 20 minus 20 is zero.

So the missing coordinate is four, zero.

If we plot zero, 10 and four, zero, we can actually draw the line of two Y plus five X minus 20 equals zero.

If you want, you could check that using graphing software and show that you've got that correct.

Well done, guys.

So much thinking going on in that lesson.

You might wanna just take a second to think about what it is that you have learned today, and to be really proud of yourself for how much progress you have made.

I really look forward to seeing you again in one of our other lessons.