Lesson video

Hello.

Mr. Robson here.

Welcome to Maths.

What a lovely place to be, especially because we're efficiently drawing linear graphs today.

And we mathematicians, we love efficiency, So let's get on with this efficiently.

Our learning outcome is that we'll be able to draw a linear graph efficiently and accurately.

Key words, linear.

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

We'll see a lot of these today, linear graphs and straight lines.

Two parts of today's lesson, and we begin by plotting linear graphs.

Aisha, Izzy and Alex are plotting graphs of equations and they come across this equation, y equals 5x minus 7.

Aisha says, "The x and y exponents are both 1, so this graph will be linear." Izzy says, "We need to find some coordinates so we can plot it." Alex says, "We can fill in this table of values in order to plot it." Do you agree with them all? Pause this video, reread their statements and tell me whom you agree with.

Welcome back.

I wonder what you said.

The truth is they're all correct, but we can look more closely at some of the language they used.

Aisha said, "This graph will be linear." That's crucial 'cause it affects what Izzy and Alex had both said.

Izzy said, "We need some coordinates." Quite vague, but we don't need all the coordinates, just some coordinates.

And Alex said, "We can fill in this table of values." It is an option.

It will work.

If we do fill in that table of values, when x equals negative 3, y equals 5 lots of negative 3 minus 7, that's negative 22.

When x equals negative 2, y is five lots of negative 2 minus 7, that's minus 17 and so on and so forth.

We populate our table like so.

We then plot those coordinates, connect them with a line, and there it is.

The linear graph, y equals 5x minus 7.

This table of values method has some benefits.

The gradient is very visible.

You can see on the graph a gradient of 5.

For every change of positive one in the x direction, we have a change of positive 5 in the y direction.

You can see that change of positive 5 in the y and reflected in our table.

We can also see the y intercept very visibly in the table.

The middle coordinate there, when x is 0, y equals negative 7.

But Aisha says, "That was a lot of work.

You know this equation makes a straight line" And Izzy says, "So we only need some coordinates.

You didn't have to find seven of them." Alex asks, "How many coordinates do I need then?" What do you think? How many coordinates are required in order to plot a linear graph? Pause this video, and have a conversation with a person next to you or a good thing to yourself.

See you in a moment.

Welcome back.

I wonder what you said.

Aisha tells Alex, "If you know any two coordinates, you can draw the line.

When x equals negative 3, y equals negative 22.

When x equals positive 3, y equals positive 8.

That's those two coordinates.

Once we have those two coordinates, we can connect a line through them and there it is.

"That's the line and in far less time," says Alex.

Alex then asks, "Could I do it with just one coordinate?" The coordinate, 3, 8.

Is it possible from there to accurately draw the line, y equals 5x minus 7.

What do you think? Again, pause the video.

Have a conversation with the person next to you or have a good think to yourself.

See you in a moment.

Welcome back.

I hope you said something along the lines of what Izzy says.

"No, Alex.

There are infinite lines that could go through the coordinate 3, 8.

You need two coordinates to confirm which one unique line this is." "Thanks, Izzy," says Alex.

Quick check that you've got what I've said so far.

What is the minimum number of coordinates we need to plot the graph of a linear equation? Is it one, two, four or seven? Pause this video.

Answer that question to the person next to you or to me at the screen.

Welcome back.

I hope you said it's answer, B.

We need two coordinates in order to plot a linear equation.

A linear graph can't be drawn with just one coordinate.

It could be any one of an infinite number of lines going through that coordinate.

But once we have two coordinates, it is this line and only this line.

If we had any more coordinates, than we would've been acting less efficiently and we mathematicians, we like efficiency.

Thanks to Aisha and Izzy, Alex is now far more efficient when plotting the equation, y equals x plus 2.

"I don't need a full table.

I just need any two coordinates," says Alex and then chooses two coordinates, 1, 3 and 2, 4 will both fit the linear equation y equals x plus 2.

Once plotted, they look like that and then we draw a line through them.

Can you see a problem here? Aisha does? Aisha says, "Whilst I love your efficiency, don't forget that accuracy is also important in maths." And Alex realises, "Oh, no negative 10, negative 7, wouldn't be on the line y equal x plus 2." Those x and y values don't fit that equation.

Aisha says, "Accuracy is important.

By choosing to closely neighbouring coordinates you lost accuracy as your line reached extremities." If you're plotting in this range of x values, choose two coordinates that have more distance between them.

and Alex follows this advice.

"Okay, I'll use negative 5, negative 3 and 5, 7." Those are those two coordinates.

Aisha says, "Perfect." Now when you join those coordinates, you'll get a clearer picture of the gradient which will enable you to extend your line more accurately.

By making the line segment between those two coordinates, we can clearly see the gradient that's happening and extend it in both directions, thus drawing the line y equals x plus 2 very accurately.

Quick check you've got that, a true or false.

When plotting linear graphs using two coordinates, we can use any two coordinates.

Is that true or is it false? Once you've decide whether it's true or false, can you justify your answer with one of those two statements at the bottom of the page? Pause this video and do that now.

Welcome back.

Let's see how we did.

False.

Two coordinates is sufficient but we need to ensure there is a sufficient distance between them for the accuracy we need.

Practise time now.

For question 1, in each case, can you fill in the missing coordinates and draw the respective linear graph? There's one axis there for drawing a, y equals 2x minus 4, and one axis for drawing b, y equals 5 minus 3x.

Fill in the coordinates and draw those lines.

Pause and give this a go.

This is question 1, part c and d.

Same task in each case, fill in the missing coordinate and draw the respective linear graph.

The only difference here, these linear equations are a little trickier.

Good luck.

Question 2.

Alex plots the new graph of y equals 2 plus x over 4.

Using these two coordinates, 0, 2 and 2, 2 1/2, they both fit that linear equation.

The line is drawn but it goes through the coordinate negative 9, 0.

Use the coordinate negative 9, 0 to demonstrate that this is wrong and write a sentence advising what they might do better.

Pause and do that now.

Welcome back.

Feedback time.

We were filling in the missing coordinates for drawing the respective linear graphs for question 1.

A was the linear graph of y equals 2x minus 4.

If x equals 0, y equals negative 4.

If x equals four y equals 4.

We can plot those two coordinates.

They're a good distance apart, which enables us to draw that line nice and accurately.

Your line should look just like mine.

For part B, the linear graph y equals 5 minus 3x.

When x equals 0, y equals 5.

When x equals 3, y equals negative 4.

Plot those two coordinates, again, a good distance apart.

We can do this really accurately with that line.

You might wanna pause now and just check your coordinates are the same as mine and your lines are the same as mine.

Question 1, part c and d.

In c, we were plotting y equals 3x minus 1 over 2.

If we substitute in x equals negative 1, y will equal negative 2.

And when we substitute in x equals 3, y equals 4.

We plot those two coordinates.

They lie there.

Join 'em with the straight line and there it is, the line, y equals 3x minus 1 over 2.

For part D, the linear graph y equals 5 over 2x minus 3 over 2.

When x equals negative 1, y equals negative 4, when x equals 3, y equals 6.

There's those two coordinates, the straight line connecting them and going beyond them in both directions.

That's the line y equals 5 over 2x minus 3 over 2.

Again, pause, just check your coordinates the same as mine and your line is the same as mine.

For question 2, Alex was plotting linear graph y equals 2 plus x over 4 using those two coordinates.

And he asked you to use the coordinate negative 9, 0 to demonstrate that they're wrong and write a sentence advising what they might do better.

When we substitute in x equals negative 9 into the expression 2 plus x over 4 we find that 2 plus negative 9 over 4 is not equal 0.

Therefore, negative 9, 0 cannot be a coordinate on the line.

In terms of advising, Alex, what they might do better Next time, you might have written, start with two more distant coordinates to increase accuracy.

The two that Alex chose to begin with are a little too close together in order to draw that line to its extremities.

Onto the second half of lesson now.

Well, we're going to look at efficiency in plotting.

We might always see linear equations in the form y equals mx plus c, where we very neatly have m describing our gradient and c helping us define our y intercept.

We often see the form axe plus by equals c.

That looks unusual when you first read it.

An example might help to clear that up.

An example being the linear equation, 5x plus 3y equals 15.

A table of values is possible but it's really inefficient.

If I just choose an x value from that table, for example, x equals negative 2, if we substitute that into our equation, 5 lots of negative 2 plus 3y equals 15, we can find that y value but it's really awkward and it's unnecessary to have this awkwardness.

There's a greater efficiency to be had.

We could rearrange it into the form y equals mx plus c.

If we subtract 5x from both sides and then divide through by 3 in our equation, we'll end up with y equals 5 minus 5 over 3x, but that's not a nice gradient to sketch.

Negative 5/3, that's quite tricky.

So what will be our most efficient way to draw such an equation? For accuracy and efficiency, there are two very obvious coordinates we can find to plot this graph.

For example, when x equals 0 and when y equals 0.

0 is a lovely number to substitute into that equation.

When x equals 0, we get 5 lots of 0 plus 3y equals 15.

Therefore, 3y equals 15.

Lovely.

Y equals 5.

We'll have the coordinate 0, 5.

When y equals zero we get 5x plus 3 lots of 0 equals 15, therefore 5x equals 15.

Lovely.

X equals 3.

We'll plot the coordinate 3, 0.

So when x equals 0, we get the coordinate 0, 5, when y equals 0, we get the coordinate 3, 0 and we plot those two.

Again two coordinates, that's all we need for a linear graph.

There are good distance apart, we can clearly see the gradient going on and we can sketch the line through both coordinates.

Quick check you've got that.

I'd like you to complete the coordinate values for when x equals 0 and when y equals 0 and thus draw the linear equation for x plus 2y equals 16.

When x equals 0, what's that missing y coordinate? When y equals 0, what's that missing x coordinate.

What will our graph then look like? Pause this video and give this problem again.

Welcome back.

Let's see how we did.

When x equals 0, substitute that into our equation.

We get 4 lots of 0 plus 2y equals 16.

That leaves us very nicely with 2y equals 16.

So y equals 8.

We'll plug the coordinate 0, 8.

When y equals 0 we get 4x plus 2 lots of 0 equals 16.

Two lots of 0, 0.

So 4x equals 16.

Therefore x equals 4.

We'll get the coordinate 4, 0.

When we plot those two coordinates and join the line through them, there it is, the linear equation, 4x plus 2y equals 16 done incredibly efficiently.

Sometimes we will see the form y equals mx plus c, but the numbers will not be integer values.

For example, the linear equation y equals 0.

4x plus 1.

2.

Again, a table of values is possible but it's inefficient.

We substitute those x values in and we get these y values.

These non-integer y values might now be quite difficult to plot accurately.

But there are two integer y values in our table that we could accurately plot.

Can you spot them? Negative 3, 0 and 2, 2.

There's two coordinates where the y values are integers.

When we plot those two, that's all we need.

There are good distance apart.

We can clearly see the gradient, draw the line, extended in both directions and there we go.

That's the linear equation, y equals 0.

4 x plus 1.

2.

Let's check you can do that.

Which of these x values will generate an integer y coordinate for the linear equation y equals 2.

5 minus 0.

25x? Is it x equals 0, x equals 1, x equals 2, x equals 6.

Substitute those x values in and see if you can spot which ones generate an integer y value.

Pause this video and do that now.

Welcome back.

It wasn't a, x equals 0.

When x equals 0 y equals 2.

5.

It wasn't b, when x equals 1.

When x equals 1, y equals 2.

25.

It was c, x equals 2.

That'll give us a y value of 2.

And d, x equals 6, also gave us a y integer value of 1.

Because 0.

25, the coefficient of x, when it's multiplied by 4, you get 1.

You know, you'll hit another integer y value for every increase of 4 in x.

So if x equals 2, gives us an integer y value, x equals 6 will give us an integer y value as well x equals 10, x equals 14, x equals 18 and so on.

It's crucial you can spot that kind of pattern.

And there they are, the coordinates, 2, 2 and 6, 1.

Join with a line and that's the linear equation, y equals 2.

5 minus 0.

25x drawn really quickly and efficiently by spotting the y integer values.

Practise time now.

Question 1, find the coordinates for when x equals 0 and when y equals 0 and thus draw the linear equations, 3x plus 2y equals 6 for part a, and 3x minus 2y equals 6 for part B.

Pause and do that now.

Part C and D of question one are very similar.

Find the coordinates for an x equals 0 and when y equals 0 and thus draw the linear equations for c, linear equation 9x plus 3y equals 27 and d, 3y minus 9x equals 27.

Pause and do those two now.

Question 2.

For each graph find two x values that will give you integer y values and thus draw the linear equations.

Y equals 0.

8x minus 3 for part a and y equals 3.

25 minus 1.

25 x for part b.

Pause and give that a go now.

Feedback time now.

For question 1, we are finding the coordinates when x equals 0, y equals 0 and join the respective linear equations.

For part a, when x equals 0, y equals 3, when y equals 0 x equals 2.

When we plot the coordinates 0, 3 and 2, 0, they lie there, connect a line through them and that's the equation, 3x plus 2y equals 6.

For part b, the equation 3x minus 2y equals 6.

When x equals 0, y equals negative 3, and y equals 0, x equals positive 2.

Plot those two coordinates, connect with a line and that line is 3x minus 2y equals 6.

I wanna pause this video, check your coordinates match mine and check your lines match mine Four part c and d 9x plus 3y equals 27.

When x equals 0, y equals 9 and y equals 0 x equals 3.

There's your two coordinates connected with a line, that line being, 9x plus 3y equals 27.

I wonder how the line 3y minus 9x equals 27 will look different.

Well, when x equals 0, y equals 9 and when y equals 0, x equals negative 3.

Plot those two coordinates and there it is, the line 3y minus 9x equals 27.

Question 2, I asked you to find the x values that generated integer y values, thus making it much easier to draw his linear equations.

For y equals 0.

8x minus 3, the coordinate 0 minus 3 worked beautifully.

When x equals 0, y equals negative 3.

Then, because if you did your multiples of 0.

8, it would go no 0.

8, 1.

6, 2.

4, 3.

2, 4.

You know that you'll hit another integer for every change of 5 in X.

You could move 5 in the negative x direction and generate the coordinate negative 5, negative 7.

You could move five in the positive x direction and generate the coordinate 5, 1.

Any two of those three coordinates would've been sufficient to enable you to accurately draw that line y equals 0.

8 x minus 3.

For part b, y equals 3.

25 minus 1.

25x.

I certainly wouldn't want to do a table of values for this one.

So, which x values generate integer y values when x equals 1, y equals 2.

That's a good coordinate to start with.

Then because the multiples 1.

25 would go, 1.

25, 2.

5 3.

75, 5, you know you'll hit another integer for every change of 4 in x.

So we could have moved positive 4 in the x direction and got the coordinate 5, negative 3, or we could have moved negative 4 in the x direction and got the coordinate negative 3, 7.

Any two of those three coordinates would've been sufficient to enable you to accurately draw that line y equals 3.

25 minus 1.

25x And here we are at the end of the lesson.

In summary, we don't always need a complete a table of values to draw a linear graph.

They can be drawn more efficiently with just two coordinates because the nature of their straight line form.

To draw them accurately, the distance between the two points should not be too small.

You should check your calculations carefully if you use less points.

I hope you enjoyed this lesson and I look forward to seeing you again soon for more maths.

Goodbye for now.