Lesson video

Hello and welcome to another video.

In this lesson we'll be looking at equivalent lines.

Now this is our final lesson, on equations of lines.

And you've made some great progress on this.

As same as always, remember to have a pen, a pencil, something to write on before continuing with today's lesson.

Okay.

Now that you have those things I'm Mr. Maseko and let's get on with today's lesson.

First, try this activity.

Fill in this Venn diagrams with some coordinates.

What do you notice? As a bit of a clue, if we look at the coordinates for the line x plus y is equal to five, Well we could have the coordinate two, three.

The coordinate two, three will be on the line x plus y equals five.

Would that coordinate be on the line y equals two x minus three? Well, two times two is four take away three that would be one.

So it wouldn't be so that the point two, three would be here, and the point two, one would be here for y equals two x minus three.

Okay.

So you try this activity.

Fill in the Venn diagrams with some coordinates.

Pause the video here and give this a go.

Okay, now that you've done this, let's see what you've come up with.

Well we already have coordinates two, three and two, one on the Venn diagram on this side.

Well let's pick another coordinate.

What about the coordinate one, four? A coordinate one, four because it lies on the line x plus y equals five, one, four but it isn't on the line two x, y equals two x minus three.

two times one that would give you two take away three, That would be negative one.

So one, negative one would be on this line.

Now let's look at that second Venn diagram.

Let's take the coordinates two, three.

It's on the line x plus y equals five.

What about the line x plus two y equals 10? Well two times two, gives you four.

two times three gives six.

Four plus six well that's 10.

The coordinate two, three is on both those lines.

What about the coordinate one, four? 'Cause on the line x plus y equals five , well two times one is two, two times four is eight two plus eight that gives you 10.

It's also on the line two x plus two y is equals to 10.

What's going on here? Well let's see.

Now when we look at coordinates on the line x plus y equals five and two x plus two y equals 10.

Now we've already seen the coordinates two, three and one, four lie on both those lines.

Should we try another one? What about the coordinate? Well what else makes five and equals to.

So what about the coordinate five, zero.

'Cause that lies on the line x plus y equals five.

So two times five gives you 10.

two times zero is zero, 10 add zero.

Well the coordinate five, zero is also on this line okay.

It's looking like, wait, all my coordinates can't be on this line and this line, can they? Okay, let's try another one.

What about the coordinate four, one surely.

Let's try this one.

Its on the line x plus y equals five.

Well, two times four well that gives you eight.

two times one that gives you two, eight plus two that gives you 10.

Well, so the coordinates four, one is also, what's going on? Well if you look at this lines, and the line x plus y equals five and the line two x plus two y is equal to 10.

What do you notice about those two equations? Exactly, this equation is just a multiple of this one.

Two x plus two y is just two lots of.

So two lots of x plus y is equal to two lots two lots of five.

So all we've done is taken, this first equation and multiplied it by two.

Those two equations are equivalent.

They are the same equation, because they have the exact same coordinates.

So any time you take an equation and you multiple it, by an integer or any value you create an equivalent equation.

And that's if you multiped every term in the equation by the same value.

Well, let's see those this work all time.

Let's explore this a bit further.

So we have the line y equals two x plus three and two y equals four x plus six.

What have we done to this equation to get this equation? Well, all we've done is we've multiplied everything by two.

Let's see, let's pick a coordinate, let's pick the coordinate.

What coordinate will land on this line? Let's say the coordinate one two times one is two add three is five.

The coordinate one, five.

When x is one, y will be five.

Well, let's see.

Four times, well if y is five, two times five that gives you 10.

Well is that equal to four times one that's four add six yeah 10 is equal to 10.

The coordinate one, five is on both those lines.

And that's the thing.

When you have equivalent lines, you're just looking at the same equation.

All you've done is scaled it up, but the equation hasn't changed.

It just looks different, but it still has the same solutions.

So in this independent task match the equivalent lines.

For question two, state a line that is equivalent to two x take away y.

Pause the video here and give this a go.

Okay.

Now that you've tried this, let's see what you've come up with.

Well the equation x plus y equals 10.

What is that equivalent to? That is equivalent to two x plus two y equals 20.

'Cause we've just multiplied that equation by two.

Y equals five x plus one, well that's equivalent to three y equals 15x plus three 'cause that's just multiplied by three.

Two x minus y equals three, well that's equivalent to four x minus two y equals six.

Just multiple the equation by two.

And, three y equals five take away two x is equivalent to six y equals 10 take away four x.

Because all we've done is taken this equation and multiplied it by two.

Now state a line that is equivalent to two x minus y equals three.

Well you could have done four x take away two y equals six.

You could have done six x take away three y equals nine.

You could have done, x take away half of y equals 1.

5.

Could just div.

We've multiplied everything get from there to there.

We've multiplied everything, by a half.

And half of three is 1.

5.

So as long as the equation you've come up with is a multiple, or a factor of this equation then they're equivalent lines.

So let's explore this further still.

To find equivalent expressions can you, make all of the pink boxes the same number? What about the purple boxes? What about the green boxes? Can you make them all the same number? Try this now.

Okay, let's see what we can come up with.

Well we've got x plus two y equals 10.

Three x plus y equals 10 and two x plus five y equals 10.

Well, let's see, we've got x, three x and two x.

We want to make the red boxes all the same number.

Well they're not red they are pink, yeah the pink boxes all the same number.

Well let's see, one, three and two.

What number do one, three and two all go into? Well we can have six, if we make that number, ops go back.

If we make, if we make that number six, well what would you multiplied x by to get to six x.

We would multiply it by six.

So six times two y that is 12y.

six times 10, that is sixty.

And then here, if we multiplied this by two, so we got to do the same thing to the whole equation.

That would be two y.

And is 20.

And then we've multiplied this by three, so times three, times three and that would be 15y plus 30.

Can you make all the, purple boxes the same? Well the same way find the lowest common multiple for two y, one y and five y.

And then see what you have to multiple each of those numbers by to get that number.

And you can try the same thing for the green boxes.

Well the green boxes are already the same number.

And is it possible to have an expression, where these numbers the same, and these numbers the same and all these numbers the same? Well explore further and if you figure it out, ask your parent or carer to share your work on Twitter tagging @OakNational and #LearnwithOak.

I look forward to seeing what you come up with.