Lesson video

Hi, and I, this is Ms. Bridgett.

In this lesson we're going to be looking at algebra simultaneous equations and in particular, we're going to be looking at checking solutions to them.

You're going to need pen, you're going to need some paper, and take a moment to remove any distractions.

Okay, let's make start.

On the screen you can see an equation.

And in that equation, two knots of x plus y is equal to eight.

I would like you to have a think about what could the values of x and y be.

Try and come up with quite a few possibilities.

You don't have to use integers you can be more adventurous.

Pause the video, and have a try.

Okay, here are the values that I came up with.

This isn't all of them.

You might have some of them, but there are certainly a lot more than this.

So I started off with x is equal to one.

Now if x is equal to one, two knots of x is two.

Which means that y has to be equal to six.

I worked systematically so then I tried out x is equal to two.

Now if x is equal to two, that means y has got to be equal to four in order for the left hand side of the equation to be equal to the right hand side.

Let's go on.

If x is three, y would have to be equal to two.

In order the 2x plus y to be equal to eight.

Now I started to notice a pattern here.

I don't know if you have.

So I went backwards instead.

X is equal to zero.

Now if I substitute x is equal to zero into there, or if I use my pattern, either way, I get that y has to be equal to eight.

Let's go into the negatives.

X is equal negative one, that means y has to be equal to 10.

Now just to be slightly more adventurous, let's finish off with something that isn't an integer.

2.

1 so somewhere in between the two and the three.

I can't use my pattern in here because I strayed away from the integers.

But I can still use substitution.

So I know that 2x's must be equal to 4.

2.

So I can work out that y must be equal to 3.

8.

So these are just some of the values that I could have.

And you'll know from the straight back glass units.

That there are an infinite amount of possibilities that you could have come up with.

Here, we've got a set of simultaneous equations.

2x plus 6y is equal to 18.

And 2x plus y is equal to eight.

We've got two equations, with two unknowns.

Now Zaki thinks he knows what the solution to these equations is.

He thinks that x is equal to six and y is equal to one.

Yasmin however, thinks something different.

She thinks that the solution to these simultaneous equations is x is equal to three and y is equal to two.

Who do you think is correct? Pause the video, and have a think.

Okay let's take a closer look at their solutions.

So Zaki thinks that x is equal to six and y is equal to one.

So let's take those values, and substitute them into the equations.

So let's substitute them into the first equation.

So if we substitute the x, the six, and y is one.

We get two knots of six, plus six knots of one, which I think is equal to 18.

Which is what we want.

The left hand side is equal to the right hand side.

So far so good Zaki.

Let's try them out in the second equation.

Remember the value of x has to apply to both the first equation and the second equation.

Let's try out the second equation then.

So let's substitute in x equals six and y equals one.

Now if we do that, we get two knots of six, plus one.

Which is not equal to eight, it's equal to seven.

So sorry Zaki, these are not the solutions.

Let's have a look at what Yasmin said.

So she thinks that x is equal to three and y is equal to two.

Let's out those into the first equation.

So we get two knots of three, six knots of two, which is six plus 12 which is in fact to 18.

So far so good Yasmin.

Let's try out in the second set.

This time we've got, 2x plus y is equal to eight.

Let's substitute in the three in and the two.

We get two knots of three plus two, which is in fact equal to eight.

So Yasmin is correct.

So she is correct because her solutions apply to both of those equations.

Zaki isn't correct because his solution only apply to one of the equations.

On the screen, you can see four pairs of simultaneous equations.

Now some of those pairs of simultaneous equations have the solution, of x is equal to two and y is equal to three.

I would like you to identify which pairs it is.

Pause the video and off you go.

Okay, hopefully you've managed to identify which equations do have the solutions of x is equal to two, and y is equal to three.

I'm just going to very quickly run through how I worked this one out.

So on the top left, I substituted in my values of x is equal to two and y is equal to three.

I found that the values of two and three for x and y satisfy both of those equations.

For all of those, for both of those equations, when I substituted in two and three the left hand side was equal to the right hand side.

So yes for that top left pair of equations the solution is x is equal to two and y is equal to three.

When we move onto the right hand side, I didn't substitute into both of them this time.

I started with the bottom equation because it looks simpler.

And I substituted x is equal to two and y is equal to three into that one.

Now what I found was that the left hand side wasn't equal to the right hand side.

So I knew that solution did not satisfy that equation.

So there was no point in me trying the second equation.

That was not the solution.

Bottom left equations started with the top equation.

X plus y is equal to five and substituted in two and three.

They did satisfy that equation.

The left hand side is equal to the right hand side.

So then I went ahead and substituted into the second equation.

The 7x plus 5y is equal to 22.

Now this time, you'll see that the left hand side does not equal the right hand side.

Those values of x and y do not satisfy that equation.

So two and three is not the solution.

And then the final equation, we substitute two and three for x and y into the top equation.

We find that they satisfy it.

And again, with the bottom one.

So two and three are the solution for the bottom right equations.

So my top left and my bottom right equations do indeed have the solutions of x is equal to two and y is equal to three.

Your final task of this lesson, I'd like you to create three pairs of simultaneous equations of your own.

Now I don't want you to just come up with any old equations, I want you to come up with simultaneous equations which all have the solution x is equal to two and y is equal to five.

Pause the video and have a go.

Okay, now I wonder what your strategy was.

I'm going to share with you my strategy.

That may be the same or different to yours.

So what I did, was I picked an expression in this case, 5x subtracts 3y.

And I substituted into that my values of x and y which I want to be the solutions.

X is equal to two and y is equal to five.

I picked another expression and I did exactly the same thing.

And I picked a third expression.

You'll notice I made it a bit easy for myself in this third expression.

So x plus y if I substitute into that two and five, I know that x plus y is equal to seven.

Now from there, I just picked out different pairs.

So my first pair of simultaneous equations, was those two.

My second pair of simultaneous equations was the top and the bottom equation.

And this was my final pair of simultaneous.

And I know that all of those have got the solution x is equal to two and y is equal to five.

Well thanks for finishing today.

So thank you very much, all of your time and all of your attention and I will see you next lesson when we look at how we find those solutions ourselves.

Thank you, goodbye!.