Lesson video

Hi everyone.

It's Ms. Jones here.

Today's lesson is all about representing simultaneous equations graphically, and it's the first of two that will show you how to do this.

Hopefully, you are now aware what simultaneous equations are and you've seen how to solve them algebraically.

And this is all about how we can represent those on graphs and use graphs to help us solve them.

However, before we can start, please make sure you have a pen and some paper, you remove any distractions from around you, and you can try and find a nice quiet space to work.

Pause the video now to make sure you've got all of that ready.

Now, we've got all of that sorted and ready to start.

Let's have a go at the try this.

The first thing I would like you to do is to write an equation to match each of the students statements.

So we've got Ben saying, I'm thinking of two numbers that have a sum of 10.

Think about that key word.

Xavier saying I'm thinking of two numbers.

I triple the first and add it to the second to get 15.

Have a think about how we could represent those as an equation.

Remember if you're thinking of a number and you don't know what the numbers are, you can use letters to represent them.

Pause the video now to have a go at this.

So hopefully you got something that looks like this.

You might have used other letters, but I've used X and Y because it's going to be linked to equations of lines in a minute.

So, Ben was thinking of two numbers, there are two different numbers.

So we needed two different letters and they have a sum of 10.

We have X, add Y cause that's what sum is equals 10.

Xavier's also thinking of two numbers, and I use the same two letters to represent them.

I triple the first, so our first letter that we used was X.

So I chip with the first 3X and add it to the second, Y which equals 15.

If you got something a little different with some different letters and that's absolutely fine, well done, but these are the equations that we're going to carry forward into the rest of the lesson.

So we've got X add Y equals 10, and we've got it represented on a graph.

So this is the line of X and Y equals 10.

Does this graph tell me the value of X and Y, have a little think about that for a second.

Does this graph tell me the value of X and Y and where would it be if it does? So this line tells us all of the different possibilities of X and Y that could make 10.

So for example, if I take this point here, I've got X equals six and Y equals four.

So six and four works.

If I take this point here, X equals two and Y was eight.

So it's all of the possible points and values of X and Y that would work for this equation.

So now you've got the red line three X and Y equals 15.

Now have a couple of seconds just to have a look at that and see whether that represents, all of the values of X and Y it could be as well.

So hopefully you've seen that yes it does, represent all of the values of X and Y that worked for this equation.

So if I take this coordinate here, I've got X equals four, and Y equals one, two, three.

So I've got three lots of four, which is 12, add three it's 15.

So that works.

If I take this coordinate here, I have got X equals one, and I've got Y equals 12.

So I've got three lots of one, which is three, add 12 equals 15.

So that works as well.

And you can see that you could take any point and they've got fractions and decimal points as well.

It's not just integers that would work for the values of X and Y.

What values of X and Y are true for both statements? We can use graphs to solve simultaneous equations, So have a little think about that.

What values of X and Y are true for both statements? Look at this graph here.

How can we use this graph to find the answer to that question? What values of X or Y are true for both statements? Where are those values, have a little think with yourself.

Hopefully, you realised that in order for X and Y to be true for both statements, they need to be on both lines.

So that means this point here, at intersection of those lines is going to give me the values of X and Y that works for both.

And this is how we use graphs to solve simultaneous equations.

Because when you're solving simultaneous equations, that is what you are trying to do, find the values that work with both of those equations.

And that is what we've done by using these graphs and finding the intersection.

It's the intersection that helps us to solve simultaneous equations using graphs.

Using this information, please pause it again now to complete your independent task.

So your independent task was building on the try this and the connect task.

So you were given two statements.

So Ben says, I'm thinking of two numbers that have sum are five and Xavier says, I'm thinking of two numbers.

I double the first and add it to the second to get 10.

The first step was to write an equation to match each of the student's statements.

So we could have written X add Y equals five, And 2X add Y equals 10 In poppy, you're then asked to plot those on a graph.

So you could have used.

Rearrange this and use the Y intercept and the gradient, or you could have plugged in some values to get some coordinators on that line then you plot it.

And then it asks you what values of X and Y are true for both statements.

So we were looking for the intersection of both of those lines.

The second question, just go straight to part C, what values of X and Y are true for both statements? You still have to follow the exact same steps because the whole point of solving simultaneous equations is to answer the question what values of x and y are true for both.

So these are the answers that we should have got.

We can see here where they intersect.

We've got X equals five and Y equals zero.

And then for the second one, depending on how you interpreted the question, you could have got different answers.

So the difference of the two numbers was one for Ben statements.

So I'd imagine most people probably wrote X subtract Y equals one, and then you would have got X equals three, and Y equals negative four when you were finding the values that would work for both of those statements.

Or you would have got X equals 1/3 and Y equals 4/3 if you decided to do it the other way around of why subtract X equals one, cause it wasn't explicit there.

So well done if you managed to get either of those amazing job, a well done if you managed to plot graphs, especially when you don't have graph paper, it's a little bit tricky, but really well done for that.

What I would like you to do now is create similar "think of a number of problems" and solve them graphically.

You can choose any wording that you like.

You can create your own think, and you can make it as complicated as you would like to, and that's absolutely fine.

You might get some answers that are decimals, fractions, negatives, something a little bit tricky, and that would be brilliant.

If you think that's going to be a little bit tricky for you, then use a template of the statements that we saw before.

So just fit in the gaps.

I'm thinking of two numbers that have a sum of something, I'm thinking of two numbers, I do something to the first I double, triple, quadruple the first and add it to the second, to get another number.

And you can fill these out as many times as you want.

I would say, do this two or three times, but if you would like some more that's brilliant and Solve them graphically, see what answers you get.

And even better, you can check your answers by solving them algebraically as well.

So using elimination or substitution to find the answers to that, as well as the graphs.

So pause the video here to have a good nap.

Hopefully you had a good experiment with those and you managed to find some great answers.

There were obviously infinite numbers of possibilities.

So we can't go through the answers, but hopefully you might actually get something and you can always check online whether you've got the correct answers to X and Y values.

Remember to take part in the quiz and complete the quiz to test your understanding, and a really well done for today's lesson.

Some of that drawing and plotting on graph could be a bit tricky without graph paper, without a teacher present, but you've done absolutely brilliantly.

Well done.