Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Superb choice to join me.

Today, we're solving graphically.

I love solving equations, so I imagine solving graphically is going to be something glorious.

Let's take a look, shall we? Our learning outcome is that we'll be able to recognise that the point of intersection of two linear graphs satisfies both relationships, and hence represents the solution to both those equations.

Many keywords in today's lesson.

One of which is linear.

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

Two parts to today's learning.

We're going to begin by looking at points of intersection.

We can draw the graphs of linear equations by calculating coordinates and plotting, or by using technology.

For example, the linear equation Y equals two X minus three.

There it is; the line Y equals two X minus three.

And you should be familiar with that.

Every coordinate on this line satisfies the equation Y equals two X minus three.

What do we mean by that? That coordinate (3, 3) is on the line.

When X equals three, Y equals three.

Let's plug those values into the equation.

The value on the left hand side is three, two lots of three minus three.

The value on the right hand side is three.

The equation is satisfied.

Pick another point: (2, 1).

When X equals two, Y equals one.

Let's plug that into the equation.

The value on the left hand side is one, on the right hand side, two lots of two minus three, that's one, and the equation is satisfied.

We could keep doing this for many more coordinates.

When X equals one, Y equals negative one.

Again, the equation is satisfied.

When X equals zero, Y equals negative three.

Again, the equation is satisfied.

Every coordinate on that line satisfies the equation Y equals two X minus three.

Remember that.

When we plot two linear equations with different gradients we get an intersection.

The linear equation Y equals two X minus three again, and the linear equation Y equals two minus a half of X.

There they are.

The intersection is the point where the two lines cross.

The point of intersection is unique in that it is the only point which satisfies both equations.

What's meant by that? This is the coordinate (2, 1), the intersection.

When X equals two Y equals one.

Let's plug that into Y equals two X minus three.

And we find we've satisfied the equation.

When X equals two, Y equals one.

Let's plug that into Y equals two minus a half of X.

When we do that, we find again we've satisfied the equation.

No other point has this property of satisfying both equations.

If I pick this point, for example, the coordinate (0, 2).

I'll plug it first into the equation Y equals two minus a half of X.

You'll see Y.

Because when X equals zero, Y equals two, that equation is satisfied.

The coordinate (0, 2) is on that line, so no surprise that that equation is satisfied.

I think you already know what's going to happen now.

When I plug X equals zero, Y equals two into the equation Y equals two X minus three, the left hand side is a value of two, the right hand side is a value of negative three.

The equation is not satisfied.

Only one equation is satisfied by that coordinate.

Contrast that with our intersection.

It is only the coordinates of the point of intersection which will satisfy both equations.

It's really important you remember that fact.

You may want to pause and write it down.

Quick check you've got that.

True or false? The point at the intersection of two linear graphs will satisfy just one of the equations.

Is that true or is it false? Upon deciding which you think it is, could you use one of the bottom two statements on the page to justify your answer? Pause, and have a think about this now.

Welcome back.

Let's see what you thought.

I hope you went for false.

"The point at the intersection of two linear graphs will satisfy just one of the equations." That's false.

Why? I hope you chose statement B to justify your answer.

The point at the intersection is unique in that it is the only point which satisfies both equations.

The intersection gives us other information.

They're the same two linear equations graft, the same intersection.

This intersection is the moment when the lines of two X minus three, and two minus a half of X meet.

Therefore, it's the solution to the equation two X minus three equals two minus a half of X.

An X input of two makes both equations equal.

Therefore, X equals two is the solution.

It's the X coordinate at that moment which provides our solution.

By X equals two being the solution, what I mean is it is two, and only two, that we can substitute into the equation in order to have a true equation.

IE: equality on both sides.

No other X value will achieve this.

Let's check you've got that.

The linear equations Y equals two X plus four, and Y equals one minus X, intersect at the point with coordinates (-1, 2).

Therefore, what is the solution to the equation two X plus four equals one minus X? Is it X equals two, X equals one, or X equals negative one? I'd like you to pause and have a go at this one now.

Welcome back.

Let's see how you did.

I hope you said it's option C.

The solution to two X plus four equals one minus X is the X coordinate where those two lines intersect.

That's X equals negative one.

The intersection gives us a solution to another equation; this pair of simultaneous equations.

Have you heard of those before? Simultaneous equations? Well, we know that this is the line Y equals two X minus three, and the line Y equals two minus a half of X.

Two equations for which we want the same solution.

We've already seen many solutions to the equation Y equals two X minus three.

We saw that the coordinate (3, 3) satisfied that equation, the coordinate (2, 1) satisfied that equation, the coordinate (1, -1) satisfied that equation, and the coordinate (0, -3) satisfied that equation.

But only one of these solutions satisfies both equations simultaneously.

Well done, you're one step ahead of me.

It's that one.

The point of intersection.

X equals two, Y equals one is the only one of those options that satisfies both equations simultaneously.

Therefore, we'd say the solution to this pair of simultaneous equations is X equals two, Y equals one.

This might look different to other equations you've seen in the past where you just found the value of one variable.

This time, we can find the value of two variables.

We'll give a value to X and a value to Y.

And this is the only solution which simultaneously satisfies both equations.

Quick check you've got that.

What is the solution to this pair of simultaneous equations? There are the lines Y equals three X minus three, and Y equals seven minus two X.

What's the solution? Is it A: X equals two? B: X equals two Y equals three? Or C: X equals one Y equals zero? Pause, have a think.

See if you can pick out the right solution.

Welcome back.

Let's see how we did.

I hope you picked B.

That coordinate (2, 3) is the intersection.

Therefore, it's the value X equals two, and Y equals three, which will satisfy this pair of simultaneous equations.

It wasn't C.

The coordinate (1, 0) is only on one of those lines.

It'll only satisfy the equation Y equals three X minus three and not the other.

It's only X equals two, Y equals three that's a solution that satisfies both equations simultaneously.

It couldn't have been option A, because our solution needs to include the value of both variables when we're presented with simultaneous equations like this.

Intersecting lines can solve more difficult simultaneous equations for us.

For example, the simultaneous equation three X minus two Y equals 12, four X plus three Y equals negative one.

Would you know how to solve this without a graph? It's possible we could solve it by manipulation.

But by plotting the graphs using technology, we can immediately see the intersection.

There are the graphs, and there's our intersection: (2, -3).

Therefore, the intersection (2, -3) gives us the only solution which satisfies both equations simultaneously.

That solution being X equals two, Y equals negative three.

Quick check you've got that.

What is the solution to this simultaneous equation? What a beautiful one.

Four X minus five Y equals negative 17.

Five X plus two Y equals 20.

What's the solution? Is it A: X equals two? B: X equals two Y equals five? Or C: X equals four Y equals zero? Pause, see if you can pick out the right solution.

Welcome back.

Let's see how we did.

I'm hoping you said option B.

Option C only satisfied one equation.

The coordinate for zero will only lie on one of those lines, not both.

We need to satisfy both equations simultaneously.

And it was our solution X equals two, Y equals five which did that.

What was wrong with A? Well, our solution needs to include the value of both variables.

Well done if you spotted that.

Practise time now.

Use the graphs to solve these equations.

There's three lines on that graph, and you'll see some intersections.

This is all the ammunition you need to solve those three equations.

Pause and see if you can find those solutions now.

For question two, I'd like you to use the graphs to solve these pairs of simultaneous equations.

The graph looks incredibly familiar.

They're the same lines.

Y equals two X minus three, Y equals three minus X, and Y equals a half of X minus six.

But this time we're solving pairs of simultaneous equations.

How will that be different, do you think? Pause and have a go at this now.

For question three, I'd like to use this graph to identify two simultaneous equations which can be solved, and find their solutions.

For part B, I'd like you to explain why we can't use this graph to find a third different pair of equations.

Why can't we find a third different pair of equations to be solved using these particular lines? You'll want to write a sentence to explain your thinking on that one.

Pause and do that now.

Feedback time now.

Question one.

Using these graphs to solve the equations.

Our first equation for part A is two X minus three equals three minus X.

We're interested in that intersection there between those two lines.

And that intersection happens when X has a value of two.

For part B, two X minus three equals a half of X minus six.

Well, those two lines intersect there when X has a value of negative two.

Therefore, X equals negative two is our solution there.

And for part C, those two lines intersect at that moment when X has a value of six.

Therefore, X equals six is a solution there.

For question two.

Using the graphs on the same lines, but solving pairs of simultaneous equations.

Again, we're looking for intersections.

For part A, the lines Y equals two X minus three, and Y equals three minus x, they intersected the coordinate (2, 1).

Therefore, the only solution that will solve both equations simultaneously is X equals two, Y equals one.

For part B, we find those two lines intersecting at the coordinate (-2, -7).

Therefore, X equals negative two, and Y equals negative seven are the only values that will satisfy both equations simultaneously.

For part C, we're interested in that coordinate of intersection (6, -3).

Therefore, our solution for part C is X equals six, Y equals negative three.

For question three, I asked you to use this graph to identify two pairs of simultaneous equations, which can be solved, and find the solutions.

One pair of simultaneous equations.

We can solve two X plus four Y equals 24, and two Y minus X equals four.

That will be that point where those two lines intersect.

Our solution: X equals four, Y equals four.

Look at the graph.

Can you see another intersection? Does that lead us to another similar pair of simultaneous equations that can be solved? It does indeed.

Another pair of simultaneous equations will be two X plus four Y equals 16, two Y minus X equals four.

They intersect there.

So the solution must be X equals two, Y equals three.

For part B, I asked you to explain why a third different pair of equations cannot be found and solved using these particular lines.

You might have said, "Because there are linear equations, and there are no more intersections; no more intersections, no more solutions for the equations we've already highlighted.

The third pair of lines, two X plus four way equals 16, and two X plus four way equals 24, they're parallel therefore they have no intersection and no solution." Onto the second half of our lesson now, where we're going to look at regions of a graph.

This'll look familiar, we saw it earlier.

The linear equation Y equals two X minus three, and some points which satisfy the equation.

Which points? These points.

We saw them earlier, and we know that they all satisfy that equation.

Why does this coordinate not satisfy the equation Y equals two X minus three? What do you think? I wonder if you are thinking because it's not on the line.

Well that would be true, but it's a slightly crude way to explain it.

A better way to explain it would be to say, "Because the Y coordinate is greater than the value of the expression two X minus three when X equals one." When X equals one, two X minus three has a value of negative one.

But this coordinate, the Y value is greater than negative one.

Y is no longer equal to two X minus three.

Y is greater than two X minus three.

So, for this coordinate we could say that this point has coordinates whereby Y is greater than two X minus three.

Let's check you've got that.

Could you write down the coordinates of another point where Y is greater than two X minus three? Pause.

See if you can write a coordinate down.

Write a few, if you can.

Welcome back.

You could have named any of these points.

There were a lot to choose from.

They are all coordinates whereby Y is greater than two X minus three.

You weren't just limited to those points; you could have chosen any of the infinite non integer value coordinates in this region also.

You didn't have to say (1, 1), you could have said (1, 1/2).

(0.

7, 1.

8).

You weren't limited to just integer values.

We call this a region.

The region where Y is greater than two X minus three.

This point is in the region where Y is greater than two X minus three, as we've just seen.

For this par of coordinates, the value of the Y coordinate is greater than the value of the expression two X minus three.

What do you think we might say, therefore, about the coordinates of this point: (2, -3)? What do you think we might say about that? Well done.

The value of the Y coordinate is less than the value of the expression two X minus three.

This point is in the region Y is less than two X minus three.

Let's check you've got that.

Could you write down the coordinates of another point where Y is less than two X minus three? Pause, see if you can spot a coordinate.

Spot two or three if you'd like.

Welcome back.

Looks familiar, but it's different.

You could have named any of these coordinates.

They're all in the region Y is less than two X minus three.

There are lots to choose from.

But the ones that I've shown you here are not the only ones.

You could have chosen any one of the infinite non integer value coordinates in this region.

If you wrote down (2.

3, -1.

8) you would also have been correct.

We call this the region where Y is less than two X minus three.

We can make more statements when we have two lines on the same graph.

Here's line Y equals two X minus three again, and the line Y equals three minus X.

What do we know about this point? We know that this point is in the region Y is greater than two X minus three.

We also know it satisfies the equation Y equals three minus x, because it's on that line.

Let's check you can do that.

Write down the coordinates of any other point where Y is greater than two X minus three, and Y is equal to three minus X.

Pause.

See if you can pick out another coordinate, or two, or three.

Welcome back.

Let's see how we did.

Any of these coordinates satisfy those conditions.

They're coordinates whereby Y is equal to three minus X, but Y is also greater than two X minus three.

Let's check you've got that.

Which point satisfies both Y equals two X minus three, and Y is less than three minus X? Is it A, B, or C? Pause and have a good think about this one.

Welcome back.

Let's see how you did.

I've solved it by a process of elimination.

Both coordinates, A and B, satisfied the equation Y equals two X minus three.

But B was the only one of those two to be in the region Y is less than three minus X.

Practise time now.

Question one.

Here is the graph of Y equals four minus three X.

Can you mark and label three points where Y equals four minus three X? Next, can you mark and label three points where Y is less than four minus three X? And finally, can you mark and label three points where Y is greater than four minus three X? Pause and try this now.

For question two, we've got two lines now.

The lines Y equals four minus three X, and Y equals X minus four.

For part A, I'd like you to mark and label three points where Y is equal to four minus three X, and Y is greater than X minus four.

For part B, I'd like you to mark and label three points where Y is less than four minus three X, and Y is equal to X minus four.

And then for part C, mark and label the one point where Y equals four minus three X, and Y equals X minus four.

Pause and do that now.

Welcome back.

Let's go through some feedback.

The graph of Y equals four minus three X; marking and labelling three points where Y equals four minus three X.

I've marked these three points.

(0, 4), (1, 1), (1.

6, -0.

8).

These are just examples.

These three coordinates all satisfy the equation Y equals four minus three X, but you are not limited to integer coordinates.

Any three coordinates on this line is fine.

You didn't have to pick the same three coordinates that I did.

For part B, I asked you to mark and label three points where Y is less than four minus three X.

I picked these three coordinates: (-2, 4), (-1, -2), and (0.

6, -1.

3).

These three coordinates are all in the region Y is less than four minus three X.

You were not limited to integer value coordinates.

Any three coordinates in this region is fine.

Any three coordinates below that line.

For part C, marking and labelling three points where Y is greater than four minus three X.

It'll be no surprise to you I've picked three coordinates that are above that line.

I went for (1, 5), (3, 0), and (2.

5, -1.

5).

These three coordinates all in the region Y greater than four minus three X.

And you weren't limited to integer value coordinates, and any three coordinates in this region are fine.

For question two, I asked you to mark and label three points where Y equals four minus three X, and Y is greater than X minus four.

I've gone for these three points here.

(0, 4), (1, 1), and (1.

5, -0.

5).

These three coordinates all satisfy the equation Y equals four minus three X, and the Y coordinates are all greater than X minus four.

You weren't limited to integer value coordinates again, and any three coordinates in this line and in this region are fine.

For part B, I ask you to mark three coordinates where Y is less than four minus three X, and Y is equal to X minus four.

I've gone for those three coordinates.

(1.

5, -2.

5), (1, -3), and (0.

-4).

These three coordinates all satisfy the equation Y equals X minus four, and the Y coordinates are all less than four minus three x.

Again, you weren't limited to integer value coordinates.

Any three coordinates on that line in that region will be fine.

For part C, I asked you to mark and label the one point where Y equals four minus three X, and Y equals X minus four.

I hope you marked that coordinate there.

There's only one coordinate that satisfies these conditions, that of the intersection in this case, the coordinate (2, -2).

Sadly, that's the end of today's lesson.

But in summary, we've learned there are infinite points on a graph that satisfy a given linear relationship.

However, the point of intersection of two linear graphs only has one coordinate that satisfies both relationships, and hence represents the solution to both of those equations.

We can use inequalities to describe regions of a graph that are created by one or more equations of lines.

Hope you enjoyed this lesson as much as I did, and I look forward to seeing you again soon for more mathematics.

Bye for now.