Lesson video

Hello, Mr. Robson here.

Great choice to join me for math today.

We're solving simultaneous linear equations graphically.

Well, that sounds wonderful, so let's get on with it.

Our learning outcome is I'll be able to solve two linear, simultaneous equations graphically, zero or one or infinitely many solutions.

Well, that sounds exciting.

Key words today.

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

We're gonna see a lot of linear graphs today.

We're gonna have two methods today to solve simultaneous linear equations graphically.

We're gonna start by looking at what it means to solve graphically without technology.

There's lots of ways to solve the equation, 3x - 2 = 4.

We could do it algebraically.

We write the equation and we manipulate.

Let's add positive two to both sides.

That'll leave us with 3x = 6.

We can multiply both sides by a third.

I'll be left with x = 2 as the solution to the equation.

It's the only x value that will satisfy that equation, but we weren't limited to doing it algebraically.

We could have solved that equation graphically.

We need to plot the line y = 3x - 2.

To do that, we'll need an axis and a table of values.

We'll populate the table of values by substituting in x = 0 and getting y = -2.

x = 1, y = 1, x = 2, y = 4.

When you put those into the table, you'll notice a pattern.

Looking at the y terms, I can see that they change by positive three each time.

When we plot those, we start to see that constant rate of change in the graph.

They form a straight line.

It's a linear graph.

The next question is when does that line hit four? Remember, we're solving 3x - 2 = 4.

It hits four there.

We read from the y axis four.

We get to that coordinate and at that coordinate, x = 2.

Oh, look, x = 2.

That's our graphical solution and it's the exact same as our algebraic solution in maths.

It's lovely to have multiple ways of doing the same problem.

The good thing about working graphically is we can use this graphical method to solve a pair of simultaneous equations.

If I said to you, solve y = 3x - 2, y = 3 - 2x, you could do it algebraically, but we can also do it graphically.

We need to graph the lines.

You've already seen the line y = 3x - 2 with that table of values and that straight line graph.

Let's do a table of values for y = 3 - 2x.

I'm gonna start by substituting in x = 0.

That's the easiest one to substitute.

three minus two lots of zero is just three.

I'll substitute in x = 1 next.

Three minus two lots of one, well, that's one.

And a pattern is starting to emerge.

Three, one, bet you the next one's negative one.

It is.

You can populate that table of values.

We'll plot those coordinates and we'll draw a straight line.

So how do we get from here to a solution? It's this moment we're interested in.

It's called the point of intersection.

It's where the two lines intersect.

This point of intersection where the two lines meet shows us a solution.

So, the coordinate one, one for this example.

Therefore, our solution is when x = 1 and y = 1.

x = 1 and y = 1, these values, they are the only values which satisfy both equations at the same time.

What do I mean by that? I mean if you substitute x = 1 and y = 1 into our first equation, it is satisfied.

One is indeed equal to three, lots of one minus two.

Let's substitute x = 1 and y = 1 into our second equation.

Again, it is satisfied.

If we test the coordinates of any other point than the point of intersection, for example, let's test three, seven.

When x = 3, y = 7, the first equation is satisfied.

Seven is indeed equal to three lots of three minus two.

But when we substitute x = 3, y = 7 into our second equation, that's not true.

Seven is not equal to three minus two lots of three.

We haven't satisfied our second equation.

x = 3, y = 7 only satisfies one of our equations.

It does not satisfy both equations simultaneously.

Therefore, it is not a solution to this pair of simultaneous equations.

The solution we saw earlier is x = 1, y = 1.

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

In order to be successful at this, we're going to need to plot linear graphs.

So for this check, I'd like you to match each table of values to the correct equation.

Pause this video and do some matching now.

Welcome back.

Let's see how we go on.

Hopefully, we matched y = 2x - 5 to the third table down.

How did we know to do so? Well, when x equal zero, y must equal minus five.

That value is in that table.

y equals three minus x of course must be the top table when x equals zero, y equals three.

We can read that coordinate from that table.

y equals four x plus eight matches to the bottom table, but of course it does when x equals zero, y equals eight and then y equals five minus a half of x is the second table down.

I hope you matched yours like this.

Next check.

Which solution satisfies this pair of simultaneous equations? The simultaneous equations y equals two x minus five and y equals five minus a half of x.

I've put those two lines on a graph for you there.

Can you pick the solution which satisfies both? Pause and choose your answer now.

Welcome back.

Let's see how we gone.

It was not a, x equals zero, y equals five.

That only satisfies one of the equations.

You'll notice that coordinate is only on one of the two lines.

It was not b, x equals five, y equals five.

Again, satisfies only one equation, satisfies the other equation this time, but it doesn't satisfy both.

It was point c and we know it's point c because that's the point of intersection.

That is the only set of values that will satisfy both equations simultaneously, x equals four, y equals three.

It wasn't the bottom one, x equals three, y equals four.

If you look where that coordinate is, you'll see it satisfies neither equation.

Your next check now, which solution satisfies this pair of simultaneous equations? The equations y equals three minus x and y equals four x plus eight.

There's four to pick from and you'll want to use that graphical representation to make the right choice.

Pause and do that now.

Welcome back.

How do we do? I hope you said it's option d, x equals negative one, y equals positive four.

Why? Because the point of intersection, it's the only moment when x equals negative one y equals positive four that satisfies both equations simultaneously.

Next, solving graphically this pair of simultaneous equations, two y plus four x equals 12, six y minus three x equals six.

A table of values, we could do it that way, but it's unnecessary and inefficient for this example.

To graph a linear equation, we only need two coordinates.

It's a straight line after all.

To find two coordinates for this equation, it's quite nice to look at the coordinate.

When x equals zero, y must be two lots of y, plus four lots of zero equals 12.

Well that's just two y equals 12, so y equals six.

Ah, when x equals zero, y equals six.

We can find another coordinate in a similar fashion.

How about when y equals zero? Two lots of zero plus four lots of x equals 12.

Well, if four x equals 12, x equals three, when y is zero, x is three.

We can find that coordinate two.

Once we've got those two coordinates and we know it's a linear equation, we draw a straight line and there's the linear graph of two y plus four x equals 12.

Let's check.

You can do that second equation.

For the equation six y minus three x equals six, can you fill in those blanks? When x equals zero y equals what? When y equals zero, x equals what? Find those values and where will those coordinates be? Pause.

Do that now.

Welcome back.

Let's see how we did.

So, when x equals zero, we get six y equals six, so y must equal one.

We plot that coordinate there, zero, one.

When y equals zero, six lots of zero minus three lots of x equals six.

Therefore x must be negative two.

We plot that coordinate negative two, zero.

Once we've got those two coordinates, we can draw a straight line.

There is the linear graph, six y minus three x equals six, which is the correct solution.

Now, that we've plotted both those lines, three to choose from.

Pause and take your choice now.

Welcome back.

Let's see how we did.

It was not option a.

The coordinate zero, one will only satisfy one of the two equations.

Did anyone say option b? It's not enough to say option b, x equals two.

Well, the x value of the solution might well be two, but without the y value we haven't solved this pair of simultaneous equations.

Our solutions must include the value of both variables, so it was c, x is two, but y equals two.

Practise time now.

Question one.

I'd like you to complete the table values for each equation, plot the lines and thus solve this pair of simultaneous equations.

Pause and give that a go.

For question two, I'd like to solve graphically this pair of simultaneous equations, y plus two x equals two, four y minus two x equals negative 12.

Notice, we haven't been given a table of values and just to remind you, we only need two coordinates in order to plot a linear equation.

Question three, the graph of y equals two x minus seven is already drawn on the grid.

For part A, I'd like you to draw the graph of five y plus three x equals 30.

Again, a reminder, you only need two coordinates to block that line.

Then for part B, I'd like to use the graphs to solve this pair of simultaneous equations.

Five y plus three x equals 30, y equals two x minus seven.

Pause.

Give this problem a go now.

For question four, I'd like you to use these lines to solve these pairs of simultaneous equations.

A, the first pair four Y minus two x equals negative eight and x plus y equals 2.

5.

Which of those three lines is four y minus two x equals negative eight and which is x plus y equals 2.

5? Once you've identified that, you need to find the point of intersection and that point of intersection will give you the x and y values that satisfy both equations simultaneously.

Once you've done that for part A, there's another pair of simultaneous equations in part B.

Pause, give these two problems a go now.

Welcome back.

Feedback time.

Question one.

I asked if complete the table of values for each equation, plot the lines, and thus solve the pair of simultaneous equations.

So, y equals two x plus seven should have given you those values and those coordinates and that line for y equals one minus x.

We should have got those values in our table, those coordinates and that line.

Then we're looking at the point of intersection.

It's negative two, three.

Is that our solution? We need to be a bit more specific.

Let's say the solution is x equals negative two, y equals positive three.

For question two, solve graphically this pair of simultaneous equations and I reminded you that we only need two coordinates in order to plot these lines.

So, for y plus two x equals two when x equals zero, y is two and when y equals zero, x is one, we can take those two coordinates and turn them into a straight line graph of y plus two x equals two.

For the second equation, we can do the same thing when x equals zero, y equals negative three.

When y equals zero, x equals six.

There's those two's coordinates and the straight line.

What's next? We're looking for that point of intersection on this occasion it's at two, negative two, so a solution is x equals positive two, y equals negative two.

For question three, we've already got the graph of y equals two x minus seven.

We needed to draw the graph of five y plus three x equals 30.

Two coordinates will help us do that, zero, six, 10, zero.

There they are, a straight line.

Then we're looking to use the graphs to solve the pair of simultaneous equations.

We want the point of intersection five, three.

Our solution must be x equals five, y equals three.

For question four, we were given the lines and asked to use them to solve these pairs of simultaneous equations, but we weren't told which equation each line was.

You needed to identify that for yourself.

I hope you spotted that the lines were thus.

Do note that y equals three, bracket x plus one is the same as the line y equals three x plus three.

That's just the bracket expanded and there you go.

A gradient of three intersecting the y axis of three, that's the line y equals three x plus three.

So, for part A, four y minus two x equals negative eight and x plus y equals 2.

5.

Where are those two lines intersecting? They're intersecting at the coordinate three and negative 0.

5.

The solution then x equals three, y equals negative 0.

5.

For part B, where did lines y equals three bracket x plus one and four y minus two x equals negative eight? Where they intersect, at that point there, that's negative two, negative three, so our solution is x equals negative two, y equals negative three.

Onto the second half of the lesson now, solving graphically with technology.

Using graphing technology allows us to solve pairs of simultaneous equations quickly and easily.

In order to do this, I'd like you to open up a web browser and go to desmos.

com.

Find and press the graphing calculator button and then your screen will look a little bit like mine.

Pause and do those things now.

Welcome back.

We can use this graphic technology to solve this pair of simultaneous equations.

Five y minus six x equals 30 and y plus x equals negative 16.

We're going to type the equations in here up in the top left of your screen on Desmos.

When you type them in, it should look something like that.

You'll notice I've changed the colour of one of my lines.

If you click and hold where the colour is, you can change the colour.

Having lines of different colours is a nice way to differentiate between the two.

Once you've typed those in, the lines will come up like so.

The lovely thing about using the graphing technology is we don't even have to read that coordinate.

We can just click on the point of intersection and it'll reveal the solution for us.

That point is negative 10, negative six, so our solution is x equals negative 10, y equals negative six.

Using graphic technology can be very helpful if the axes are not initially drawn long enough.

If we're solving y equals three x plus eight, y equals five x minus 10, we could type into Desmos and look what happens if with an axis all up to positive 10.

On the x and y axis, we still can't see the solution.

If we can't immediately see the intersection, we can scroll or zoom to locate it.

If you scroll by holding down your mouse button and dragging your graph, you'll find the required region.

I'll scroll all the way up to there.

I can no longer see the x axis, but I can see the point of intersection, and it's at coordinate nine, 35.

One click tells me that coordinate.

So, this solution must be x equals nine, y equals 35.

An alternative would've been to zoom out to spot the point of intersection.

To zoom out, hold the shift button on your keyboard and click and drag your mouse.

When you do that, your screen will look a little bit like this and we can see our solution again, x equals nine, y equals 35.

Quick check, you've got that.

I'd like you to zoom or scroll to find the solution to this pair of simultaneous equations.

You should definitely use Desmos to do this.

Don't try drawing these.

x plus four y equals negative 24, y equals 10 minus a half of x.

Type those into Desmos and do some zooming and scrolling and find the solution.

Pause and I'll see you in a moment.

Welcome back.

Let's see how we got along.

Hopefully, if you zoomed out, your screen looked a little bit like that.

You found the point of intersection at 64, negative 22.

Therefore, the solution x equals 64, y equals negative 22 And even Lucas are trying to solve this pair of simultaneous equations.

x plus three y equals 10 and y equals seven minus a third of x.

Andeep says, "I need to zoom out or scroll to find the intersection." We often do so that's logical, but Lucas says, "I don't think you can." Whatever might he mean? Who do you agree with? Hmm, pause.

Have a think and I'll see you in a moment.

Welcome back.

I wonder what you discovered.

I wonder what you thought.

Andy noticed something.

He zoomed out on Desmos and he's discovered they're parallel lines.

There's no intersection.

What a good discovery, Andy.

Well done.

Lucas says, "That's right.

There is no solution for this pair." If we rearrange x plus three y equals 10, we'll see why.

If I add negative x to both sides and then divide through by three, I get the equation y equals 10 minus a third of x and look, the x coefficient is negative a third in both cases, a common gradient of negative a third.

When two lines have the same gradient but different y intercepts, they will be parallel.

There will be no intersection, there will be no solution.

Next, Andy and Lucas are trying to solve this pair of simultaneous equations.

Eight x plus four y equals 34, y equals 8.

5 minus two x.

Andeep says, "I need to zoom out or scroll to find the second line and the intersection." And Lucas says, "I don't think you need to." Whatever could he mean? Who do you agree with? Have a pause, have a look on Desmos, see what you notice.

See you in a moment.

Welcome back.

I wonder what you discovered.

I wonder what you noticed.

Andy zoomed all the way out and then questioned.

"Does the second line even exist?" Lucas says, "They're the same line." In this case, there are infinitely many intersections and solutions.

They're the same line because we can rearrange that left-hand equation to read y equals 8.

5 minus two x.

Oh, look, they are the same line.

They lie on top of each other.

In that case, there's infinitely many intersections.

That means that this pair of simultaneous equations has infinitely many solutions.

Isn't that amazing? An equation with infinitely many solutions.

Quick check.

You've got that.

How many solutions does this pair of simultaneous equations have? Is it zero solutions, one solutions or infinitely many solutions? Pause and you'll definitely want to use Desmos to check this and I'll see you in a moment for the answer.

Welcome back.

I hope you went for option A, no solutions.

If you graph the two of them, you'll see that they're parallel lines and the lines are parallel.

There is no solution.

Another check.

I'd like you to select the pair of simultaneous equations that have infinite solutions, three different equations there, and if you pick the right pair you'll find a pair with infinitely many solutions.

So, pause, use Desmos, have a little play, and I'll see you in a moment for the answer.

Welcome back.

How did we do? I hope you picked out A and B.

When you graph those, you'll find they're the same line.

They lie on top of each other.

If they're the same line, they have infinitely many intersections, infinitely many solutions.

If you had plotted line C, you've noticed they had one solution with either of the two equations.

Practise time now.

Question one, I'd like you to graph and, therefore, solve these pairs of simultaneous equations.

Use your graphing technology, plot both lines at once, find the intersection, thus find your solution.

Pause.

I'll see you in a moment.

Question two.

This time I'd like you to decide how many solutions each pair of simultaneous equations has.

I don't want to know the solution, I just want to know how many solutions, zero, one, or infinitely many.

Once you've decided which it is, I'd like you to write a sentence to justify your answer.

Pause.

Give these three problems a go now.

For question three, Alex is trying to solve this pair of simultaneous equations and he says, "The lines don't intersect, so this pair has no solution." Alex is wrong.

I'd like you to explain why and give Alex some advice as to what they could do differently.

You'll want to write at least two sentences for that moment.

Pause and do this now.

Feedback time now.

For question one, I ask you to graph and solve these pairs of simultaneous equations.

Your first graph should have looked like so and you should have noticed a point of intersection four, three.

Therefore, your solution x equals four, y equals three.

For part B, the graph should look like so and you find that point of intersection negative 4.

5, negative 7.

5, so our solution x equals negative 4.

5, y equals negative 7.

5.

For part C, when you graph it, you really have to do some zooming out to find this point of intersection.

That point of intersection is the coordinate 200, negative 243.

I'm glad I had the technology to help me with that one.

Finding the solution x equals 200, y equals negative 243.

For question two, I asked you to decide how many solutions each pair of simultaneous equations has? For part A, your graph would've looked like that and you should have noticed this pair has no solution.

You might have written, "If the lines are parallel, they will not intersect.

Therefore, there is no solution." You might also have written, "The gradient of both lines is five." That's why they're parallel, there's a common gradient.

For part B, your graph would've looked like so and you should have said, "This pair has one solution." You might have written, "These lines have one intersection, therefore there is only one solution which satisfies both equations simultaneously." For Part C, your graph would've looked like so and you would've noticed they're the same line.

This pair has infinitely many solutions.

You might have written, "The two lines are the same, so lie on top of each other on the graph, they have infinitely many intersections and infinitely many solutions." Question three, Alex said, "The lines don't intersect, so this pair has no solution." And I asked you to explain why and give Alex some advice as what he could do differently.

You might have written, "The lines do not have the same gradient, therefore they will intersect." You can notice that just from the equations of the lines.

You might also have written, "Zoom out and you will see the intersection and find the solution." You might have done that zooming out, found that intersection, and helped Alex find that solution, x equals 12, y equals 24.

Sadly, that's the end of lesson now.

What we've learned is that whilst we can solve two linear simultaneous equations algebraically, we can also solve them graphically.

We can calculate, coordinate pairs, plot the lines, and identify the intersections to find solutions.

Alternatively, we can use graphing technology and wasn't it wonderfully helpful.

That's all for our folks.

I should look forward to seeing you again soon for more maths.

Good bye for now.