Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Super choice to be here.

We're solving linear equations graphically with technology today.

That sounds pretty exciting, so let's get started.

Our lesson outcome, so we're gonna explore how to solve linear equations graphically via dynamic software.

Keywords for today, equation and solution.

An equation is used to show two expressions that are equal to each other.

The solution to an equality is the value which, when substituted, maintains the equality between the expressions.

You're gonna hear those words a lot today and become very familiar with them.

Two parts to our learning today.

We're gonna start by solving equations with unknowns on one side.

To get started, I'll ask you to open up a web browser and go to desmos.

com Once there, can you find and press the graphing calculator button? It'll be in the middle of your screen.

Your screen should look just like mine.

Pause and get to this point.

With Desmos open, I'd now like you to click on the Graph Settings menu in the top right of your screen.

It's the little spanner icon just there.

Press that now.

The Graph Settings menu looks just like this.

A few changes we're gonna make so that your screen looks like mine as we explore the solving of linear equations.

Firstly, turn off minor gridlines.

We might use these later, but, for now, let's turn them off.

It's always important to label your axes.

We'll do that now by typing in x in the X-Axis label box and y in the Y-Axis label box.

Finally, change the step on both axes to one.

Later, we might change this to something different, but, for now, say a step of one and your screen will look just like mine.

Pause and make sure your graph settings are the exact same as mine here.

Right, with Desmos looking a little something like this, one more skill we're gonna need is to zoom in and out.

We can do that using the addition and subtraction signs in the top right of the screen.

Alternatively, you can zoom in and out by holding the Shift key down on your keyboard, clicking and holding your mouse button down, and then moving the mouse back and forth.

You'll zoom in and you'll zoom out.

We won't always want to look at the same region of our graph.

We're gonna need to scroll around the place.

You do that by clicking and holding your mouse button, and then you can drag your axes all over the screen to look at other regions of the graph.

Can you pause and practise those skills now? Zooming in and out via both methods, and then dragging around so you have a good look into the distance in the first quadrant, the second quadrant, the third quadrant, the fourth quadrant.

Pause and play with those two things now.

We're now ready to explore the solving of equations.

Zoom and drag your screen so it's scaled like this.

I can see -5 to +5 on both axes.

Then I'd like you to type y = 2x - 1 here.

Just click in that area, and you can type that in.

Desmos will draw the line y = 2x - 1 for you.

It'll look just like that.

Pause and make sure your screen looks just like mine.

Right, next question for you is, what integer value coordinates can you read from this section of the graph of y = 2x - 1? I've given you one coordinate, the coordinate 1,1.

Which other coordinates can you pick that are on the line y = 2x - 1? Once you've typed them in, if you click and hold that point there, you'll be able to change those coordinates to crosses, label them, and adjust the text size so that label's nice and clear.

You can see exactly what coordinate that is.

Right, pause now and label as many coordinates as you can, or integer value coordinates as you can, on that line, in that region.

You should have this set of coordinates, the coordinates, 3,5, 2,3, 1,1, 0,-1, -1,-3, -2,-5.

But what do these coordinate pairs mean? It's crucial to understand the answer to that question if we're gonna use this software to solve equations.

This is the set of coordinates that we generated or found on the line y = 2x - 1.

What do they each mean? The coordinate 3,5 means when the x value is 3, the y value is 5.

At coordinate 2,3, when the x value's 2, the y value's 3.

When the x value's 1, the y value's 1.

When the x value is 0, the y value is -1.

Can you guess what I'm gonna say next? Absolutely, when the x value is -1, the y value is -3.

And that last coordinate, when the x value is -2, the y value is -5.

That's what those coordinates mean.

But is there anything else that we can pick from that coordinate? When the x value is 3, the y value is 5.

y is 2x - 1, so we could also say, if x = 3, then 2x - 1 = 5.

We could then turn that sentence around and say, if 2x - 1 = 5, then x = 3.

And this is a glorious moment for us.

2x - 1 = 5.

Well, that's a linear equation.

x = 3 is the solution to that linear equation.

We've used technology to find the solution to an equation.

What a joy.

The solution to an equality is the value which, when substituted, maintains the equality between the expressions.

If you substitute x = 3 back into our equation, 2x - 1 = 5, well, two lots of 3 - 1 does indeed equal 5.

That solution, and only that solution, x = 3, maintained the equality.

Any other x value would not have worked.

Quick check you've got that.

I told you for the coordinate 3,5, when the x value is 3, the y value is 5.

If x = 3, then 2x - 1 = 5.

If 2x - 1 = 5, then x = 3.

Can you give me those same three statements, those same three crucial bits of information, for the coordinate 2,3 by reading these sentences aloud and filling in those blank spaces with a number? Pause and do that now.

I hope you said, when the x value is 2, the y value is 3.

If x = 2, then 2x - 1 = 3.

Turn that sentence around.

If 2x - 1 = 3, then x = 2.

Well done.

You solved the equation 2x - 1 = 3.

You found the solution x = 2.

Can you do that again but for this coordinate, -2,-5? Pause and do that now.

I hope you said, when the x value is -2, the y value is -5.

If x = -2, then 2x - 1 = -5.

If 2x - 1 = -5, then x = -2.

One advantage of technology is that it deals with large numbers really quickly.

I could ask you to scroll and find the solution to 2x - 1 = 283.

Notice I haven't changed the 2x - 1.

You still have the line y = 2x - 1 drawn.

Would you be able to scroll and find the solution to that equation? If you think you can, pause and give it a go.

If you're not ready to go alone just yet, then go through it with me.

I need to scroll all the way up here, way off into the distance in that first quadrant, to find the coordinate 142,283.

You can add that coordinate, label it, and then what does it tell us? It's the coordinate 142,283, which means if x = 142, then 2x - 1 = 283.

Turn that around.

It's where 2x - 1 = 283.

The specific x value x = 142.

That's our solution, x = 142.

x = 142 is the only value which maintains equality in the equation 2x - 1 = 283.

No other x value will maintain the equality for us.

Let's try another one now.

How about the solution to the equation 2x - 1 = -687.

Pause and give that one a go.

We need to scroll right the way down into that third quadrant to find this coordinate, <v ->343,-687.

</v> Once we've found that coordinate, it tells us if x = -343, then 2x - 1 = -687.

Alternatively, if 2x - 1 = -687, then x = -343.

We've found the solution to our equation.

The x value of -343 is the only x value that will maintain equality in that equation for us.

There's an easier way to solve equations than to have to find coordinates all the time.

So, solve 3x - 7 = 5.

We're gonna start by thinking of that equation in a different way.

We could read it as, when does the line y = 3x - 7 meet with the line y = 5? It's that moment which will give us our solution.

There's the line y = 3x - 7.

There's the line y = 5.

You can find that meeting point now by drawing those two lines.

If you click on the meeting point, which we also call an intersection, where those two lines intersect each other, Desmos will reveal that coordinate to be 4,5.

Pause, draw those two lines, and click on that intersection for me.

Did this happen? The coordinate revealed as a point, and the label 4,5 comes up.

Our solution, therefore, is when x = 4.

It's the x-coordinate of that intersection.

You can always check that by substituting 4 back into the original equation.

3 lots of 4 is 12, minus 7 is 5.

Aha, we have equality.

x = 4 must be a solution.

Let's check you've got that now.

What does this image tell us? Hmm.

Does it tell us that the solution to 11 - 2x = 3 is x = 5, or the solution to 11 - 2x = 5 is x = 3, or is it the solution to 11 - 2x = 5 is x = 5? Which one of those three is it? Pause and have a think.

I hope you said it's option b.

It shows us that the solution to 11 - 2x = 5 is x = 3.

The expression 11 - 2x is only equal to 5 the one moment it crosses that y = 5 line when the x-coordinate is 3.

Next, I'd like you to plot the line y = 4x - 5.

And it looks like so.

Then I'd like you to plot the line y = 2.

And it looks like so.

Now, I'd like you to solve 4x - 5 = 2.

You know what to do.

Click on the intersection, read the x value.

Pause and do that now.

Did you get that intersection, the coordinate 1.

75,2? If you did, well done.

Within it lies our solution.

If x = 1.

75, then 4x - 5 = 2.

Or if 4x - 5 = 2, then x = 1.

75.

So x = 1.

75 is the only value which maintains equality.

That's our solution.

But how does this solution compare to when we solve without graphing? If I ask you to solve 4x - 5 = 2 with just pen and paper, I'd like to think you would add positive 5 to both sides and then multiply both sides by 1/4.

You might think of that as dividing through by 4 on both sides.

Therefore, we're left with x on the left-hand side and 7 divided by 4 on the right-hand side, which we would just write as 7/4.

In written notation, we'd leave the solution as a fraction for simplicity.

However, Desmos will always give you that solution as a decimal.

So, do note, x = 7/4, well, you would say, 7 1/4, 1 3/4.

Oh, 1 3/4, that's the same as 1.

75.

It's the exact same solution, just communicated differently.

Just be aware, Desmos will always give you a decimal.

Next, plot the line y = 6x - 5.

And it looks like that.

Plot the line y = 2.

And it looks like that.

And use those two lines to solve the equation 6x - 5 = 2.

Pause and do that now.

Did you get that coordinate? 1.

167,2.

If you did, does that mean if 6x - 5 = 2, then x = 1.

167 is our solution? Well, that would make sense because that's in line with everything you've told us so far, Mr. Robson.

But notice on this occasion, I've left it framed as a question rather than a factual statement.

I wonder why.

When solving equations, it's good practise to check your solution by substituting it into your original equation.

So we think x = 1.

167 is the solution to 6x - 5 = 2.

So let's substitute back in.

6x - 5.

If x is 1.

167, 6 lots of 1.

167 - 5 = 2.

Hold on.

It didn't equal 2.

What's happened here? What's happened is x = 1.

167 is not the exact solution.

It's incredibly close, but it's not exact.

Let's solve without the graphing software.

6x - 5 = 2.

Let's add positive 5 to both sides.

Let's divide through by 6, x = 7/6.

I wonder if you know immediately something special about that fraction.

It'll give us a recurring decimal.

7 divided by 6 is 1.

1666666.

The 6 recurs, so we communicate that as 1.

16 recurring.

However, Desmos doesn't compute like that.

It will round to three decimal places, so need to be aware of rounding errors.

If I said 7 divided by 6 to three decimal places, you would say 1.

16, and then that last 6 rounds up to a 7.

So it's 1.

167 to three decimal places.

We have to just be aware that Desmos is limited to three decimal places, and it'll round correctly, but it will lose the exactness of the solution when it rounds.

So, true or false, this image shows us that the exact solution to 3x - 4 = 10 is x = 4.

667.

Is that true, or is it false? You might wanna pause and have a look at that image 'cause it's about to disappear.

I want you to justify your true or false answer with one of these two statements.

x = 4.

666 is the exact solution, or x = 4.

667 is an approximate solution accurate to three decimal places.

So, is it true, is it false, and which of those statements will you use to justify? Pause and have a think about that now.

It's false.

Because x = 4.

667 is an approximate solution accurate to three decimal places.

The actual exact solution was 4.

6 recurring.

Practise time now.

Question one, I'd like you to draw the line y = 3 - 2x and use it to solve the below equations.

You don't need to draw multiple lines for this one.

Just that one line y = 3 - 2x will do it.

You're going to read some coordinates from that line, and those coordinates will enable you to identify the solutions to those four equations.

Pause and do that now.

Question two.

This time, for each equation, draw two lines on Desmos.

Those two lines will help you find the solutions to these equations.

Pause and do that for all three of those equations now.

And for question three, Lucas uses these two lines and draws a conclusion.

What two lines? He's drawn y = 430 - 11x, y = 132.

He's found the intersection, an x-coordinate 27.

091, a y-coordinate of 132.

So Lucas concludes the solution to 430 - 11x = 132 is x = 27.

091.

Seems logical.

What I'd like you to do is write a sentence explaining to Lucas why this might not be entirely accurate.

Pause and write that sentence now.

Feedback time now.

Four equations to solve, and the solutions lie on the line y = 3 - 2x.

If you identify these four coordinates, they'll help you to solve these four equations.

The coordinate 1,1 helps us with equation a.

When does the line 3 - 2x = 1? Well, when the x-coordinate is equal to 1.

That's our solution for part a.

For b, the coordinate 4,-5, when does 3 - 2x = -5? That's when the x-coordinate is 4, so our solution is x = 4.

The coordinate -1,5 helps us with part c.

When does 3 - 2x = 5? When the x-coordinate is -1.

Our solution, x = -1.

Lastly, 0,3, when does the line 3 - 2x = 3? When x = 0.

For part two, I asked you to draw two lines and find the solutions.

If you drew the line y = 5x - 7 and y = 13, if you click that intersection, Desmos will tell you it's the coordinate 4,13.

What does that tell us? It tells us that those two lines intersect when the x-coordinate is 4, or rather, the solution to the equation 5x - 7 = 13 is x = 4.

For part b, I'm glad I had the technology to help me with that one.

278 - 1/2x = 105.

I can do it, but I can do it quicker with technology.

There's the lines y = 278 - 1/2x and y = 105.

They intersect at the coordinate 346,105.

That's a solution x = 346.

For part c, there's our two lines intersecting at the coordinate -0.

52,2.

608, giving us a solution x = -0.

52.

For question three, Lucas concluded that solution to the equation was x = 27.

091.

Hopefully, you said something along the lines of, Desmos is limited to three decimal places, so when we get an answer to three decimal places, we're not sure it's entirely accurate.

It might be rounded.

x equals 27.

091 is an approximate solution correct to three decimal places.

You might have written the equation again, solved without the software, and found the exact solution is x = 298/11.

On to the second half of the lesson, solving equations with unknowns on both sides.

We can use Desmos to solve equations with unknowns on both sides, such as the equation 3x - 7 = x + 1.

If I said solve without technology, you would start by writing the equation, maybe add -x to both sides, and then add positive 7 to both sides, and then multiply both sides by 1/2.

x = 4, plug that in, 3 lots of 4 - 7.

That's 5.

On the right-hand side, 4 + 1 is 5.

Aha, x = 4 is definitely the solution.

Look at that second line, 2x - 7 = 1.

That form looks familiar.

We could plot the two lines, y = 2x - 7, y = 1, and we could see the solution from there, couldn't we? Absolutely.

Look at where those two lines intersect, the coordinate 4,1.

When x has a coordinate of 4, that's our solution, x = 4.

Could I have done it with the next line? Plotted y = 2x, y = 8.

Look for the intersection.

Absolutely.

The lines are different, naturally, but it's the same solution, x = 4.

So same solution, different method.

Does that mean, could I have just plotted these lines? The lines y = 3x - 7 and y = x + 1.

What do you think? I hope you think, absolutely, Mr. Robson.

We could, and there they are.

The lines y = 3x - 7, y = x + 1 intersecting at the coordinate 4,5.

x = 4, the exact same solution.

So, graphing software means we don't have to rearrange in order to solve equations with unknowns both sides.

We just plot the expressions and identify the intersection.

Quick check you've got that.

True or false, when solving equations with unknowns both sides, we need to manipulate so that the unknowns are on one side before solving.

Is that true, or is it false? I'd like you to justify your answer with one of these two statements.

With Desmos graphing software, we don't need to rearrange.

We can plot the two expressions and identify the intersection.

Or, we need to isolate the unknown terms on one side in order to solve equations.

What do you think, true, false, and how are you gonna justify your answer? Pause this video and have a think now.

I hope you said false with the justification, with Desmos graphing software, we don't need to rearrange.

We can plot the two expressions and identify the intersection.

Izzy is using Desmos to solve the equation 2 - x = 3x + 10.

There's the two lines.

<v ->2,4 is where the lines intersect.

</v> Izzy says, "That was really quick and easy.

"I can see that the solution is x = 4." Do you agree with Izzy? 4 is not the solution.

It's the x value -2 that is the solution.

The 4 is the value of both expressions when x = -2.

Quick check you've got that.

Which of these is the solution to 2x + 3 = 4x - 1? Is it x = 2, y = 7, or x = 7? Pause, have a think.

Maybe have a conversation with the person next to you.

I hope you said option a, x = 2.

x = 2 is the only solution.

When x = 2 both expressions have a value of 7.

Aisha and Sofia are using Desmos to solve equations with unknowns both sides.

"This technology is brilliant.

"I just solved 2.

3 - 0.

4x "= 7.

6x + 14.

7 in a few seconds." And there's Aisha's screen.

Sofia says, "Yes, I see.

x = -1.

55." That's some pretty awesome solving of equations with dynamic software.

Sofia says, "Can you help me? "I'm stuck trying to solve 4x - 5 = 4x + 11." Can you see why Sofia has gotten stuck? Pause and have a little think.

Why has Sofia gotten stuck? This is why she's gotten stuck.

There's the lines y = 4x - 5, y = 4x + 11.

What do you notice? They're parallel lines.

They have no intersection.

They will never intersect.

Without the intersection, there is no solution.

So this equation, 4x - 5 = 4x + 11, has no solution.

I can zoom out a long, long way and show you they do not intersect.

Therefore, there's no solution.

Quick check you've got that.

Which of these equations have solutions? Four equations there.

Some have solutions and some don't.

You might know just from looking at them which ones do and which ones don't.

You might want to use Desmos and check them.

Pause and do that now.

The answers, the first one does.

There's the two lines, 5x - 1, x + 5.

They have an intersection.

They've got different gradients.

They have an intersection.

This equation has a solution.

The same was true of b.

When I plot those two lines, they've got different gradients.

Therefore, they have an intersection.

This equation has a solution.

For c, no solution.

If you plot them, you'll see they've got the same gradient.

They're parallel.

Therefore, they will never intersect.

This equation has no solution.

That was also true of d.

5x - 1, 5x + 7.

Same gradient, no intersection.

This equation has no solution.

Practise time now.

Question one, I'd like you to plot pairs of lines and find the intersections to solve the below equations.

Three equations.

Pause, graph them.

Find those solutions now.

Question two.

Alex says, "This graph shows that the solution to the equation 3x + 9 = -2x - 1 is x = 3." Alex is wrong.

Identify the correct solution and write a sentence explaining Alex's error to him.

Pause and do that now.

And for question three, one of the below equations has no solution.

Identify which one it is and show why it has no solution.

Pause and do that now.

Feedback time.

Question one, part a.

That's the pair of lines you should have drawn, finding the intersection 3,14.

If they intersect at 3,14, the solution is x = 3.

For part b, there's the two lines.

Zoom quite a way out to identify the intersection at 177,2210.

Therefore, the solution is x = 177.

For part c, there's the two lines.

We can even type that bracketed expression in, straight into Desmos.

They intersect at 2,-7.

75.

Therefore, the solution, x = 2.

For question two, you might have written x = -2 is the solution.

3 is the value of both expressions when x = -2.

For question three, which one of the below equations has no solution? It was c.

In terms of part a, different gradients.

Therefore, they have an intersection.

That equation has a solution.

The same was true of b.

But for c, if you expand that bracket, you'll notice 6x - 11, 6x + 8.

Both got a gradient of 6.

Same gradient.

No intersection, no solution.

That's the end of the lesson now, sadly.

In summary though, we can use dynamic software to solve linear equations graphically by identifying the x-coordinate of the intersecting points of the lines.

I hope you enjoyed today's lesson as much as I did, and I hope to see you again soon for more mathematics.

Goodbye for now.