Lesson video

Hi.

I'm Mrs. Dennett.

And in this lesson, we're going to be looking at how solutions to equations are linked to their graphs.

Let's remind ourselves of some of the types of graphs we will see in this lesson.

First of all, we have linear graphs.

These are straight line graphs.

They can have a positive or negative gradient, so they can slope to the right or to the left.

They are usually of the form y equals mx plus c.

But as you can see with y equals six minus x, this form can vary.

Next, we have quadratic graphs, with their distinctive curve, a U shape or an N shape.

These are of the form y equals axe squared plus bx plus c.

Let's look at circle graphs.

These have equations such as x squared plus y squared equals 16, where 16 is the radius squared.

You can see the radius of four here.

Next, we have reciprocal graphs.

These look like this and can often be recognised by their equations, y equals one over x or x times y equals a constant a.

Or in this case, we've got xy equals 10.

You may have come across other graphs, such as cubic or other polynomials, but we won't be looking at these in this lesson.

We can use graphs to help us solve simultaneous equations.

Here, we have the graph of two equations.

One is quadratic, and one is linear.

What do you notice about the points of intersection on the graph where the line crosses the circle and the two pairs of solutions are x and y? Here are the coordinates of the points of intersection.

Did you notice that we have the same numbers? Each pair of solutions can be written as a coordinate x, y.

So when x is four, y is three.

We can write four, three, and this corresponds to one of the points of intersection on the graph at the point four, three.

The same is true for our other pair of solutions.

When x is zero and y is minus five, we write the coordinate zero, minus five.

You can see the other points of intersection at the bottom of the graph.

We can write our solutions to simultaneous equations as pairs of coordinates, and these correspond to the points of intersection on the graph.

Here is a question for you to try.

Pause the video to complete the task, and restart when you're finished.

Here are the answers.

The points of intersection are three, nine and minus one, one.

Your answer can be written as x equals and y equals or as a pair of coordinates.

Here is a question for you to try.

Pause the video to complete the task, and restart when you are finished.

Here are the answers.

If we substitute x equals six and y equals zero into each of these equations, we find that they both work.

Six squared plus zero squared is 36, and six minus two times zero is six.

This solution can be written as a coordinate point.

And if we wanted to draw the graph of each equation, we would find a point of intersection at six, zero.

Here is another question for you to try.

Pause the video to complete the task, and resume once you're finished, Here are the answers.

When we substitute x equals minus three and y equals two into the first equation, we do get 13.

Well, this doesn't work in the second equation.

Minus three plus one is minus two, not two.

So minus three, two is not a point of intersection.

To find the x value, we have to substitute y equals three into one of the equations.

We solve it to find the value for x, and we get x is equal to two.

In some pairs of equations, it is difficult to clearly read off the points of intersection because the solutions are not integers.

In this scenario, we will be asked to estimate the solutions.

Here's the graph of a quadratic and a linear equation.

We can see the points of intersection here.

But even if we zoom in, it is difficult to read off the coordinate points.

This is where we need to get our rulers out and, as accurately as possible, estimate the solution.

Draw yourself a vertical line to estimate the x solution.

Look carefully at the scale.

Our answer is between one and two.

The scale is split into five little squares between each whole number.

So each square must be worth 0.

2.

My vertical line is between 1.

2 and 1.

4.

It's just a bit more than 1.

2 but less than halfway, so less than 1.

3.

So depending on how accurate the question wants us to be, to one decimal place, the x value will be 1.

2.

But to two decimal places, we could say the x value is anything up to 1.

25, perhaps even anything up to 1.

3.

We'll go for 1.

2.

You will always have a window of accuracy when answering these questions anyway.

Now let's look for the y-coordinate.

We draw our horizontal line.

This looks like it is right in the middle of three and four, so 3.

5.

So we write our solution as a coordinate.

X is 1.

2, and y is 3.

5.

Now we do the same for the other point of intersection.

Draw in your vertical and horizontal lines with a ruler.

You can see we get minus 3.

2 and minus 5.

5.

Here is a question for you to try.

Remember to use a ruler to guide your solutions, and interpret the scale carefully.

Pause the video now to complete the task, and restart when you're finished.

In this question, we are asked to estimate our answers to one decimal place.

Notice that the scale is going up by 0.

25 each time.

Don't worry if you were a little bit inaccurate.

Give yourself an extra 2/10 either side of the answer stated here.

So for example, for x equals 1.

7, it is acceptable to have anything from 1.

5 to 1.

9.

Here's a question for you to try.

Pause the video to complete the task, and restart when you are finished.

Here are the answers.

Did you notice that the equations only have one point of intersection which is a tangent to the curve? If we solved these equations algebraically, we would see that we get a repeated root for the x value.

So there is only one pair of solutions.

If we wanted a pair of equations with no solutions, we would have to have no points of intersection.

There are many answers to this question, such as the one I have given here, xy equals minus 16 and y equals x.

But if you want to check your answer, you can ask a parent or carer to put your equations into an online graphical calculator, such as the one I have used in this lesson, Desmos, to create my graphs for this lesson.

That's all for this lesson.

Remember to take the exit quiz.

Thank you for watching.