Lesson video

Hello, my name is Mr. Clasper, and today we are going to be looking at different aspects of proportionality and investigating some problems. It's time to put everything together and look at examples where we have three variables.

Y is directly proportional to x.

X is inversely proportional to z squared.

Y is equal to 20 when x is equal to five and z is equal to two when x is equal to 10.

Find the value of y when z is equal to 40.

Well, the first thing we need to do is find an equation with y as a subject written in terms of x.

So we know that y must be equal to kx, as y is directly proportional to x.

We can substitute 20 and five and we find that we have a constant of proportionality of four.

This means our equation would be, y is equal to 4x.

Let's consider our next equation.

So with this equation, we're going to need to write x in terms of z.

So we're going to need this information.

Our formula will be x is equal to k over z squared.

And now we can substitute z is equal to two, and x is equal to 10 into this.

This means that 10 is equal to k over four, and therefore the constant and proportionality must be 40.

And therefore we have an equation of x is equal to 40 over z squared.

Now to find the value of y when z is equal to 40, we're going to need an equation for y in terms of z, or in other words, we need an equation with y as the subject and an expression with only the variable z on the right hand side of this equation.

Let's have a look at our two equations.

We have y is equal to 4x, and x is equal to 40 over z squared.

Well, if we know that x is equal to 40 over z squared, we could look at the first equation, and instead of writing y is equal to 4x, we could write y is equal to four ups of 40 over z squared, as this is equal to x.

And if we multiply, this means that y must be equal to 160 over z squared.

From here, we can substitute our value of 40 for z, and this will give us a value of 0.

1 for y.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So remember to solve these problems, we're going to need two separate equations.

So we're going to need y in terms of x, and we're going to need x in terms of z.

Once you have these, we need to make sure that we have an equation, which is y in terms of z, which in this case will be y is equal to 10z squared.

And once we have this, we can substitute the value of z given, which is 12, and it should give us a value of y of 1,440.

And here is your last question, pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So once again, we've been given two pieces of information.

We have an equation which we'll link g and h and we have another equation which will link u and g.

So we need to make sure that we work with these first of all.

And then we need to find a formula which will connect u and h.

And this will be done by substituting part of one of the formulas that we've been given.

Once you've done this, it's a case of substituting your value for h, for part b.

So substituting h is equal to 100 in your formula from part a, and you should find that the answer will be 2.

5.

And that brings us to the end of our lesson.

So you've been solving problems with inverse and direct proportion involving three variables.

Want to give the exit quiz a go to show off your skills? I'll hopefully see you soon.