- [Instructor] We're told
a city bike rental service

charges customers based on how
long they rent the bicycle.

The table shows the total
cost for renting a bicycle

as a function of the
number of rental hours.

So they say,

"Complete the equation to
model the hourly rental cost."

So what they really want us to do

is have an equation of Y being,
we could say a function of X

that can describe what is going on here.

So pause this video and have a go at it

before we do this together.

Alright, so let's just look
at the data a little bit

and think about, okay, is
this a linear relationship

or is this something else?

So when we increase our
rental hours by two,

it looks like here, or when
we go from one to three,

we're increasing by two and
our cost is increasing by,

let's see, to go from 12 to
30, it's increasing by 18.

See, when we go from
three to five, once again,

that's two more hours, and
it looks like every two hours

we do indeed increase our cost
by 18, let me check again.

Hours increase by 2 and
cost indeed increases by 18.

So if we are increasing
by $18 every 2 hours,

that's the same thing as
you have a change in Y of $9

every time you have a
change in X of one hour.

So it's $9 per hour.

Now change in Y over change in
X might look familiar to you.

That is the slope of a line.

So this is a linear relationship.

It's going to have the form Y = MX + B,

where this is the slope and
this is the Y intercept.

We just figured out the
slope, it is $9 per hour.

So we could say Y = 9X plus
whatever the Y intercept is.

The Y intercept would be the minimum

that they're going to charge you

before they even bill you based on hours.

And to figure out that,

we just have to substitute
one of these points.

We can say, okay, when X is 1, Y is 12.

So let's just substitute
that Y is 12 when X is 1.

So 9 x 1 + B or 12 = 9 + B.

You could do this in your head

or you could subtract
nine from both sides,

and you get 3 = B.

So our equation, this
right over here is three.

So we get Y = 9X + 3.

One way to interpret this is even if you,

just renting a bike before they
even charge you the hourly,

they're gonna charge
you $3 just to do that,

and then they're going to charge
you $9 per hour after that.

And you can double check that.

You could say,

"Okay, well if I had to rent
this bike for three hours,

I'm gonna pay that $3 and then
I'm gonna pay an extra $27

for the hourly amount that I'm using it."

27 + 3 is indeed $30.

You could try out any of these other ones.