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

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charges customers based on how
long they rent the bicycle.

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The table shows the total
cost for renting a bicycle

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as a function of the
number of rental hours.

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So they say,

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

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So what they really want us to do

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

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that can describe what is going on here.

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So pause this video and have a go at it

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before we do this together.

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

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and think about, okay, is
this a linear relationship

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or is this something else?

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So when we increase our
rental hours by two,

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

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

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

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

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

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

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Hours increase by 2 and
cost indeed increases by 18.

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

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

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

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So it's $9 per hour.

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

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That is the slope of a line.

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So this is a linear relationship.

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

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

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

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

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The Y intercept would be the minimum

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that they're going to charge you

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before they even bill you based on hours.

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And to figure out that,

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we just have to substitute
one of these points.

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

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

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

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You could do this in your head

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or you could subtract
nine from both sides,

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and you get 3 = B.

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So our equation, this
right over here is three.

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So we get Y = 9X + 3.

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One way to interpret this is even if you,

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

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

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

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And you can double check that.

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You could say,

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

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

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

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27 + 3 is indeed $30.

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You could try out any of these other ones.