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## Constructing linear models for real-world relationships

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# Modeling with linear equations: gym membership & lemonade

## Video transcript

In this video, I want to do a
couple more word problems dealing with graphs of lines. So here we have a
gym is offering a deal to new members. Customers can sign up by paying
a registration fee of $200 and then a monthly
fee of $39. This is registration. How much will this membership
cost a member by the end of the year? So let's figure out an equation
that determines how much total we will pay. p is equal to the amount that
we're going to pay in total for our membership. So no matter how many months we
use the gym, just to start using the gym, we have to pay
$200 in registration. I'll just write-- everything
we'll assume is in dollars --so I'll write $200. And then we're going to have
to pay $39 for every month we're there. So then we're going to take
the number of months we're there and multiply
that times 39. Notice if we stay there 1 month,
we'll have to pay 1 month times $39. And we would have already paid
the $200 registration fee. So it'll be $239. If we stay 2 months, we pay the
$200 registration fee and then we pay 39 times 2 months,
which is what? Like 78 or something. So it would be $278. So just to tie this altogether
with linear equations and graphs of them, let's
graph this relation. Remember the graph of
a line can be y is equal to mx plus b. That's one of the forms. So to
put this line in this form or this equation in this form, we
can just rearrange the 39m and the 200 and you get p is
equal to 39m plus 200. So what's the slope and what's
the y-intercept? You might get confused and say,
hey there's an x there and a y, but now you're doing
it with p's and m's. Just remember this is the
independent variable and this is the dependent variable. Here this is the independent
variable. How many months? You pick a number of months
and I'll tell you what the total cost of your membership
is going to be. It's the same thing. This is like the
x right there. This is like the y
just like that. Just using our pattern match,
this right here is the-- we could say it's the vertical
intercept or the p-intercept or the-- I'm tempted to call
it the y-intercept. But we're really intersecting
the p-axis instead of the y-axis there. This right here is our slope. So let's graph this function. I won't do it too accurately. I just want to do a hand
drawn graph just to give you an idea. We could just stay in
the first quadrant. We're not going to stay negative
months and the gym is never going to pay us money. So right off the bat,
we're going to have to pay the gym $200. $200 for 0 months. Then for every month, we're
going to have to spend an extra $39. So the slope is 39. Let's say this is 1
month right there. This is in months. And this axis is price,
the p-axis. So this is like the
p-intercept or the y-intercept. So after 1 month, how much are
we going to have to pay? Well our slope is 39, so if we
move 1 month forward, we're going to go up by 39. So this will, right here,
that will be 239. If we go another month,
it'll be 278. This is kind of a weird labeling
of an axis, but I think you get the idea. So the graph of how much it'll
cost us as per month will look something like this. So they say, how much will
a membership cost by the end of the year? 12 months. We would have to go 2, 3, all
the way out to 12 months, which might be here. So then our graph is going
to be out here someplace. But we could just figure
it out algebraically. At the end of the year,
m will be equal to 12. When m is equal to 12, how
much is our membership? The price of our membership
is going to be our $200 membership fee plus 39
times the number of months, times 12. What's 39 times 12? 2 times 9 is 18. 2 times 3 is 6 plus 1 is 70. I have a 0. 1 times 9 is 9. 1 times 3-- we want
to ignore this. 1 times 3 is 3. So we have 8. 7 plus 9 is 16. 1 plus 3 is 4. So the price of our membership
is $200 plus 39 times 12, which is $468. So it's equal to $668 at
the end of our year. So if you went all the way out
to 12, you would have to plot 668 someplace here on
our line, if we just kept going out there. Let's do one more of these. Bobby and Petra are running a
lemonade stand and they charge $0.45 for each glass
of lemonade. In order to break even,
they must make $25. How many glasses of lemonade
must they sell to break even? So let me just do
it with y and x. y is equal to the amount
they make. Not max-- the amount
they make. Let x is equal to the number
of glasses they sell. What is y as a function of x? So y is equal to-- well for
every glass they sell, they get $0.45 --so it's equal
to $0.45 times the number of glasses. There's not any kind of minimum
fee that they need to charge or they don't say any
kind of minimum cost that they have to spend to
run this place. How much in order to
break even for each a glass of lemonade? They need to make $25. So in order to break even,
they must make $25. So how many glasses of lemonade
do they need to sell? y needs to be equal to $25. How many glasses do
they need to sell? Well you just set
this equation. You say 0.45x has to
be equal to 25. We can divide both
sides by 0.45. On the left-hand side, you're
just left with an x. You get x is equal to-- What
is 25 divided by 0.45? It is equal to-- They would
have to sell exactly 55.55 glasses or 56 if I round. 55.5 repeating glasses. But you can't sell half a glass
or we're assuming you can't sell half a glass. So the answer to that, they
must sell 56 glasses. Because you can't sell half
a glass, I'm assuming. So they need to sell 56
glasses to break even. Just to graph this. Once again, we'll hang out in
the first quadrant because everything is going
to be positive. Every glass they make $0.45. Let's say that they sell--
So this is the number of glasses, x. This is how much they make. Let me go by increments of 5. 5, 10, 15, 20, 25. Actually I need to go by even
larger increments to get to the point that we're
talking about. Let me go increments of 10. 10, 20, 30, 40, 50, 60. So that's the number
of glasses. When they sell 0 glasses,
they make $0. That's their y-intercept. y is equal to 0. When they sell 10 glasses,
they make $4.50. So this is 4.50. This is 9. Actually let me just
do it like this. Let me just mark only
the multiples of 9. Let's say 9, 18, 27, 35. When they sell 10 glasses,
they're going to make $4.50. 10 times 0.45. That's right there. 20 glasses, they're going
to make $9.00. We can keep going there. 40 glasses, they're
going to make $18. You see their slope. When you move 10, you're going
to have to go up 4.50. This graph is going to look
something like that. Should be a straight line. Then if you want to see their
break even, their break even has to be $25, which is
right about here. Their break even is $25
right around there. Let me draw the line a little
bit better than that. The line is going to
look like this. If the break even is $25-- it
would be right there --you see that they have to sell
about 56 glasses. Obviously the way I drew this
isn't the super neatly drawn graph, but hopefully it gives
you the general idea.