Modeling with linear equations: gym membership & lemonade

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.