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in this video I want to do a couple more word problems dealing with graphs of lines so here we have a Jim is offering a deal to new members customers can sign up by paying a registration fee of $200 so they pay $200 and then a monthly fee of $39 $39 per month this is registration 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 so what we pay so let's 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 and registration so we're gonna have to pay $200 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 one month we'll have to pay one month times $39 and we would have already paid the 200 dollar registration fee so it'll be two hundred and thirty nine dollars if we stay two months we pay the two hundred dollar registration fee and then we paid 39 times two months which is what like 78 or something so it'd be two hundred seventy eight dollars so just to tie this all together with linear equations and graphs of them let's graph this relation and 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 P is equal to 39 M plus 200 so what's the slope and what's the y-intercept and you might get confused say hey there's an X there and a Y but now you're doing with P's & M's just remember this is the independent variable that is independent independent and int this is the dependent variable here this is the independent variable how many months you pick number months and I'll tell you what the total cost of your membership is going to be so it's the same thing this is like the X right there and this is like the Y just like that and so just using our pattern match this right here is the we could say it's the vertical intercept or the P intercept or the well you know I'm tempted to call it the y intercept but we're really at intersecting the P axis instead of the y axis there and this right here is our slope so let's graph this function and 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 we're going to pay the gym $200 just like that $200 for zero months then for every month we're going to spend an extra $39 so the slope is 39 so that's let's say this is one month right there that's this is in months this is months and then this axis is price the p axis so this is as I said this is like the P intercept or the y-intercept so after one month how much are we going up to pay well our slope is 39 so if we move one month forward we're going to go up by 39 so this will right here that will be 239 and if we go another month 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 per month will look something it will look something like this and so they say how much will a membership cost by the end of the year twelve months so we would have to go to three all the way out to 12 months which might be here so our graph is going to be out here someplace but we could just figure it out algebraically at the end of the year we will 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 and what's 39 times 12 39 times 12 2 times 9 is 7 is 18 2 times 3 is 6 plus 1 is 7 T F 0 1 times 9 is 9 1 times 3 we want to ignore this 1 times 3 is 3 so if 8 7 + 9 is 16 1 + 3 is 4 so this right here so the price of our membership is 200 + 39 times 12 which is 468 dollars so it's equal to 668 dollars at the end of our year so if you went all the way out to 12 you'd 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 45 cents for each glass of lemonade 45 cents for a glass 45 cents per glass of lemonade in order to break even they must make 25 dollars they must make 25 dollars how many glasses of lemonade must they sell to break even so let's let's let me just do it with y + X Y is equal to the amount they make amount they make and they max the amount today make and let X is equal to the number of glasses they sell they sell so what is Y is a function of X so y is equal to well for every glass they sell they get 45 cents so it's equal to 45 cents times the number of glasses right and there's not any kind of like minimum fee that they need a charge or they don't say any kind of minimum cost that they have to spend to run this place so how much in order to break even for each glass of lemonade they need to make 25 dollars so in order to make break even they must make 25 dollars so how many glasses of lemonade do they need to sell so that this thing needs Y needs to be equal to 25 dollars so how many glasses do need to sell well you just set this equation you say point zero five I saw point zero four or 0.45 X has to be equal to 25 it has to be equal to 25 we can divide both sides by 0.45 0.45 0.45 on the left hand side you're just left with an X you get X is equal to what is 25 what is 25 divided by 0.45 is equal to so at 50 they would have to sell exactly fifty five point five five glasses or five six if I round so fifty five point five repeating glasses but you can't sell half a glass what we're assuming you can't have sell half a glass so the answer that they must sell fifty six glasses because you can't sell half a glass I'm assuming so they can just sell fifty six glasses to break even and 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 forty five cents so let's say that they sell so this is the number of glasses X this is how much they make let me go and buy and we go by increments of oh I don't know let me go by increments of five five ten fifteen twenty twenty-five actually need to go by even larger increments to get to the point that we're talking about let me go by increments of 10 10 20 30 40 50 60 so that's the number of glasses and when they sell zero glasses they make zero dollars that's the y-intercept it's at Y is equal to zero then when they sell 10 glasses they make $4 fifty so this is 450 this is nine actually let me just do it like this let me just let me just mark only the only the $9 the multiples of nine so let's say 9 18 27 35 and when they sell 10 glasses they're going to make $4 50 10 times 0.45 so that's right there 20 glasses are going to 9 dollars 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 450 so this graph is going to look something like that should be a straight line and then if you want to see their breakeven their breakeven has to be $25 which is right which is right about here $25 the break-even is $25 right around there we do it a little bit let me draw the line a little bit better than that so the line is going to look like this and that the breakeven is $25 be right there you see that they have to sell about 56 glasses and obviously the way I drew this isn't a super neatly drawn graph but hopefully it gives you the general idea