hey everybody this is Donald Trump and this lesson is modeling linear relationships let's go ahead and get started here so um a department store offers a frequent buyer card for um to earn rewards for purchases uh customers make at the store each transaction is worth 12 points and the customer automatically earns 25 points when they sign up okay so let's write an equation for the function that gives the card value based on the number of transactions that have occurred Okay so what units this is always what we have to ask ourselves what units will be associated with the variables in this okay well the units are points you guys so the variables are card values and transactions and the units for the card values are going to be the points right there okay so let's go ahead and complete the verbal model for the frequent buyer card function including units okay so the card value points equals the initial value uh points which is that 25 points for signing up plus you get 12 points for each transaction so the purchase value which is points per transaction that's the 12 times the number of transactions and we're going to call that t in this case so write the function rule for the card value function C so this is the cost or the card value for the number of transactions that's what T is it's going to equal our initial sign up which is that 25 right here the initial points Value Plus the 12 points for each transaction so this will be 25 + 12T now can you see this right here y equals over here is our mx + b right there okay so we can graph that and this would be our Y intercept right here this would be our slope or a rate of change okay so for each 100 points the customer receives a gift card woohoo and then so how many transactions will it take for the customer to earn the first gift card well okay so we're asking when is this going to equal 100 so we just set this equation equal to 100 and then we solve for T and that'll tell us how many transactions so here we are subtracting 25 from both sides then dividing by 12 and we get 6.25 transactions well what does that mean we can't have 6.25 transaction so it's going to be more than six transactions so on the seventh transaction they're going to earn their first gift card because that's when it finally goes over 100 right there so 100 points so what is the Y intercept for this linear function function and what does that represent well the Y intercept was that + 25 so it's that 25 and it represents the beginning value where we have zero transactions in the points on a frequent buyer card so when we first sign up we get 25 points so what is the slope for this linear function well that was that 12T and what does it represent well the slope was 12 and it represents the rate of change or the rate of increase in points per transaction because you get 12 points per transaction so that would be the rate of change all right let's create an interpret a linear model right here okay so here's another one uh fundraising so the band Boosters club is selling T-shirts and blanket wraps to raise money for a trip the band director is asked the club to raise at least 1,000 bucks okay all right so uh the Boosters club president wants to know how many t-shirts and how many blanket wraps the club needs to sell to meet their goal of 1,000 bucks okay so the t-shirts cost $10 $ each so this will be 10t and the blanket wraps cost $25 each so we'll call this 25b I think so let's write a linear equation that describes the problem and then Graph the linear equation all right easy enough and well maybe not but we'll go through we'll get through this so how can the Boosters club president use the sales price of each item to meet their goal okay woohoo all right here we go so let's identify the important information okay so t-shirts cost how much your t-shirts cost 10 bucks and blanket wraps cost 25 bucks okay and the Boosters club needs to raise 1,000 bucks all right so easy enough we're going to go ahead and put that in right there okay so the total amount of Revenue earned by selling uh t-shirts is I think it was 10 bucks and the total amount uh Revenue earned by selling the bracket blanket wraps is 25 bucks so these two results can be added and set equal to the sales goal to find the number of t-shirts and blanket wraps that need to be sold to reach our $1,000 okay so let's graph this function to find all possible combinations of t-shirts and blanket wrap so this is like 10 x + 25 uh y = 1,000 okay or but they're going to do 10 t + 25b = 1,000 okay just think of X and y's okay so we're trying to get that th000 bucks right here so so um uh looks like we're going to put the blanket wraps first so this is going to be 25 right here plus the t-shirts this is going to be 10 that's going to equal a th000 right there okay all right so let's go ahead and uh graph that right here so it looks like you guys um that you can choose whichever one you want to be uh your your x axis and your Y axis but they put the T right here so this right here is always our X column and this is always our y column so T is right down here so this is for t-shirts right here so right in here is going to be the number of t-shirts and and then B is going to be our y column and this is going to be the number of our blanket wrap so those got to be distinguished first right there so we got to put that in there okay all right so when uh when uh T equals z here's right here then 25 goes into a th000 um uh we get 40 times 25 goes into 100 four times so into a th000 is 40 times notice I plotted 040 right there all right and then um let's do this one next you guys when uh T when b equals Zer so we'll make this uh zero so 10 goes into a, 100 times right there so uh we're going to graph 100 Z right there well that's easy enough so we can um go ahead and connect that line right there and then this one um I mean you can put in 50 for T right here so this would be uh 500 right here so we subtract 500 and divide by 25 or now that we have the gra we can go over 50 and figure out it's going to be uh that the blanket wraps is going to be 20 to get that matching pair right there so it's 20 right there all right okay so uh there's our equation right there 10t + 25b = 1,000 I don't know why they switched it we could have written this right there but they wrote 10t first right there probably because the t's down here and the B's going right there that comes first right there all right so the X intercepts represents the number of here's the x intercept right here it represents the number of uh t-shirts that are needed to be sold if there are no blanket wraps right there and the Y intercept right here is the number of blanket wraps that need to be sold if there are zero t-shirts being sold right there okay all right and then the Boosters club uh president can use the the line right there to find the possible outcomes of t-shirts and blanket wraps that's going to reach our $1,000 right there okay just filling in some blanks right there so technically you guys the graph of possible combinations and t-shirts of blanket wraps uh that reach the goal of $1,000 should be discret well discret just means they're they're just points you guys they're not aign so so let me back up right here this is not a discret graph right there if I took out this line right here and just had these points right here it would be a discret graph discrete just means it's just separated by points it's not connected by a line but for conv vence the graph is shown as a connected line so it helps us you know find other points right there so explain why the solutions to this problem would only be points on the line that have whole number coordinates well you can't sell you know decimal blanket wraps or decimal T-shirts they can only sell a whole number of t-shirts and whole number of blankets so they they'd only be uh points on the graph not the line connected but the line just makes it convenient for us to find things right there all right let's try another one the sandwich shop sells sandwiches for $5 each and bottles of water for $1 each the owner of the shop needs to earn a total of 100 bucks by the end of each day write a linear equation that describes the problem then Graph the linear equation and make sure to label both axes with appropriate titles okay and then use the graph to determine how many sandwich shops must be sold uh must uh sell uh if no water is sold all right so right here the S stands for sandwich and W stands for water so this is going to be the number of water bottles this going to be the number of sandwiches sold okay all right there we go and then so our equation is five bucks for the sandwiches plus $1 for the water equals 100 right there okay well so um we can go ahead and graph right here if if uh s equals zero then W is going to have to be 100 right here and then so we'll graph 0 100 right up here and then if W equal 0 5 goes into uh 120 times so we'll graph 0 or 20 Z so there's 20 let's go ahead and connect them up with that line right there again that would be a a discret graph but this is a non-discrete graph an indiscrete graph so um technically it wouldn't be a line it would be a bunch of points but this line helps us solve some things right here all right so let's answer the question right here so then use the graph to determine how many sandwiches uh the shop must sell if no waters are sold so if zero waters are sold okay that would be right here then the number of sandwiches would be 20 they'd have to buy 20 sandwiches so if there's no water 20 sandwiches must be sold okay all right so how can the graph of a linear function be used to answer the real world problems well there's the answer so the points along the graph of a linear function are the answers to a real world problem and so what is the first step when modeling the linear relationships well we got to uh determine the units of these things right there okay gang if you're in my class that would be your homework sorry for the long lesson and see you in the next one