BAM! Mr. Tarrou. In this Algebra lesson we

are going to learn how to model a real life scenario, or word problem, with a linear function.

And through the lesson we are going to be using this example of a car salesman that

makes part of their income based on salary and another part of it on commission. And

we are going to identify what the independent and dependent variables are, what are the

units of measure. And like many of us, if you have a hard time creating an equation

after you read a word problem, then we are going to go back and rely on something you

have been doing much longer than Algebra. That is basic arithmetic. We are going to

make a table here where we are calculating the pay of… well this is in terms of you

are the car salesman… so we are going to calculate your pay based on whether you sell

0, 1, 2, or 3 cars. We are going to allow the pattern we see in that arithmetic to allow

us to come up with the equation, this linear function, that is modeling your salary. And

we are then going to graph it and maybe you have or have not graphed lines yet, but because

this is a real world setting we will need to discuss how you label the x and y axis.

We will also determine what would be an appropriate scale. We are going to be discussing why was

a linear function a good model for this scenario. Why is the equation and the graph of this

line describing how much you are going to make as a salesman. Then we are going to do

another lesson based on this same topic, and I will put a link to that in the description

of this video, that is going to have a little bit of a different look… a little bit more

difficult to hopefully help you through more of your homework problems. So we will be discussing

again how can we identify that a linear model is a good fit for a word problem. Then we

are going to answer question of how many cars do you need to sell to make a certain amount

of money. And if you know how to find an x intercept letting y equal zero, and a y intercept

letting x equal 0… and if you know that slope is rise over run that is all good for

graphing the equation of a line. But since this is going to be applied to a word problem,

you know… How do we interpret those values in the context of your problem? So let’s get

started. We have got that you are working at a car dealer that pays a monthly salary

of $1000.00 dollars. No matter whether you go in for 40 hours for the month or 200 hours

for the month, you are going to make that flat income of a thousand dollars. So what

do you want to do? You want to sell a lot of cars because the dealer is also going to

pay you $200.00 in commission for every car that you sell. An income that is based on

how much you are selling for the company. What is the independent and dependent variable?

Basically… and why? Basically I am asking you…If you have graphed lines before you

know there is an x axis and a y axis. What is going go along that x axis? And what variable

is going to go along the y? Well what are we even looking at in this problem? Let’s

see here. You are working at a car dealer that pays you… Ok, it is something about

pay. You are making money. And you are making money, yes you have a salary, but really and

hopefully if you are really successful yo are going to sell a lot of cars. That is going

to increase that pay. So it is a relationship between your income and how many cars you

sell. Well, an independent variable is an input value. It is the number that is going

to go into an equation and give you out an answer, an output variable. And in some situations

it is kind of hard to tell what should be the independent and dependent variable. If

your word problem has time involved in it, almost always the time is going to be the

independent variable. And is there a variable that you can think of that you would have

some control over? Well, you are going to have some influence in how much you make by

how hard you work. But how hard you work and how you present yourself to your customers

is really about the number of cars you are going to sell. So the number of cars is going

to be the independent variable. How many hours a week do you work? How are you approaching

your customers and dealing with them to entice them into wanting to buy a car. The independent

variable are those cars. And the dependent variable,

selling those cars and selling more cars is going to have an influence on your pay. Well,

I don’t want to say salary because I have the phrase salary of $1000.00 as there is

a flat value of a thousand dollars a month. So your overall pay is going to a combination

of your salary and your commission. So your pay is your dependent. And I did not write

out why, but I kind of verbalized that. The independent variable is often a variable that

you have some control over and the dependent variable is going to be the output, or the

answer, that you get. And in case you forgot another vocabulary word that goes along with

independent is domain. Ok. So your domain value, that is d-o-m-a-i-n if you were not

sure of that, is also going to be your value that is going to go along the x axis. And

your dependent variable, another vocabulary word that goes along with that is range. So

Domain and Range. And they generally.. when you are thinking about the Cartesian plane

with the x and y axis, independent is along the x axis the the dependent is along the

y axis. What are the units of measure? Sometimes this is a little bit difficult to connect

with your quantity… or your variables. I mean your independent variable is the car

but you naturally say it is the number of cars you sell. So the unit of measure for

the cars is simply, let’s put this in blue, you know… the number of. The number of cars.

The units of cars maybe is a language you might want to include. Now the pay is a little

bit more obvious. I mean if I live in America… well I do live in America. I get paid in dollars.

Maybe if you are in Europe or England it is the Pound. I am just spacing out here about

the Euro. Well… ok. Instead of taking time to think of it, where we live is going to

determine the currency we are paid in. Right now I am going to be saying our unit of measure

is in dollars. Ok. So we have got our independent and dependent variables identified reminding

you which is going to land on the x and y axis. We talked about the unit of measure

for each of these. Write the calculations required to compute your pay if you sell zero

car, one car, two cars, and three cars. And again this is only going to be really necessary

if the textbook asks for this or you are not sure how to write the equation based on the

word problem. So the quantity. The independent variable we already discussed is going to

be the cars. And the unit of measure, that is kind of… The unit… That is some language

that some people use, the units of cars… or I sold this many cars… or 17 units in

a month. I am just going to write “the number of” and run out of space. And the dependent

variable is that pay which is going to be in dollars. Now what if you sold no cars at

all. Just… I don’t know. Just for some reason for an entire 30 days you did not sell a single

car. I am not sure how long you are going to keep your job. But the way this problem

is worded, you would still get a $1000.00 check so you can at least pay some basic bills.

Now if you sell one car that is going to be $1000.00 plus our $200.00 commission. If you

sold 2 cars, that was $1000.00 plus$200.00 plus $200.00. And if you sold 3 cars that

would be of course $1000 and three $200’s. Now if these numbers are simple enough to

do in your head very very quickly, and a lot of my students… They can’t come up with

the equation. Then I start asking them questions like this and they start rattling off numbers.

No, I am making the numbers easy for us to work with in class.. But it is the recognition

of the pattern that is there. Write out that arithmetic so you can see the pattern forming

to help you develop and equation even if you can do the numbers in your head. What happens

a problem that you cannot do in your head? So what is going on here? What do all of these

lines have in common? Well your pay is based on every single one of these lines regardless

if it was 1, 2, 3 cars has $1000 dollars written in it because that is your flat salary. And

then you can see here that.. you know… what is varying is the 200 dollars for 1 car. I

have 200 showing up twice for 2 cars. 200 dollars showing up three times for 3 cars.

So if we put a general value of x, or c, and actually I kind hate to just write x and y

when I am introducing these type of problems to Algebra students for the first time. I

kind of like to just write the word. You know… pay. My expression, if you will… Some textbooks

will call it an expression. How many cars have I sold? Pay is 1000 plus 200 times the

number of cars. That is like a general expression. Now that you recognize the pattern. 1000,

1000, 1000, 1000, ok no matter how many cars I have sold I got paid a $1000 once. And then

the $200 starts to vary by the number of cars sold. And this is an exact match. 1… 1,

2…2, 3… 3. So it is $200 times the number of cars. Some textbooks when they put tables

like this would maybe on the side say, “Give me an expression… A general expression that

describes this pattern that you are developing in the arithmetic.” Then the function that

is actually going on, if we use function notation is your pay is based on the number of cars.

That is equal to what we see here, a thousand dollars in salary plus 200 times how many

cars you sell. Or you might see this is y=1000+200x. But with the words it is much more obvious

where all the numbers belong and mean. So now let’s get to the next screen where we

actually… let’s see here. We already have the function written, so we are going to create

a graph that represents the relationship between the number of cars and the pay. [nananana]

Well we are going to graph that function making sure the label the x and y axis. Of course

we hopefully remember that the x axis is the horizontal axis and the y axis is the vertical.

And include a scale. Now you will notice that I have only included this first quadrant area

where we have positive x’s and positive y’s. That only makes sense in the context of this

problem. You cannot sell a negative number of cars. And I don’t believe we are going

to have any kind… You can’t sell a negative number of cars and you cannot have an income

that is negative. So we are just going to have a graph up here in quadrant 1. The upper

right hand quadrant of our x y axis. And we already discussed how the cars are going to

influence how much we earn. The car is the independent variable. So the number of cars

whether you write number or units of those cars is only going to go along the x axis.

And we are going to say this is going to be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and so

on. Putting the numbers on part of that scale. Those cars are going to influence our income

or our pay. And that is going to be in dollars. That is a little bit too sloppy. Let’s try

that again. Our pay is going to be in terms of dollars. And now our income is starting…

the lowest it is ever going to be is going to be $1000. [dogs barking] But before I do

that let’s go and try and get my dogs to stop barking! Ok. Yes, I feel very safe from the

salesman that was at my door. Now our pay is ranging from… The lowest we are ever

going to make is a thousand dollars. So having my vertical scale start at zero does not really

make a whole lot of sense. Now some teachers are going to want you to start this at zero,

and then put a little spacer here. Then count from 1000 and from here it is natural that

we just count by $200. So 1000, 1200, 1400, and 1600, and so on. But when you… Or, when

you are making graphs it is acceptable to just let the lowest value along your y axis

be appropriate for your word problem. So maybe I could let this lowest value be $800 and

then put a tick mark at 1000 dollars, another tick to represent 1200, 1400, 1600, 1800,

2000 and so on. But do what your teacher wants. If they want you to put a zero here and then

that little squiggly line gap to indicate that you skipped a portion of that scale before

you start counting, then by all means do what your teacher is asking you. What is really

important though with making a graph is once you start a scale you don’t change that scale.

I cannot count by 200, 200, and then go.. Oh, ok. I am going to count by 500’s. That

is very wrong. Ok, so we have 800, 1000, 1200, 1400, 1600, 1800, 2000, and so on and so on.

And we have got a point, an (x,y) coordinate… and independent and dependent values of (0,1000),

(1,1200), (2,1400), and this is not on graph paper so it is not going to be perfect. (3,1600)

And it looks like if I connect the dots we are going to indeed get that linear function

as the title of the lesson of course implies. (Points should not be connected in this example)

Well, there is the graph. And we have our scale. Probably would be best if maybe I continue

that on. We have our units. Our scale is our numbers. We have the labels on the x and y

axis. The independent and dependent axis. Our value… our quantity if you will and

the unit of measure. And all that would be left maybe is to write on the top a title

for this graph, Our Income as a Salesman based on the number of cars sold. How can we identify

this scenario as one that can be modeled by a linear function. Well, we have the graph

now and we can see if the graph is straight. So that is one way. If you have that at your

disposal or if you are able to make a graph, your graph is linear. Now another… and I

am going to verbalize this and then step off and write it just to speed the video up a

little bit. Depending on your textbook another thing to realize here as far as the language

they use, that you are…. This might be called your consecutive points, or the difference

between your points which are consecutive in order. Of course the differences between

your consecutive points (0,1000), (1,1200), (2,1400), and so on… that those differences

between your consecutive points are equal. Now when they say consecutive points, they

are talking about this list of your domain values… your x values are counting up by

1. And indeed any time you have a linear function, if the difference in your x coordinates are

equal you will see a difference in the output values, the range values, your dependent values,

your y values are going to be equal as well. Now one of the textbooks I have taught from

also calls this, when these values of x go up by one, that those first differences are

going to be equal. So let me step off and just write to speed this up a little. Now

the last way of recognizing that, or another way of recognizing that a linear model is

a good fit for your word problem, is it is not just these differences between successive

points your first difference are going to be equal. But a line is straight. A line does

not change direction. And we know, hopefully you have studied it already, and if not I

am going to talk about this right now… The slope between any two points is going to be

equal. So in the next lesson I am going to do with this type of topic, the values in

our table are going to be a bit more scatters. We see that for each increase of 1 in the

domain value, the x value, the range value.. the y values.. are increasing by a fixed amount.

But indeed, you know…. I guess what I am trying to say is slope is right over run,

right. It is the change in y over the change in x. The real definition of slope is the

change in y per one unit of change in x when you want to interpret it in terms of a word

problem That is how this table is set up. But indeed, you know, we can see the slope

is 200 here and I should be able to pick any two points in this table and get a slope of

200. So the slope between two points… or the slopes between any two points… all of

those points that you can possibly do are equal. Ok, well let’s just try that. The slope

is y sub 2 minus y sub 1 over x sub 2 minus x sub 1. I am going to let this be my second

point and this be my first point. And we have got 16-1200 over 3-1. That is going to be

400/2. And yes we actually did once again, as we see here, get that slope of 200. And

now I am pointing to the value in front of the x because this equation is in slope intercept

form of y=mx+b. It is just that the order is turned around with the way that I have

it written. Ok, so linear graph, differences between successive points or our first differences

are equal, and the slopes between any two points that you picked out of that table…

you keep getting the same answer. That is indication that a linear model is a good fit.

Now one more thing. All of these first differences are 200. 1200-1000, 1400-1200, and so on.

Then when we just pick two random points I got again a slope of 200. And if you go back

to the wording of this problem…. the wording of the problem said that you are getting paid

200 dollars per car. You get a commission of $200 for each car you sold. Well, a lot

times in math the word “per” is a division symbol. So if we write this as 200 dollars

per car, there is your change in y over your change in x… or at least an indication of

what should be on the y axis, the range values or dependent variables… That is what I wanted

to say… over your.. y over x… your domain values or your independent variable. Again

slope is, and a different way of writing it yet again, change in y over a change in x.

But you don’t see delta really until you get to PreCalculus or Calculus textbooks. Now

one more question here before we start interpreting the slope meaning in context of this problem

and the x and y intercepts. How many cars must you sell to make 3100 dollars? So you

want to make… you want your pay… you want your salary, not your salary but your income

to be $3100. Well it can be a little bit confusing to know where you put $3100 when all you see

in your equation are a bunch of x’s and y’s. But if you see here, your pay is based off

the number of cars you well. Well I want my pay, I want my income, to be $3100. So 3100

is equal to 1000 plus 200 times the number of cars which you could put x. I am going

to write cars because I think Algebra is clearer when you have words. We are going to identify

where the variable is. The variable is on the right hand side of the equation now. So

we are going to have to move everything away from the c, x, or the cars. And I have an

addition of 1000. We are going to subtract both sides by 1000 to undo that addition of

1000. So now I guess we are doing a little review of how to solve two step equations.

Then our last step is to… again the question is, “How many cars must you sell?”, so we

need to get the 200 away from the variable of cars. That is connected by multiplication,

so we are going to divide both sides by 200. And that is going to be… well it looks like

it is going to be… These zeros are going to cancel out. And 21divided by 2 is 10.5.

Now, there are a lot of applications… a lot of scenarios where decimals or fractions

are acceptable and really expected. But we cannot sell half of a car. (Why the points

in graph should not be connected) ok. I guess salesman… Two salesmen can work together

to sell a car to a customer. But I don’t want to get into that. We are talking about one

person selling cars. You cannot sell half a car. If you sell 10 cars, you are not going

to make… you are going to be a little bit below 3100. And if you sell 11 cars you are

going to make over 3100. So maybe… First of all we are going to have to say that the

number of cars you are going to have to sell… you have to sell. Let’s put this in context.

Really your answers when you are dealing with word problems should be at least an expression

if not a full sentence. You need to sell 11 cars. Because if you sell 10 cars you are

not going to make that $3100. You are going to be a little bit below. There is dust flying

around. And if you sell 11 cars, that almost looks like I wrote a parallel sign, but if

you sell 11 cars you will be a little bit over that. But having a little bit too much

money is much better than not having enough. Now if I wanted to change this just a little

bit and say how many cars must you sell to make at least $3100. This why videos go a

bit long. I am always thinking well what if it is, and what if it is that. What if you

see that in your homework? If this is the expression that your income from the dealership

and… So I want this expression to be at least 3100. So that expression of your pay

can be equal to or more than 3100. I can just bring that down, and that down, and now instead

of saying you need to sell 11 cars to make $3100, you need to sell 11 cars

or more to meet the requirements that are given in this particular question. Ok, so

now I am going to clear all of this off and we are going to interpret… Oh! You know

what? I am going to include something else at the end of this lesson. You can do this

graphically as well. We have the graph which is our pay. Pay is 1000 dollars plus 200 times

the number of cars. We have the graph of line representing the income. We can also say going

back to the equal part, that I want to just make $3100. That can be equal to my pay, right.

That is what I want my pay to be. And my pay is a value along the y axis. So I could enter

into a graphing calculator y=1000+2x and y=3100, and I am going to go off the scale because

I only when to 2000, but I can graph y is equal to 3100 and then use the intersection

function of my calculator to find the coordinate where those two graphs intersect. I will do

a little snippet at the end of our lesson to show you how you could have done this graphically

with your calculator as opposed to Algebraically. Alrighty then! Interpret the slope in the

context of this setting or scenario. Well I have already done that in words. I just

wanted to write it. For every car your sell, your income will increase by $200. For every

change in… for every unit of change in x, one unit of change in x… for every car that

you sell, your income will increase by 200 dollars. Because yes slope is rise over run,

yes it is change in y over change in x. But in terms of word problems you need to focus

on the definition of slope as the change in y, 200 dollars of income, per one unit of

change in x. In that case it is every car. And when you figure out what slope is and

you simplify it to a point where the denominator is equal to 1, that will also be called Unit

Slope. So basically the Statistical or word problem definition of slope is basically Unit

Slope. Interpret the y intercept in context of this problem. I have already done that

in words as well. Luckily we have it in our table. The y intercept is where the graph

is going to cross the y axis. Well, when you are crossing the y axis the x is zero. And

we have a pay for when you sell zero cars a month. Now in this particular problem the

y intercept has some real context. It is a very unsuccessful salesman, but it is not

unreasonable or ridiculous to interpret this value of making a $1000 based on a sale of

zero cars. That is what it is. Interpret the y intercept. You earn, or if it was a Statistical

problem that was modeling real data, then you would say that you expect to earn $1000

dollars when you sell zero cars. Now I also teach AP Statistics besides Algebra, and PreCalculus,

and Calculus in my high school and I have never had a Statistics book ask you to find

or interpret the x intercept. But I noticed in one of the recent algebra books our school

bought to use, it does. So we are going to find the x intercept and interpret it in the

context of this problem. Now often the y intercept will not have any real life meaning. Let’s

see if the x intercept does. Well to find the x intercept, that graph needs to cross

the x axis. I already have you a hint earlier that I am thinking that as this line continues

down and it seems like it is going to cross the x axis on the negative side. x is the

number of cars sold. We are probably not going to have a real life context, or meaning, for

this particular problem. But y, in that case the y axis is pay, if our pay is zero that

is going to be equal to… Our base salary is equal to $1000 plus $200 times the number

of cars that we sell. We are going to subtract both sides by 1000. Get a

negative 1000 is equal to 200… I am going to put… well I was going to say I was going

to put x. But that would just because I do not feel like writing the word cars. We are

going undo that multiplication of 200. That is what is connecting that coefficient to

the variable cars. Inverse math operation there. We get negative 5 is equal to cars.

I am running out of room to write a sentence. Let me just verbally just say it. Our x intercept

is when y=0 when x=-5. So if I were to sell a negative number of cars, if I were to sell

negative five cars my y axis which is pay… my income would be equal to zero. Well I already

said that you are going to make $1000 a month period unless the place fires you. And it

is impossible to sell a negative number of cars. So for this particular problem that

would be how you would write it in context of the problem. If I sell -5 cars, then I

will make 0 dollars. That is how you write it as far as interpreting the x intercept

in context of this problem, but that does not mean that it has any real meaning. That

is how you model real life situations with a linear function. I have another lesson coming

up like this. I will have a link in the description. But for now I am done. i am Mr. Tarrou. BAM!

Go Do Your Homework! Ok, so here we have our INSPIRE calculator up. We are going to graph

those two functions. Hit the Toggle button here until we get to the graphing window.

And our first equation was 1000+200x. Hit Enter. And we get what almost looks like a

vertical line because of the scale of the window for the application of this word problem.

Now we have already done the algebra so we need to sell 2.5 cars which is unrealistic…

or 11 cars. We are going to go to Window Setting. There is no reason to have negative x values

when the x values represent cars that are sold unless you just want to see the access

line on your screen. Maybe that is not a bad idea so let’s make that -5. And Calculator

is probably going to have a problem because I hit the subtraction key. You have to be

careful with that negative sign. Negative 5, there we go. We are going to toggle down.

Make that say 15. Toggle down some more. Let’s let that y minimum be -100 just to see the

axis line. And the vertical scale is going to be…. Let’s set that to 3400 since we

are looking for an income of 3100. Hit Ok. Now we are going to see using those negative

numbers we included the x and y axis. Hitting Tab to open up that entry line again. Let’s

put in y is equal to 3100, that income that we are looking to make in a particular month.

We see that the graphs like I drew on the chalk board are intersecting at a particular

point. We are going to go to menu, Analyze the graph, and find the Intersection. If you

are working with a TI-83 or TI-84, you can do Second… there is a second button over

here… and hit the Calculate button underneath the viewing window of your graphing calculator.

You will find that Intersection option there as well. Where is the Lower bound? Let’s hit

Enter. Let’s put the cursor to the right of the upper bound of the intersection point.

Basically just move to the right of the intersection [point. There we see we have the same x coordinate

we found algebraically of 10.5. And because this is an application problem and you can’t

sell 10.5 cars we are going to say that we need to sell those 11 cars to make the $3100

dollars of income. This is in scientific notation, 3.1 times 10 to the 3rd power. So one last

time, I am Mr. Tarrou. BAM! Go Do Your Homework:D