Math for Economists - Lecture 1

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Math for Economists - Lecture 1
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Focus (optics) Free group Forcing (mathematics) Multiplication sign Student's t-test Open set Mereology Graph coloring Open set Grothendieck topology Fisher information Mathematics Fisher information Lecture/Conference Moving average Social class
Frequency Radical (chemistry) Goodness of fit Computer animation Cube Lecture/Conference Multiplication sign Social class Scheduling (computing)
Point (geometry) Quantum state Gradient Interior (topology) Bending Maxima and minima Binary file Mathematics Computer animation Lecture/Conference Cylinder (geometry) Physical law Figurate number Social class
Euler angles Witt algebra Multiplication sign Gradient Maxima and minima Volume (thermodynamics) Mortality rate Student's t-test Mass Fisher information Degree (graph theory) Mathematics Goodness of fit Fermat's Last Theorem Computer animation Lecture/Conference Different (Kate Ryan album) Term (mathematics) Universe (mathematics) Matrix (mathematics) Figurate number Maß <Mathematik> Social class
Point (geometry) Computer programming Observational study Multiplication sign Strut Modulform Maxima and minima Student's t-test Ordinary differential equation Vibration Grothendieck topology Wave packet Mathematics Lecture/Conference Term (mathematics) Different (Kate Ryan album) Well-formed formula Bellman equation Operator (mathematics) Maß <Mathematik> Social class Area Process (computing) Computability Lattice (order) Complete metric space Statistical hypothesis testing Fisher information Computer animation Morphismus Order (biology) Configuration space Convex hull Figurate number
Lecture/Conference Different (Kate Ryan album) Musical ensemble
Mathematics Lecture/Conference Equaliser (mathematics) Order (biology) Division (mathematics) Theory
Fisher information Axiom of choice Predictability Mathematics Lecture/Conference Model theory
Complex (psychology) Multiplication Many-sorted logic Lecture/Conference Meeting/Interview Mortality rate Rule of inference
Point (geometry) Time zone Lecture/Conference Multiplication sign Square number Mortality rate Figurate number Physical system
Point (geometry) Fraction (mathematics) Zirkulation <Strömungsmechanik> Multiplication Lecture/Conference Well-formed formula Geometric series Physical law Calculus Mortality rate
Statistical hypothesis testing Multiplication Observational study State of matter Archaeological field survey Linear algebra Calculus Variable (mathematics) Mathematics Lecture/Conference Well-formed formula Different (Kate Ryan album) Social class
Point (geometry) Positional notation Lecture/Conference Set theory Social class Wave packet
Positional notation Lecture/Conference Kälteerzeugung Different (Kate Ryan album) Image resolution Content (media) Archaeological field survey Genetic programming Algebra
Geometry Lecture/Conference Linearization Sheaf (mathematics) Line (geometry) Nichtlineares Gleichungssystem Algebra Physical system
Logical constant Linear equation Functional (mathematics) Multiplication sign Equals sign Line (geometry) Variable (mathematics) Numerical analysis Power (physics) Lecture/Conference Linearization Nichtlineares Gleichungssystem Coefficient
Latent heat Lecture/Conference Real number Universe (mathematics) Right angle Nichtlineares Gleichungssystem Variable (mathematics) Set theory Power (physics)
Latent heat Finitismus Lecture/Conference Universe (mathematics) Coefficient Variable (mathematics)
Linear equation Real number Barrelled space 3 (number) Line (geometry) Variable (mathematics) Dimensional analysis Power (physics) Wave packet Frequency Latent heat Latent heat Arithmetic mean Lecture/Conference Linearization Nichtlineares Gleichungssystem Coefficient Intercept theorem
Linear equation Standard deviation Goodness of fit Positional notation Lecture/Conference Model theory 1 (number) Variable (mathematics)
Point (geometry) Linear equation System of linear equations Real number Variable (mathematics) Numerical analysis Variable (mathematics) Latent heat Positional notation Lecture/Conference Universe (mathematics) Coefficient Algebra Physical system
Statistical hypothesis testing Lecture/Conference Curve 1 (number) Genetic programming Circle Power (physics)
Functional (mathematics) Observational study Wage labour Multiplication sign Model theory Physical law Insertion loss Mortality rate Lattice (order) Mereology Mathematical model Theory Substitute good Product (business) Mathematics Radius Lecture/Conference Term (mathematics) Different (Kate Ryan album) Linearization Negative number Energy level Circle Social class
System of linear equations Linear equation Multiplication Observational study Multiplication sign Infinity Mereology Variable (mathematics) Thermodynamic equilibrium Category of being Goodness of fit Lecture/Conference Different (Kate Ryan album) Nichtlineares Gleichungssystem Physical system
Point (geometry) Uniqueness quantification Line (geometry) Mortality rate Variable (mathematics) Limit (category theory) Thermodynamic equilibrium Leak Mathematics Lecture/Conference Term (mathematics) Nichtlineares Gleichungssystem Physical system
Point (geometry) Plane (geometry) Lecture/Conference Meeting/Interview Point (geometry) Saddle point Line (geometry) Variable (mathematics) Dimensional analysis
Point (geometry) System of linear equations Linear equation Multiplication sign Line (geometry) Variable (mathematics) Dimensional analysis Recurrence relation Lecture/Conference Different (Kate Ryan album) Negative number Nichtlineares Gleichungssystem Coefficient Algebra Set theory Physical system
Point (geometry) Process (computing) Lecture/Conference Energy level Line (geometry) Physical system
Point (geometry) Expected value System of linear equations Lecture/Conference Multiplication sign 1 (number) Genetic programming Mortality rate Variable (mathematics) Substitute good
Area Gaussian elimination Process (computing) Lecture/Conference Correspondence (mathematics) Negative number Price index Nichtlineares Gleichungssystem Mortality rate Resultant
Fisher information Gaussian elimination Lecture/Conference Combinatory logic Order (biology) Water vapor Line (geometry) Nichtlineares Gleichungssystem Algebra Substitute good
Multiplication Lecture/Conference Multiplication sign Line (geometry) Nichtlineares Gleichungssystem
Lecture/Conference Multiplication sign Nichtlineares Gleichungssystem Mereology
Point (geometry) Multiplication Building Matching (graph theory) Sheaf (mathematics) Line (geometry) Variable (mathematics) Numerical analysis Explosion Fisher information Meeting/Interview Lecture/Conference Nichtlineares Gleichungssystem Algebra Physical system Social class
Gaussian elimination Lecture/Conference Combinatory logic Nichtlineares Gleichungssystem Variable (mathematics) Numerical analysis Theory Substitute good
Gaussian elimination Group action Exclusive or Lecture/Conference Order (biology) Nichtlineares Gleichungssystem Variable (mathematics) Substitute good Euklidischer Ring
Point (geometry) Lecture/Conference Uniqueness quantification Multiplication sign Selectivity (electronic) Nichtlineares Gleichungssystem Variable (mathematics) Numerical analysis Physical system
Mathematics Focus (optics) Lecture/Conference Nichtlineares Gleichungssystem Variable (mathematics) Mathematician Leak Physical system
Point (geometry) Lecture/Conference Negative number Linearization Line (geometry) Nichtlineares Gleichungssystem Variable (mathematics) Algebra Physical system
Euler angles Multiplication sign Line (geometry) Mereology Variable (mathematics) Frequency Many-sorted logic Lecture/Conference Term (mathematics) Operator (mathematics) Nichtlineares Gleichungssystem Set theory Physical system
Point (geometry) Blind spot (vehicle) Line (geometry) Parameter (computer programming) Term (mathematics) Variable (mathematics) Mereology Causality Lecture/Conference Negative number Nichtlineares Gleichungssystem Physical system Social class
Linear equation Mathematics Presentation of a group Focus (optics) Lecture/Conference Term (mathematics) Real number Parameter (computer programming) Line (geometry) Physical system
Point (geometry) Focus (optics) Moment of inertia Line (geometry) Variable (mathematics) Substitute good Dimensional analysis Fisher information Category of being Gaussian elimination Plane (geometry) Lecture/Conference Negative number Nichtlineares Gleichungssystem Arithmetic progression
Point (geometry) Lecture/Conference Term (mathematics) Equaliser (mathematics) Negative number Lie group Numerical analysis Set theory
Lecture/Conference Numerical analysis
Lecture/Conference Multiplication sign Sheaf (mathematics) Food energy Open set
The welcome the mat for this is meant for 8 Yet again parts I therefore I so this is our Web page on Tripoli if not a student here you know a Tripoli is maybe that anybody in that situation
We understand believes this where you will get mostly information for the course the most important thing is that homework and stuff like that is posted their head focus
Jason groundwater letter me this is my office and Roland half Over across the park And then those amount about erect before class 1 come quest for you walk over here It's OK notice more in a film of course is not just today goal and so if you don't wanna be on film you don't want them in an hour during their early in the legend won't be included in the bill would oppose this has an Open course open-source course As for the whole world for free after I done filming So that's the reason for the cameras So that's kinda what's what's gonna happen in this course we got S 6 weeks we have setting here Very ratios of all times because pact but it's a small cluster really hear me in your life Me get on the site about grading forces get all Most important thing such a one-and-a-half how are things rated again after due for the colors so a couple things years all the homework for the whole course and this is an enemy collected homework is not part of that degrading is just suggested this could still sold posted on the Web page is 98 OK that's OK
Request to there's discussion right after class and that's where you had your quizzes and things like that and then ask questions about the homeworkers electric questions and then also lectured himself will have a mid-term and while a final and then use the scheduled so This is just a tentative schedule if I don't get everything in 1 male Pick up where they left off And I get a little ahead and that's OK to is kind of attended schedule that is to our classes Since were filming amounted to a break in the middle but I am also probably not going take the whole time to go over the income account on adding a little break before discussion because all the little buddy you can play on the classes being at least an hour and a half if I get through all the material I need to For the day then all stop in today's world trade and sometimes only go goal period is India's through something I wanna leave off in the middle so I just a little flexible on day of there's only 3 things a union immigrated on quizzes and terminal finals
The good news and bad news The good news is there aren't that many things they agreed on but the bad news is that if you mess up on any 1 of those things you don't all our room to make up Want that
Take the quizzes seriously they should be really easy you should be able to get good grades on all quizzes diet that the design anyway internal below more challenging in the final 2 we wanna try to get your points in the quizzes and then Of course introduced you can watch the final yes not suddenly scandal here so the first one is Friday of this week is only 6 Total us alive Friday there's 1 than Wednesday Friday with
We have our winters and and this is a holiday songs have quiz the day after the holidays The 6 Total Figure each quizzes 5 per cent a year You don't really wanna blow off a Although it is survivable if you do that That's adjusted there's the grading on the book that were using his UCI has treated at a special version of a Polish founded by mines a little so it
There's a lodger book here we just printed up a few Chapter 7 5 through 13 hearing books a year entitled to purchase the whole book it you won't find it online or something that all you need is the 1 that's provided bookstore printed of And although homework questions are Quebec is the cylinders The Defining itching to Tripoli yearn even have to login You can't just go there Class websites and glass websites math near misses a change to turn him if their dismal economists in the there's all the homework
So soon list talking about the book and forced to talk about why this class exists in the wife is there the matter economist
Well The economists in the UCI they'd been complaining over the years that when when they wanna teach economics students are hampered by their math and so they Asked that we provide a course that Supplementary knowledge so that they don't have to to spend their time teaching math they want to economics was an understandable that there's this common ground Ojeda happen ordered understand economics but it but if you don't have that they need to spend their time teaching you that 1st and so they don't end up being the teach Uh what what they want to the asses the teachers course we've been doing for a lotta years The 1st thing I wanna community This is a math class I'm in the math department classes designed by the math department we get approval from on apartment so we want to make sure that we cover certain topics that day thinkers important but the gist of the classes this is this is a matter not any come however it's got economists in the title we got this book this is mathematics were Economics so as a scanner how little bit economic you can't really I just do the pure but that the idea is that you wanna get good at the mass of the way you go do the economics its solid in your mind so why is that important Well what'll economist do think than in all of you have to assume is anybody here not any major most people are some people are so you guys are all potentially budding economists prey to getting ready to become a harness they might not be a professional economist images majored in economics and then move on to something else which is probably pretty normal But the design of the university is this is this is a center of knowledge this is where research happens this is where people Come to share ideas and further the knowledge of humor of the human race that's the idea of a university so what we're here to do it we were here to prepare people to do that which means that we have to design the classes to prepare people go to graduate you may not be going to graduate school in fact most of you are that's the that's the the problem here is that it's designed for people of graduates works with the degree at at you is for so then when you're finished here you're capable of then applying to graduate school and then being successful that's the designed so you have to ask yourself what it is that you're here if you here for that Great you're in the right place if you're not here for that some hybrid of that maybe you want the knowledge but maybe you just agree this is just a stepping stone for something up she'd identified for yourself what is it you're doing here not just in this class at the University want you identify which is doing here then you can have a plan of attack so if your goal is a good graduate school usually China learn everything you possibly can you you really into it you should be so curious and end Still motivated to learn every possible detail you can because that's what's gonna be required when you get to graduate school she star that mentality now if that's not your goal in your goal is to get a certain grade For the classes and then move on it is just a stepping stone this is just 1 piece in the degrees you gotta get past this year to the next class Thatcher attitude that's fine that's the normal attitude remained probably 90 per cent you have that I'm not picking on that I'm just saying they want you identify that then you have a different approach rate you want what you wanted to use you wanna you wanna get the information that's outside of your head into your head to the you can take exams you can pass them successfully hopefully you can retain the information but more reportedly you can still preserve your time for others because it's not your main goal to learn all this stuff and be totally dedicated to So what this was a math class a roomy studying matrices so I thought I'd had such appropriate that I Poeppel clips from the movie The Matrix because after all memory learning math and we have to take a 2nd 1st figure out how do you learn let's say you are that person that's just trying to get through this is a stepping stone then the question is how do you get the information at the outset of your head India ahead in the most efficient way so honored Will watch a little clip here the matrix of a helicopter is that it The hope as volumes terms of volume in in years also see this
I was a dot-com enough
They just about toward
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Right That
A that's all I wanted to see to see that download now will here
Would you like to build a walk in the exams and the out for mid-term downloads of getting ready to go with that nite so that the goal obviously were not quite there yet but that the goal so what I wanna do 1st of all it changes thinking my thinking has been changed when I saw that movie triggered something for me as it As a teacher I was thinking when I saw that You know I'm not really teaching and the students are really learned I wanna put that in a 21st century mind-set what I what I'm really doing is I'm a biological computer This is my hard drive for my operating system and what I do is I get this training program me and I can't take my eyes and I can scan his books and then it goes in the and my computer and I have certain training certain software that Bloated and Then what I do is I can mix it around and then I 3 comes out it comes flying out my mouth vibrations sound vibrations and it comes out of my hand on the board with all my hand in certain configurations and talk goes on the board and the polite from that goes into the room and it goes into your eyes and you download the information that I uploaded from that sore orders biological computers and more or there's information that's outside of our hard drives that we wanna get you so now when you start thinking it in those terms what's the most important issue when it comes to download is it Quality of understanding what is What is the issue was the main issue it down what makes 1 download better than another So now you might think of learning as You can you're downloading information now talk about speed how you do it quickly I get this done had I get the information in there is little effort as possible so that I can go off and do these other things that I like doing Enough to spend all my time studies that the So for those people who are not trying to get every little knocking trainees learn every little detail about every little thing here at the school on China memorize the library now you look at it in terms of just how quickly can I get this information in my head That everyone's gonna be different for instance someone like my like to download at midnight and someone might like to download at 6 AM so my leg to download at the library so my elected downloaded a cafe was bitten into individual but you need to decide what's the most efficient way so that general has nothing to do with this course If you want knowledge that's outside your figure out how get inside your head so what the waiters where the entry points lawyer I is ready Look it's that you're downloading it you listen this stuff and also tackles those are the 3 main of course you could smell stuck with nite cannot help you with studying and you could certainly tried Eat the paper I know that's not going to be So the main the 3 3 cents site the sound of the tackles So ideas all 3 years said things out loud when you study have you ever just copied stepped out of a book or a copied you know that's all the download so you'll have to figure that out for yourself is 1 of just put that idea They're so then you can take it and run with it if you you feel motivated to broker so that's just in general not specifically how you do that for because We do that a lot And that in this class so when he'd spend a little bit of time talking about how do you download what I've found is 2 years let's say I do a problem on the board And then you can't look at it OK just that none which try the homework but you can't even do the problems that he watched me do Would that be the better place to start So when you're downloading math information which you always want to start with something that's ridiculously simple and injures started adding onto it and builders and a build until you're now doing something that's really complicated but it but you view it as having been built up from something simple so that always the way you wanted a simple way to get this job done is if you have a problem was said do a problem lecturing you copy can you have the complete solution take clean piece of paper and until you can go from the beginning to the end without looking at the solution then you're not ready to move on that's that's a simple as that make it now it and you can't do the problems then ask yourself did you do that interim that's how you learn that I wanna learn how this problem there is there's a solution clean piece of paper but I got stuck woes that I was that's that piece of paper away clean piece of paper start at the beginning and go from beginning to end without looking for solutions to there's your formula for how you learned so now what I'm gonna do on the upload a lot of problems on the board during a write them down your 1st time learning that Material will be when copied banning class but you can expect much dad that just let it happen let it wash over you then you go only China reproduced you might be much more skilled at somebody else then at a meeting in ages read that I do recommend reading them so that the goal he wanted download information as efficiently as possible on the secret and you can do that other subjects to More comfortable talking about man And other subjects OK Now what about economists why they need now but slipped convince ourselves for the colonists really need man What do Before I do that we figure area this week
Brief But But Alright tap So an economist is somebody that is fascinated by The economy generalization but I think it's pretty much true but what is at Minot fascinated with the economy economy it is
Vast subject there's you could be fascinated in the history of the war you could be fascinated by the social implications of Anaconda or you could be fascinated by the monetary policy of different countries Whatever it is there's all these different aspects and what happens is as an economist you look out in the world you find something that you're fascinated with And you start to study it and quickly will you realize is that you can understand it unless you put it in mathematical then you construct understand Solos taken example here but in in 1964 and then in 1960 something happens
Which I think is fascinated so as as an amateur economists I look out into the end into the world and that maybe a still history can like history and I see what happened here is But we haven't silver dollars a year and silver dollar here where Silverbell so here it was still in here was not just 1 180 use woke up in in the coin was minted wasn't silver and theory for was incidents in but in 1865 the Quinta sitting there together in economy and they were equal so he could stay in 1965 we had a situation where the new food was
Equal to the total gives you an opportunity to see what's going on in the economy in the last 40 years
Well vocalist that's fast forward ballots let's check out today if I take those same 2 silver dollars a rapidly people now this 1 is worth have to spend 20 dollars to get so now in the new situation B the old Silver takes 20 of the New so we are an opportunity to do a little matter understand what's gone on the economy let's do division so the new money is equal to 1 in 20 of these old money will orders some more advanced math I can replace that with the desolate version
So then OK what's important what's important is the economist looks into the world see something interesting choice of model in using some mathematics and then there's a really important thing that has happened you have to really make a prediction The duty any good we are you know what happened in the past were living in the presence of getting information about the futures not really useful Tillis check it out so our new money
Today ls back our new money has lost 95 per cent of its value so wary think things there had Webster prediction is eager to go back to this But in a 0 let you make your concludes
But you see that's how you understand what's going on Earth chief says to another war maybe you've heard of the money multiply the money multiplier
So what is the money multiplier it's the amount of money they gets created after you just create 1 multiple so the idea is that Better reserve makes 1 you end up with 10 dollars in the account has that work the look at that so so you create 1 a white the whole economy clean it either way that's what economists do is these simplify things because
The real economy is very complexes all these things going on so if you wanna understand 1 aspect usually which you do Jaeger simplified contrite understand in the simplest case so let's White the whole economy clean let's have 1 a day just heard the very 1st dollars Her Now the rules of the road and play by his distract show reserve making so the bank must keep 10 per cent of deposits That's the rule must keep 10 per cent of the Ford So that means they can land The implications that still and I'm not very happy cheat 10 per cent of deposits will make it to land 90 per cent of take a look at all of us should take a look at all alone So the ferry Ferdinand this is this is this is sort of a side note with the most amazing thing is this original gets feared that gets life that's impressive dollars and they either to the government or to bank and there's interest charged on To percentages at the end of the year someone ozone DOLLAR to use only 1 in the economy Service issue kind or Elisa's take care us take a look at the launch of the very 1st loan 1 that goes on the banks the Federal Reserve and then the debate Castor keep 10 per cent said they must keep 10 per cent of them they have 90 per cent of available the land that rate
Now what happens is you learn that out somebody and it goes into the economy and then it moves around and people buy stuff with that but eventually zones Itanium push in the base rate at some point go back and so was warmup for some of the specter that you might not be all at once but eventually 90 cents a portion of a good portion of it will be back in the banking system so that the next loan will be 90 per cent of that 90 cents that went back in . 9 square so now that's the amount that they just now land . 2 people so now it's full around the country but then it is back in the bank sometime time and they can only guess what 90 per cent of the next loan is . 92
2 . 9 and president back deposited the thing you just don't feel it's still a mafia
Well it's 1 . 9 plus . 9 squared than the next 1 . 9 2 pages keep going this figures
This is a prerequisite to the in calculus learn about geometric series the son of
Are and is equaled 1 world leaders are as long as our listen so that that's this this If I make this out this is all want Our squared feet so that We have a lawyer Arturs point of this is the same as . 9 and watches
The same 1 Answering 9 over 10 And now or in that situation so I can use the formal it's 1 or 1 minus 9 can that People could So we create 1 and possesses circulation and honor that banks create 10 dollars in law
Let's see how he didn't understand the matter the who what ends up happening so so what you learned in Economics is just give for formula they say the money multiplier is 1 over F for that is the fractional reserve rate so
I just gave you that formula at the beginning I would say will they they have to keep 10 per cent of the deposits so that would be wonderful for 110 can so that's easier way to compete at
You know in understanding the math to reveal exactly what's going on here you just memorizing you don't really question it you just go OK yeah money multiplier and get it right on the test who what is wires ceremony multiple That's why we see that's why map is useful for studying That's not the way you really have to be convinced that he'd had enough economics classes you probably know that is useful but she had a depth of understanding rather than memorizing formulas
Bush talk about what the goal so this class Goals for this class organist survey certain math topics that have been traditionally used in economics
There's a list of those states that folly not 1 of the really would be eventually duty stated that the finance class or something like that we'll eventually study that stuff before this glass or going to 1st thing is have surveying up due to different branches of mathematics 1 linear algebra and not just calculus calculus but it's more than that multi variable
All key variable Who will be going to different topics And this is the 1st hour of the courses Through midterm 1 and this is the 2nd half of the course on the The other thing that I think about We'd like to get out of a class like this is a mathematical amateurs will I mean by that I mean that he'd at some point you need to stop being intimidated by notation so we need a practice that is if you if you just You don't have a course like this where we will we take simple settings and work on notation then you get thrown into economics courses and they have all kinds of notation and then you get Bogged down on that he gets stuck thinking about the notation so you need to just get used to it OK so that means notation of abstraction thinking about things abstractly generalizations and may be improved training lectures all mathematical maturity you get a
But Back The
3rd things that like 3 days to get out of this glass so just too little experiment here was practice on your biological computer the see how it works a little so I'd like three-day stalled imagine the contents of your refrigerator door
Show Now watch what happens when when I said that did you What what popped in your mind so somewhere there's a file on your brain when I see refrigerator door a file and maybe you retrieve that file you you can you can see Europe earlier during a scandal maybe the resolution isn't so good that you can sit there and look at everything the different people at different villages on that but still had think probably just the general picture of the feeling of your Georgia showed up in your mind you could just can't go through it with their very few of you if any had a piece of paper pop in your mind with a list of
So what that shows you is this year Mind prefers pictures it stores details in pictures right if you in a retreat something quickly your mind can do it Really quickly it's a picture taken mood around a do amazing things that said recite all the contents of your refrigerator door that would take you while you sit there and say the different things so What I'm revealing to you is that your mind Really relies on the visual really likes the visual because it's instantly so when I say that you have that we want developer tuition what we want have when I put mathematical symbols on the war that when you look at it pictures pop in So then you can move around in your head on the pictures while you're doing this stuff on on the algebra or whatever it is were working out so that the 3rd reduced survey these subjects along the way we're gonna work on notation abstract shouldn't generalization little bit approved fretting about alive and you'll see that big deal but it sounds and then the other thing I think is probably more important is that you have some pictures in your mind
Mathematical concepts For her trip They may have a questions for me but better start with sections 7 . 1 and the book this is solving aimed solving systems of equations the 1st thing we need to do is to remind you that you've already salt systems of equations in your life he did it in an algebra before geometry where you had no to 2 lines and you see where the intersection of 2 lines is this but I ahead of me add a word here because I the focus on vocabulary she is as important Linear pocket so the 1st thing
We've got some vocabulary here and call up all challenges that when when you don't understand something and now it's usually you don't understand that almost always case you really wanna put a little time in understanding what the word means so that when
The word pops up in front of you you get something that is conjured in your mind that you can relate to
OK so 1st of all a linear equation understand that carried comes up all the time you may already be comfortable with that let's make sure everybody's on The linear equation that I have an equal sign us with the word equation and in your mind it should trigger line but life is limited because that's only when you have 2 variables wonder When you start have more variables you still have the concept of linear but is not always circuitry aligning warned I always so linear equation let's Let's say OK we kinetic any number of variables that's not restricted but The variables informally have powers of war said That's why it's the most basic functions in constant functions linear till powers of work and then the coefficients
Can be any real number of any real them if it was hoof if would be a problem
But we
Right that's go through it So it any variables solicitors go through different levels of that I have 1 best examples to For worldwide 3 so those earlier years X equals 2 years equals 3 Losers linear so even if you just have 1 variable see the variable has a power of 1 thing you get a lot better to variables this Hilliard familiar setting here that I have 2 variables to add plus why he want this is specific here this is an example of something specific Patty right down let's write down every 2 variable Leary equation in the universe
We a finite life isn't really meant that many of them get all I came to see the thing that defined this It is just these numbers and from these coefficients so If I wanna write them all down in the universe this is what I mean by generalization I wanna down mall random all down a one-time that's a and B
By sea And I don't specify what a B and C are so that's everything in the universe
But this is gender Show But right down 3 variables well with it starts sealed Repetitive here I have that 3 variables
Women face walleye me and then to make it a specific at put some real numbers as coefficients in front
As a matter of 2 3 negative war have equal something otherwise is not an equation focused on how should we visualize this what happens only have 3 barrels here you could visualize that 1 is
It's got a slope a negative to discover why intercept 1 their there I can visualize what that picture as consumers have the training I had this off or downloaded when you were younger you could look at that with Verizon conjure pictures practices who we do about this what about the 3rd period OK it's still linear because it satisfies is conditionally has powers of wanted this is really a line is where we depart from linear means line where I want you to do it
He think of each variable as being a dimension so I have to variables I have 2 dimensions wife to dimension
Now I have 3 variables now I have X Y and Z and the way to do this in your mind is if I'm gonna added dimension just grab and just poured out from a wall You can do that your mind just think that the fallout from a war so what I get is a plank solar plane is the linear equation And Treatment and that as far as you can go Donor comes to see it in your mind because only 4 variables and it's in 4 dimensions and you can see that
Because we live in 3 Chips Looking for a 2nd foreign go 1 step further this is about that this is the standard that 1 variable 2 variables 3 variables but we we can do more hands The reason I wanna mention moral because this now gets into the notation what happens only start 10 variables say we have 27 very well If we had 26 letters of the alphabet then we run our letters So that would this would be an appropriate way labeling of variables 27 vary so what happens is once you get beyond 3 you usually stick to subscripts I have BUYING variables escape
For just go straight to 5 years Then you will write a B C D ES some like that will really do This is more appropriate for economists is you label the variables was subscripts like good ones that set X of 1 good to his next to the 3 so that it you want to get in your mind when you're doing this summer the economics models seen so for 5 variables Matson like X want minus X 2 . 2 3 For explore last 6 5 per 7 A perfectly good linear equation in 5 variables you don't really have any hope of seeing that
So at some point the visualization stuff breaks down you have to rely on your algebra So we need were going after work on that quite a bit on the altar of 5 variables that wouldn't looks like
This is linear and then how about The 2 of the more of this notation and generalization straight down every linear equation in the universe So I have to I can't say that has any specific number of variables has been very will then how I do that I get ahead X 1 and got next to get But then I could have a seat over here I had when I wanted to write down the general 1 I put in the letters here indicate that is to be any real number but Again on around The letters warning is a B and C were on a designer label the coefficients also within city's so that the coefficient because of this guy is a 1 and then this is 82 down and then had an accident and then it's got something so they could be So this is that this is all that Linear equations Because if I won a million variables than others but a million million the 1 billion that's fine I want to All the stopper and notice if I stop with 2 that looks just like this 1 here next were next to The of what pockets of that Linney letters Hanoi vocabulary linear equations so now we're very interested in solving systems of linear equations so now you know have to wonder what that word system is doing there while I was there system
Take a look at that but that of non-linear steps to compare the amount so rude X plus 1 equals 3 Dad power that not 1 of those is non-linear had been hidden in your is they've got some curve too
Citizens nonlinear got some curve Can you can you picture that your mind how strong is you're computer have updated his it can you look at the symbolism on that page with clear eyes and have some the pop India made such as little testers will work
Do the same with them Non linear and non-linear because the powers not 1 its negative ones not 1 of his nonlinear and then something like X squared plus Y squared posed for a picture that
New picture with This is a circle And then this guy here rude the justices go after died here Why equals ex wives rude X important hidden in economics we can't answer that question was you could see it in your head and you see it in your head you my ability the question will assist
Were what does it look alike were earlier seen that she's in economics
The loss was a law that skirt that A lot diminishing returns rates when your model Thomas looks out of a world that is that's interesting I wanna model what I see when they noticed this diminishing returns in a 1 0 Inc. their mathematical model theory use functions with fractional powers because that's going to be ready in London meeting returns is not a linear function of fractional power that will get more into that later about 1 over X visitors is a fundamental Grabbed her economists Why is this is 1 of the books that have the negative our I can drive lightly because in math that's true graft but in economics you really deal with negative mounted the negative production and stuff like that so negative labor rate would have negative 3 workers The McConnell and economic models a really positive anywhere where seeing that shape in economics class comes up all the time and I don't know what level you there's a at part of the actor look is that comes in utility functions rate compliments and substitutions so this is a this is an indifference curves difference that term So that's why these grass so when you put an Exxon Nevada treats the shape and that model something that they see in the real world so they use function of you not familiar with their when you have study with them you get bogged down in trying to deal with the matter instead of learning the economics non-linear a circle radius to include his coming up later in the class man 1st introduction to them
How So OK now go back to that word system so the system part that they been over there
This just means I have more linear equation multiple will critters Now we put the minister's sacking of beer we spent a lot of time Do you know what a linear equation is now stepped a bunch of So we need to do is we need to go back to when you 1st saw that practice that get comfortable with that OK so here's the possibilities by 12 1 is it had a unique solution you could have no solution or you could have infinite when we start getting into the category of 3 variables 5 variables then Figuring out each 1 of these situations It is increasingly complicated the more variables rehab so we wanted to go back to really familiar situation where we have all these things happen study that had been built But That's the goal of the half of it The variables so the 1st situation is the the most common and this 1 also when you're thinking about these systems of linear equations what should you be thinking about in terms of economic where he seems systems of linear equations what's the setting you should be keeping the back your mind were doing that Supply-demand equilibrium that's a more general way of saying just equilibria when you're finding equilibria then here Often solving systems of linear equations supply equals demand 4 different good supply Costa Manigat your system of equations and you find the 1
A point where the economy is in equilibrium talk at so for too
Variables unique solutions that should be familiar to you Rate for supply and demand that the The supply and demand equilibrium of this a unique solution and that so again when you read unique solution What is that conjure up in your mind does that meaning to that just words to to be really careful that when you're reading math terms that you don't as glaze over You don't just say Oh yes leaks Alicia with any way we users were unique with what's going on what's solution filets this translate that that means you have a 1 point satisfy both that's what the unique solution not telling anything are you now you haven't seen before his China refresh your memory and maybe conjure up new thoughts about That's the most common the 2nd 1 is when you have no solution Wasn't look like 1st what I tell you a linear system is no solution can you conjure up the limit You knew the images parallel that's no solution because there isn't what what's a solution it's a point that satisfies both equations there's no point that satisfies both equations that No solution here We have parallel line and then the 3rd case this is to hear you bring up the trustee case so if you have
If that is the 3rd 1 there is infinitely many solutions I have 1 Glad to variables the matter second-line that's exactly the same elected us this struck as 2 lines on the same of this is infinitely many of these are the same line that is a degenerate cases but on where they don't want to do is see now for each of these
Let's add a variable their 1st opportunity Practice with more variables so for each these pulleys at Add the dimension peso here if I pull that out I have 2 planes intersecting what happened so that point pull saddle wall at that point becomes whole winery of I have 2 planes intersecting then you'll get a line is a solution that infinitely many solution simple cases if I pull this out
This like having to play stacked on each other donors and then here well is good 2 planes sitting next on top of this auto crop
Without some examples see what we need to do here is what we always working on his intuition that up there anymore list intuition now we need to go and work on the algebra where where where the equations they give you these settings the first one plus most shoe minus 1 0 0 then to quickly recover to merge
But there's a system Equations a system of linear equations into variables and that the solution is right there had that 1 point that weeks
What would it look like it that parallel lines will then I lost wife to so what has happened by apparel lines these days
These coefficients is thinkers as it would define the slope and was why but now I put it to nearly the same values but something different like before then their parallel is all that did was change the Weiner said I have 1 that has a slope of negative on passing through Chu and 1 the slope native 1 passing through 4 And then what is it look like if I have the exact same blind man While the simple thing is to ever that's that's has you're right because they're exactly the same really in practice if you had to have the same lines you could identify giving it because it was disguised a little bit so was disguise so at 1st glance you might think you have to different equations there but when you realize that if you just divided this 1 by 2 of the original 1 of this But say the same for the says infinitely many says no solution and this 1 has a unique solution well the title of this was solving systems equations of what I did was 1st used Rueter linear equations that little time on that day we said OK what's a system of linear equations now we spent little time on that just in 2 variables so now we need to start solving them the idea is to find a solution for China Solomon you get those cases as degenerate situations command comes up is often but when get to more variables they start so that's why we dimensions
But how
OK the first one is reminding you win you learned this when you were younger the 1st thing they did was a touchy photograph lines and they give you a couple lines and they say grapples 11 and find that point of intersection a simple example is to remind you could still do this but I also wanted to it to illustrate that it's not that satisfying is not very effective all OK so it by autograph this lying here is the same as x equals to draft that vertical line that's the first one and then this is why equals X plus 1 Weiner said of 1 and the slopes of 1 and I would discover that the point of intersection that's the point 2 3 that's the unique solution not hard to see that if if I added variable here we just pull this out from the wall the United the draw that figure out what the intersection infected infinitely many solutions to its Arkady is an effective way to solve systems when you get to a higher levels so this is nice and everything career high school other recourse but we're really need something more sophisticated job
The If you recall when you learned that when you were younger 1 of the methods it you learned was a substitution that will tell the example of that you're my earlier so here's a system of linear equations
The goal is to find the point x y that satisfies both simultaneously in your mind when you see this you know that an accurate picture you just in your mind you go OK them looking for that point sees that teddy developed in tuition when I see the symbolism and picture like that in my mind
That I'm looking for OK So the substitution method would be to solve for 1 of the variables and then substitute hence the name of that year by sulfur X here I have a quota 3 why financing team and then what we do is we take this version of X and substituted rate there so now I have 1 3rd Exs written 3 why minus 18 minus 2 I was 11 so evoke that I get why minus 6 murders To work negative 11 or why died And the ones I have white equals 5 I can go now back substitute I know Exs wise Gold 5 and X 3 wide team that implies that The 3 times supplies When his team were negative 3 so might unique solution is A negative 3 A simple substitution method was to variables However fortunate I have to deal with these
What if I better at it would get the next method they you may remember learning the elimination of the word elimination corresponds to the process of eliminating a variable remember if you could eliminate a variable than you would have 1 left consult for that's the idea foods have an example of that
The idea areas
This is how you were trained hopefully you look at that XEC you know I could eliminate that Texaco wanted to how would I do that I would multiply this by negative for rate multiplied both sides by negative for then would I as a result I would end up with a negative data X here I'd end up with a bunch of other stuff and modified but ultimately 2nd had equations and indexes will go away and I just have 1 with a wife console from now have you think of it this way when you're
When you're adding this equation this 1 think of it as incorporating this information into the equation for solar just about what I have here a negative tactics Last 12 why 20 and this the equation here X plus 7 y equals 1 So when I'm doing this adding I know it is going to be adding Inez's about arithmetic but intuitively what you're doing if you taking the information from this equation in your incorporating it into the equation the 2nd place that those have got both Lines in in morning Why negative 19 so why negative 1 and that's how we got wide and then what we do is we back substitutes in order a combination of elimination and I take this value here and substituted back in and find out What Exit Why negative 1 and the information they said But said while I was 1 That gives me a networks said our 1 of 8 8 1 so we have ours are solution Bonds 1 water a or 0 works
Talk at is quickly go and look at those 2 degenerate cases see with those look like algebraic
Generous cases The 1st was parallel lines who want below look at these equations and and identified as airlines without having to grab that was 1 of the 2nd case would be fed parallel lines The typical equation by 2 x plus 3 4 and 6 plus 9 why but so I can look at that immediately know there's no solution at their peril talk and I do that on a couple of things when I see that this guy is the same multiple Of the Sixers through his 3 times to a 90th 3 times so when these are the same multiple I know the same slow same slope means saying parallel lines unless they have the exact same line but the fact that they're not a multiple of 3 means that we have Caroline's instead of saying this the 12 years may be exactly the same villages write that down here we have
Multiple 3 and here they're not so look what happens when you have a solid is
The important part is that when you go to solve this I I could multiply this by a negative 3 I get a negative 6 x allowing us go over decided to this song to negative 3 times to have relied
3 For multiplying both equations by native 3 both sides of equation native 3 Internet negative 6 x minus 9 wide and 12 little put that bottom agrees that add so I get 0 equaled limited 1 19 others nonsense that's no solution algebraically when union no solution you should get some kind of nonsense when you're trying to solve to get fired equals 7 or something like that the doesn't make sense that's what should happen
But if he OK the most see what happens with the the same wine throat
We have to do it Closed 3 1 4 6 6 plus 9 12 years the multiples are consistent with multiple about 3 on side And the same of our reviewer so that's going to create the same line I can reproduce the top 1 by multiplying by negative 3 Sabres over here just making a match so I can eliminate variables that gives me negative 6 x minus 9 while I was 12 and the other equation 6 explodes climbed 1 pools 12 now We add those we get something different the elder abuse of 0 . 0 0 0 you compare that to 0 equals made 1 of the distinction when you get something like this is not nonsense what this is is using get find what the variables are which means that there are infinitely many Of half of so every point on this line is a solution
Book filled with natural pointillistic a pause for what we've done I just did all year high school algebra stuff again so hopefully will follow at me when it comes to systems of duty-free Schaefer now orders building on so like I said do you think the way we wanna learn this we wanted download this information is a star was something you really comparable with remind yourself but get thinking about it again and then start adding to that making it more complicated that's that's the the design that's style that I've tried Inc. this class Right Trip He faces sections 70 . 2 is about values Valley more variables and and what happens is there's there's 2 things we consider there's the number of equations and the number of variables so when you didn't high school you just stuck to to the simple situation where you had to variables into equations Mexico
So even with this would build it up slowly size let's go start with a standard example Just a kind of will have 3 variables and 3 but as I write that imagining your mind we could have all the different combinations we could out 5 variables and 3 equations erected at 3 variables into the equations er 5 equations into variables
All different combinations like you have to have the same number will start off OK so here's a example X plus Y 0 x minus why is
2 Negative X plus what might choose Tory start this is This is the problem is that when you are in 2 variables in June equations you'd see it before and you'd probably even have a method in your mind he liked best maybe substitution of elimination theory that's more going on here is that 3 equations that 3 variables so what's the matter who had eased start
Where would you do If I ask everybody my Okamoto was something different so unfortunately the answer to your question why where you start is a testament to why I look at this satellite experience so I can know what to do now all I can do is try to impart haven't experienced you go as best they can but no matter what it's hard and even if you can handle this The a group of 4 variables 5 variables where we can keep making it more more complicated so let's just Still leaking due here remembered the methods that we learned were substitution elimination we could try graphing it but that his case he argued here OK so the 1st images of his I can notice that
If I were to add these 2 equations I would eliminate 2 variables you notice that that's why I say it's largely experience I look at that looking for that kind of thing I could see that those who could be be eliminated by just that these 2 wage so if I had been EDS There were I get I get to x equals to which says that equals 1 of us could carry out 1 of the variables and then I could take these 2 equations and add them noticing that look Exs or go away and wise will go away An order said Aziz endemic and find out Woods years so Bad that these equations that Negative Z equals negative Was 3 so that doing the elimination method worked pretty well I got to
The variable still need to know why it is so now I could pick Any 1 of these equations substitute Aziz that I know and find out what what wise 1 here taxes to 1 plus Y minus Z which is 3 equals 0 so why to
Yeah I'm doing that 1 here During the these 3 putting those numbers in there now I know why so now my solution by unique selection It is a four-day here x equals 1 wife to
Should be feeling a little uneasy at this point because Tokyo fine to which it did I saw those steps in that you want we ask yourself how what I know to do that could I have known to do that and what would arises when you start thinking that way ways they could you make it harder What I couldn't do that had been doing this I was pretty convenient the weather variables just eliminated Reddaway as pretty convenient maybe I rigged it this way So we think it is a good 1 of It's not read that way toward need A method for that cannot be satisfied with just every time we get a system of equations just trying to invent the system and the method right there looking added we need a method Will eventually get Right now we just need to practice Do So let's go to is degenerate cases that's a unique solution but got a 1 where there's no solution but we have More equation so again more vocabulary and over determined system over determined what when you have a sister Leary equations it's the equations that determine the system I have a pile of equations sitting there that's what determines my system over determined that means at too many equations of that
Celebrate over like said before when you try and understand math vocabulary pattern say this before but usually when you understand math concept is because you are understandable kept so when You focus on the vocabulary Matt think about the that
The mathematicians who discovered this stuff they pulled words that party had definition From the English-language they made upwards of myriad call this you know Super Fresh Calif rudderless sticks system Leak words that word that they had me so when they attached the or is there something in there for you learn and usually contributes to your intuition focus over determined that means they have to go a man I have more equations and varied more equations and and there's an implication that what was that meet so that start with an example we have 3 equations so I have more equally Asians and variables are at 3 equations and either to variables are 1 of the 2 variables aware that look like half does is get picture now mind paint what would be
Used to variables linear systems then if I have to weekly Asians that This picture we very had that so there's the solution are ready if I had 2 lines is already this potential unique solution so what are the chances that about a 3rd line passes through that not very good pages draw a random line is is not gonna pass through that point you have to be pretty precise and make it happen so I just have 3 equations into variables it's possible I could have a solution but the most likely scenario is that I have no solution vessel Osika look at that would see how that plays out algebraic X plus white for and negative X plus 3 while was
And have 2 X minus 6 1 5 days year System is over determined system I get too many Equations operative fine Nice solution of what we did last time we just try to eliminate burials I was convenient if I had those 2 equations get rid of X gives me for for why he tells me while stew and it flies to go back and plug-in find out what exodus employers equals because the attitude period X equals to look at called the system may even pay attention to the 30 quid so that this phenomenon here if you have 2 lines chances are they intersected a point and if the 3rd 1 is gonna be part of the solution has to pass through there so now I can just check to who doesn't The sort of the plug 2 years I don't get a solution can I ran at its base 3 Here I had to do to sick too Equals I've been at minus 5 day-to-day no solution even I got a solution for the 1st 2 It's gonna be a solution whole system and so our conclusion years that there is no now it's not always the case that over determined systems have no solution but you are For our practice practice now if I give you go over there system chances are the solution is if I wanted to be something that as a solution I will give you know over OK so that's 1 of the degenerate cases In a different setting now we need to do a similar thing
Created in the opposite If we have a word called over determined And that means we have more equations and variables then we should have a word under deter without English-language were use the word over as the prefix chances are there's a word with under the prefix that the office So that's the next 1 was a little bit more of under determined that when there is less that's kind of like that parallel the lines there this year the quiz the same line just that come up that much is even tho you should know what it was under determines the stone does come up by often with his arm my exams right this damage so an undetermined system should be the opposite of what's get that straight vocabulary under terms that means I have fewer equations very votes out the simplest example of this is 1 year familiar with except you just never talked about it and the familiar example
It is but 1 Eco Asian and 2 variables So if I have 1 equation into variables was that look like that's negative UXO plus why cause what 1 equation 2 variables to what we do with this scrap 1st of all the driver to white equals 2 X plus 1 if you like there's the grass to less trained to do As if I give you this equation you're able to grab that what I've saying is that every point here is a solution to this system to assist 1 equation for what I wanna do is a practice writing these solutions so are infinitely many solutions to this system filling every point on this line is a solution may ask you can you see this affair right down here as part of the board the rooms Were you get to every point there's a solution so how we can organize a solution this is This is good enough but I wanted to use a new way of organizing it because only get 2 more complicated said things were gonna need this other way to muzzle practice here doesn't hurt OK so how I 1 organize a solution already down such a way that it really shows the infant infinitely
Any solution so that too was really use what's called a parameter to writers if this a new class for the film is here Clashes removed Ahead from her up
Tho No less intimidating because if you're not really comfortable with math who when I say was just go ahead and presenter solution user parameter to go ahead We're not talking about that that's what I mean by the vocabulary you don't know what this means is that you know what to do great start from the beginning focus are the parameter Kewaunee analyze the language year Parra paralegal paramedic parallel its It's a parallel measure and that doesn't really help you that anyway 281 must study the language there that's what we're doing so so what I wanna do is only use 1 letter to organize the text worldwide so here's what I do organist said X of the And then I know why isn't terms a teacher because wise to plus 1 of then why is equal to 2 So my solution can be written like this is what I mean by presented the solution using a parameter will write solution like this X and Y equal to 1 and now the solutions are any team real numbers so you give me help plug it in there now find a solution to that 1 line that once that system 1 linear equation
So that's writing things in terms of parameters which enemy Now for the next labeled earlier more complicated but OK perp
Now let's have 3 variables But only to clean that falls under the under determined category because that is at fury equations and variables X plus Y V 2 negative at closer to But see what 3 Focus of Dollar still work on intuition here it I didn't have Z year images textbooks you just 2 lines and be looking for the point that the lines intersect so soon as I add a V you wanna pull that picture out from Walpole EX out from the wall of what that gives you his got 2 planes intersecting and they intersect outline That's our goal was defined that line will refine all infinitely many solutions were those 2 planes inertia so here's the picture here this is injure rather than having the intersection of 2 lines which is what you're familiar with now at the intersection of 2 planes and we can draw that Chavez Drive so the 2 planes intersect had intersect intersect he line that's a lot of fun to find the equation will notified who lower than a woman do is rural wanna raided this way because this is a complicated
It's a plane it's a line that goes in 3 dimensions so 1 I have is that all set up so that you tell me a value T I can give you point out that the purpose of learning to do that privatization focus on how we do this well I like that elimination method that that's eliminating variables tends that help us make progress and we have elimination we have substitution to Phyllis try something here on let's just take this in sulfur Z Excuse me Z to minus X minus why so the very least they could reduce it 2 variables disease equal to minus sex murders why now want to incorporate that information into the 2nd equation Inc. and now I have a negative X from Y plus 2 minus X minus while I 3 limited support by that
I've got a negative vaccinated X native to let's get to Y and negative lies so I have plus Y I and and I have a 2 years to bring them over to the other said that's what If you believe that You believe me Look and then this is not how we like things usually like white somewhere else 1 thoroughly assess familiar with his head like equals to exports
So let's see Now do this parameterization which we just practice on the simple case now let's do this and a real-life example with 3 years OK so what I wanna do as I want presented the solution what I've got years a relationship between XOY wouldn't tell you about seat you see this is an incomplete solution disease equal and wires equally so we're running numbers renewed fears were really need that parameterization again but that set
X He said that so it a starting point take 1 variable said equality and now the other 1 should be in terms of case so if I said Texaco to just like in the previous example have why called to 90 plus 1 of those 2 were taken care of and then Z is to minus X minus So Societe is evil 2 actress So the 2 and then why is this 2 close 1 so now seem words equal to To minus 1 1 the net negative teammate itching to me that
So to right down there Complete solution if a a But the half So the could so intuitively I can understand that there is
Infinitely many pay but Harry ready down we got there is will organize it had taxi why she plus 1 1 3 and this is for TV serial number so of you give me T equals 1 then won the 3 negative too is a solution that we have exit polls on July 3 equals 0 1 ministry is negative to this is a solution but we We will get 1 of these solutions for any Wasn't so you give me a value to you are give you a solution and or find that solution by this relationship literature need to learn is a great example for you to do I was telling him The following new to you take that example take a clean piece of paper at home and reproduce
Although they hadn't without looking at stuck due to the solution a that's been crumpled Newspaper star over to you go from the beginning to the end without looking at the solution you are not ready to move on that followed the hardest thing we've done today is We built that A lot of energy into doing some old stuff so that we could build a this this example pockets Soleidae here I am will pick up here on Wednesday
I haven't quite finish section 7 . 2 assault pick up there That will move on to a chapter 8 use the rest of the time I CA here yet is what so they're UTA and then add 100 years any questions you come and ask him