Linear Models and Systems of Linear Equations

1 Chapter 1 Linear Models andSystems of LinearEquationsContents1 Linear Models and Systems of Linear mathematical Models .. Functions .. mathematical Modeling .. Cost, Revenue, and Profits .. Supply and Demand .. Straight-Line Depreciation.. Systems of Linear Equations .. Two Linear Equations in Two Unknowns .. Decision Analysis .. Supply and Demand Equilibrium .. Enrichment: Decision Analysis Complications . mathematical ModelsAugustin Cournot, 1801-1877 The first significant work dealing with the application ofmathematics to economics was Cournot sResearches intothe mathematical Principles of the Theory of Wealth, pub-lished in 1836. It was Cournot who originated the supplyand demand curves that are discussed in this section.

2 IrvingFisher, a prominent economics professor at Yale Universityand one of the first exponents of mathematical economics inthe United States, wrote that Cournot s book seemed a fail-ure when first published. It was far in advance of the methods were too strange, its reasoning too intricate forthe crude and confident notions of political economy thencurrent. Application: Cost, Revenue, and Profit ModelsA firm has weekly fixed costs of $80,000 associated with themanufacture of dresses that cost $25 per dress to firm sells all the dresses it produces at $75 per the cost, revenue, and profit Equations ifxis the numberof dresses produced per week. See Example 3 for the will first review some basic material on functions. An intro-duction to the mathematical theory of the business firm with somenecessary economics background is provided.

3 We study mathemat-ical business Models of cost, revenue, profit, and depreciation, andmathematical economic Models of demand and supply. We will onlyconsiderlinearrelationships, so you may wish to review materiallocated in the Algebra Review chapter on straight FunctionsMathematical modeling is an attempt to describe some part ofthereal world in mathematical terms. Our Models will befunctionsthat show the relationship between two or more variables. Thesevariables will represent quantities that we wish to understand or de-scribe. Examples include the price of gasoline, the cost of producingcereal or the number of video games sold. The idea of representingthese quantities as variables in a function is central to ourgoal ofcreating Models to describe their behavior.

4 We will begin byreview-ing the concept of functions. In short, we call any rule that assignsor corresponds to each element in one set precisely one element inanother set mathematical Modeling1-3 For example, suppose you are going a steady speed of 40 miles perhour in a car. In one hour you will travel 40 miles; in two hoursyouwill travel 80 miles; and so on. The distance you travel depends on(corresponds to) the time. Indeed, the equation relating the variablesdistance (d), velocity (v), and time (t), isd=v t. In our example,we have a constant velocity ofv= 40, sod= 40 t. We can viewthis as a correspondence or rule: Given the timetin hours, the rulegives a distancedin miles according tod= 40 t. Thus, givent= 3,d= 40 3 = 120. Notice carefully how this rule isunambiguous.

5 Thatis, given any timet, the rule specifies one and only one rule is therefore a function; the correspondence is between timeand the letterfis used to denote a function. Thus, using theprevious example, we can writed=f(t) = 40 t. The symbolf(t)is read f of t. One can think of the variabletas the input andthe value of the variabled=f(t) as the output. For example, aninput oft= 4 results in an output ofd=f(4) = 40 4 = 160 following gives a general definition of a of a FunctionAfunctionffromDtoRis a rule that assigns to eachelementxinDone and only one elementy=f(x) inR. SeeFigure caption is here, if neededThe setDin the definition is called thedomainoff. We mightthink of the domain as the set of inputs. We then can think of thevaluesf(x) as outputs.

6 The set of outputs,Ris called helpful way to think of a function is shown in Fig-ure Here the functionfaccepts the inputxfrom the conveyorbelt, operates onx, and outputs (assigns) the new valuef(x).Figure letter representing elements in the domain is called thein-dependent variable, and the letter representing the elements inthe range is called thedependent variable. Thus, ify=f(x),xis the independent variable, andyis the dependent variable, sincethe value ofydependsonx. In the equationd= 40t, we can writed=f(t) = 40twithtas the independent variable. The dependentvariable isd, since the distancedependson the spent are free to set the independent variabletequal to any numberof values in the domain. The domain for this function ist 0 sinceonly nonnegative time is that the domain in an application problem will always bethose values that are allowed for the independent variable in theparticular application.

7 This often means that we are restricted tonon-negative values or perhaps we will be limited to the case of wholenumbers only, as in the next mathematical Modeling1-4 Example 1 Steak SpecialsA restaurant serves a steak special for $12. Writea function that Models the amount of revenue made from selling thesespecials. How much revenue will 10 steak specials earn?Solution:We first need to decide if the independent variable is the price ofthe steak specials, the number of specials sold, or the amount ofrevenue earned. Since the price is fixed at $12 per special andrevenuedepends on the number of specials sold, we choose the independentvariable,x, to be the number of specials sold and the dependentvariable,R=f(x) to be the amount of revenue.

8 Our rule will beR=f(x) = 12xwherexis the number of steak specials sold andRis the revenue from selling these specials in dollars. Note thatxmustbe a whole number, so the domain isx= 0,1,2,3, .. To determinethe revenue made on selling 10 steak specials, plugx= 10 into themodel:R=f(10) = 12(10) = 120So the revenue is $120. T Technology may wish to see Technology Note 1 forthe solution to the question using the graphing (see Appendix A) that lines satisfy the equationy=mx+b. Actually, we can view this as afunction. We can sety=f(x) =mx+b. Given any numberx,f(x) is obtained by multiplyingxbymand addingb. More specifically, we call the functiony=f(x) =mx+balinear of Linear FunctionAlinear functionfis any function of the formy=f(x) =mx+bwheremandbare 2 Linear FunctionsWhich of the following functions are Linear ?

9 + 2x= 1/x+ : is a Linear function. The slope ism= and they-interceptisb= this function first as,5y 2x= 105y= 2x+ 10y= (2/5)x+ 2 Now we see it is a Linear function withm= 2/5 andb= mathematical is not a Linear function. Rewrite 1/xasx 1and this showsthat we do not have a termmxand so this is not a Linear raised to the second power and so this is not a linearfunction. mathematical ModelingWhen we use mathematical modeling we are attempting to describesome part of the real world in mathematical terms, just as we havedone for the distance traveled and the revenue from selling are three steps in mathematical modeling: formulation, math-ematical manipulation, and , on the basis of observations, we must state a questionor for-mulate a hypothesis.

10 If the question or hypothesis is too vague,we need to make it precise. If it is too ambitious, we need to re-strict it or subdivide it into manageable parts. Second, we need toidentify important factors. We must decide which quantities andrelationships are important to answer the question and which canbe ignored. We then need to formulate example, each important quantity should be representedby avariable. Each relationship should be represented by an equation,inequality, or other mathematical construct. If we obtain afunction,say,y=f(x), we must carefully identify the input variablexandthe output variableyand the units for each. We should also indicatethe interval of values of the input variable for which the model ManipulationAfter the mathematical formulation, we then need to do some math-ematical manipulation to obtain the answer to our original might need to do a calculation, solve an equation, or proveatheorem.