Transcript Week 7
Systems of two equations (and more) Solving systems of several equations Supply and demand Systems of two equations Today, we use algebra to solve several equations with as many unknown variables Although in theory this can be used to solve an arbitrarily large system, we’ll limit ourselves to 2-3 equations/unknowns This is basically just an extension on what we saw last week: The aim is to modify the system of equations into a series of single – variable equations that we know we can solve Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand Equations with several unknowns Last week we saw the notation used for unknowns inside and equation: 2 x 6 We also introduced the idea that several components of the equation could be unknown, including parameters 2 x a With such an equation you can’t find a solution for “x”: you need more information Equations with several unknowns This extra information is provided by a second equation, which helps to specify “a” a 23 Replacing in the first equation, one can now solve for “x” 2 x a As a result, you have the value of both “x” and “a” Equations with several unknowns There are a few elements of notation to consider: There is no distinction between unknown variables, parameters, etc: all are “unknowns” Unknowns all have the same notation, typically “x,y,z” in mathematics (not necessarily so in economics) The system of equations is indicated by an “accolade” 2 x y y 23 Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand Solving a simple system of equations The system considered in the previous section is rather simple: 2 x y y 23 In particular the 2nd equation is trivial!! What about a more complicated system? x 10 4 y 5 2 x 3 11y Solving a simple system of equations x 10 4 y 5 2 x 3 11y This system can be solved by isolating an unknown in one equation, then substituting it in the other equation You then have a single equation with a single unknown This method (the substitution method) is the simplest, and it works best for small systems (2-3 equations) For larger system, other (faster) methods are used Solving a simple system of equations x 10 4 y 5 2 x 3 11y Step 1: isolate one of the variables. Lets isolate “x” in the 1st equation x 4 y 15 2 x 3 11y Step 2: replace in the other equation x 4 y 15 2 4 y 15 3 11 y Solving a simple system of equations x 4 y 15 2 4 y 15 3 11 y We now have a single equation (the 2nd) with a single unknown (y) Lets rearrange and solve the 2nd equation for y: x 4 y 15 8 y 30 3 11y x 4 y 15 33 3 y Solving a simple system of equations x 4 y 15 y 11 Step 3 : replace in the 1st equation This gives us again a single equation with unknown x x 4 11 15 y 11 x 59 y 11 Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand Supply and Demand Supply and demand on a market provide a good example of how systems of equations can be used in economics On a market (say the market for computers) economists want to know 2 variables: The quantity of computers available (Q) The price of a computer (P) Supply and demand provide the 2 equations required to solve the system Supply and Demand Supply : There is a positive relation between the quantity supplied and the price: The higher the price, the more computer manufacturers will want to sell Q s 400 P Demand: There is a negative relation between the quantity demanded and the price The higher the price, the fewer computers people will be willing to buy: Qd 1600 2 P Supply and Demand The system is completed by a 3rd trivial equation: the market equilibrium equation Qs Qd The full system is: Q s 400 P d Q 1600 2 P Q s Q d