Systems of two equations (and more)

1. Systems of two equations (and more) Solving systems of several equations Supply and demand

2. 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

3. Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand

4. Equations with several unknowns • Last week we saw the notation used for unknowns inside and equation: • We also introduced the idea that several components of the equation could be unknown, including parameters • With such an equation you can’t find a solution for “x”: you need more information

5. Equations with several unknowns • This extra information is provided by a second equation, which helps to specify “a” • Replacing in the first equation, one can now solve for “x” • As a result, you have the value of both “x” and “a”

6. 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”

7. Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand

8. Solving a simple system of equations • The system considered in the previous section is rather simple: • In particular the 2nd equation is trivial!! • What about a more complicated system?

9. Solving a simple system of equations • 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

10. Solving a simple system of equations • Step 1: isolate one of the variables. • Lets isolate “x” in the 1st equation • Step 2: replace in the other equation

11. Solving a simple system of equations • We now have a single equation (the 2nd) with a single unknown (y) • Lets rearrange and solve the 2nd equation for y:

12. Solving a simple system of equations • Step 3 : replace in the 1st equation • This gives us again a single equation with unknown x

13. Systems of two equations Notation: Equations with several unknowns Solving a simple system of equations A practical example: supply and demand

14. 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

15. 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 • 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:

16. Supply and Demand • The system is completed by a 3rd trivial equation: the market equilibrium equation • The full system is: