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Systems of two equations (and more)PowerPoint Presentation

Systems of two equations (and more)

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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:
- 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

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”

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”

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:
- In particular the 2nd equation is trivial!!

- What about a more complicated system?

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

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

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:

Solving a simple system of equations

- Step 3 : replace in the 1st equation
- This gives us again a single equation with unknown x

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

- 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:

Supply and Demand

- The system is completed by a 3rd trivial equation: the market equilibrium equation
- The full system is:

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