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Lesson 1: Systems of Linear Equations (slides)
1. Lesson 1
Systems of Linear Equations
Math 20
September 18, 2007
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Problem Set 1 is on the course web site. Due Wednesday,
September 26.
Office Hours: Monday 1–2, Tuesday 3–4, Wednesday 1–3 in
SC 323
3. Economies are complicated
things, because of their
interconnectedness. Wassily
Leontief (Professor of
Economics at Harvard) used
linear algebra to analyze
economies, and was able to
make correct predictions
about the growth of
economies. He was eventually
awarded the “Nobel Prize” in
Economics.
We will study very simplified
economic models by hand.
Real analysis would require a
computer.
4. Example
Consider an open economy with two very basic industries: goods
and services.
To produce $1 of their products,
The goods industry must spend $0.40 on goods and $0.20 on
services
The services industry must spend $0.30 on goods and $0.30
on services
Assume also that during a period of one week, the economy has an
final demand of $75,000 in goods and $50,000 in services.
5. Example
Consider an open economy with two very basic industries: goods
and services.
To produce $1 of their products,
The goods industry must spend $0.40 on goods and $0.20 on
services
The services industry must spend $0.30 on goods and $0.30
on services
Assume also that during a period of one week, the economy has an
final demand of $75,000 in goods and $50,000 in services.
Question
How much should each sector produce to meet both internal and
external demand?
6. Formulating the equations
Let x1 be the dollar value of goods produced and x2 the dollar
value of services produced.
7. Formulating the equations
Let x1 be the dollar value of goods produced and x2 the dollar
value of services produced.
The total dollar value of goods consumed is
0.4x1 + 0.3x2 + 75000
8. Formulating the equations
Let x1 be the dollar value of goods produced and x2 the dollar
value of services produced.
The total dollar value of goods consumed is
0.4x1 + 0.3x2 + 75000
The total dollar value of services consumed is
0.2x1 + 0.3x2 + 50000
9. Formulating the equations
Let x1 be the dollar value of goods produced and x2 the dollar
value of services produced.
The total dollar value of goods consumed is
0.4x1 + 0.3x2 + 75000
The total dollar value of services consumed is
0.2x1 + 0.3x2 + 50000
If we assume that production equals consumption, then we get
equations
x1 = 0.4x1 + 0.3x2 + 75000
x2 = 0.2x1 + 0.3x2 + 50000
12. Solving 2 × 2 systems
Solve the system of linear equations
2x + 3y = 7
5x + 4y = 14
13. Methods
There are two ways to do this:
substitution Solve one equation for one variable in terms of
the other and put it in the other equation
elimination use a combination of the two equations to get one
variable alone.
16. Consistency
Example
Solve
2x + 3y = 7
4x + 6y = 15
Solution
Impossible!
Definition
A system of linear equations is called consistent if it has a
solution, otherwise inconsistent.
