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Math 115
Finite Mathematics
Topic 2: Systems of Linear Equations — Session 1
Outline
I. Systems of Equations in Two Unknowns
A. Simultaneous Solution
B. Geometric Interpretation
1. Intersection of Lines
2. What can the intersection of two lines be?
3. One point
a. unique solution
b. The empty set
i.
parallel lines
ii.
inconsistent system
c. A line
i.
equations are equivalent
C. Solving
1. Substitution
2. Elimination
D. Examples: (pp.56 - 59) #2, 18, 30
II. Applications
A. Supply and Demand
1. Equilibrium
B. Examples: #36, 42, 48, 50
III.
Systems of Linear Equations with Three or More Unknowns
A. Geometric Interpretation
1. Graph is a plane in 3-dimensional space
2. What can the intersection of three planes be?
a. A single point
i.
b. A line
unique solution
Math 115
Finite Mathematics
i.
infinite solutions
c. A plane
i.
planes are identical
d. The empty set
i.
inconsistent system
ii.
planes are parallel
B. Solving
1. Substitution gets harder to manage
2. Elimination is really no harder
a. Reduce problem to simpler case
3. Example
x  2z  1
2x  y  z  1
3x  2y  7
Math 115
Finite Mathematics
Topic 2: Systems of Linear Equations — Session 2
Outline
I.
Matrices — Systematizing the Elimination Method
A. What is a matrix
B. Coefficient and Augmented Matrices of a System of Equations
1. Example
x  2z  1
2x  y  z  1
3x  2y  7
2. Elements Designated by Row and Column
3. Diagonal Elements
4. Applying the Elimination Method with the Augmented Matrix
a. Row Operations
i.
Interchange any two rows
ii.
Multiply any row by a constant
iii.
Replace any row by a linear combination of rows
b. Pivoting
5. Diagonal Form
6. Gauss-Jordan Elimination
7. Examples: #6,14,26,36,38,54
II.
Gauss–Jordan Elimination and Non-Square Systems
A. Gauss–Jordan Elimination Worksheet
B. Reduced Echelon Form
1. Leftmost non–zero entry (lead entry) of each row is a 1
2. Lead entry of any row is to the right of the lead entry the previous row
Math 115
Finite Mathematics
3. If a column contains the lead entry of some row then the entries in that column
of the other rows are all 0.
III.
Quiz #2
IV.
Reduced Echelon Form and Parametric Solutions
A. Systems with More Equations than Variables
B. Systems with More Variables than Equations
1. Parametric Solutions
C. Worksheet
D. Examples: #1 - 8, 16, 22, 24, 48, 64, 78
Math 115
Finite Mathematics
Topic 2: Systems of Linear Equations — Session 3
Outline
I.
Matrix Arithmetic (Sections 2.4, 2.5)
A. Dimensions
1. Number of rows X Number of columns
B. Equality
1. Elements in corresponding positions are equal
C. Matrix Addition
D. Subtraction
E. Zero matrix
F. Scalar Multiplication
G. Dot Product
H. Matrix Multiplication
1. Identity Matrix
I. Examples: (Sec. 2.4) #2, 6, 8, 10, 16, 38, 48
(Sec. 2.5) #2, 8, 14, 16, 20, 34, 58, 62
II.
The Inverse of a Matrix (Section 2.6)
A. Definition
B. Multiplying by the inverse of a matrix is quivalent to “dividing” by that matrix.
C. Examples:
1 2
1
and


0 3

0


3 
2
1
3
1 0 2 
1
3 2 4  and 3 2


0 0 1 

0
0 2
1
1 
2
0 1 

D. Using the inverse of the coefficient matrix to solve a system.
1. Matrix Equations
2. Example (p. 119) #42
E. Finding the inverse of a matrix
1. Equivalent to solving several systems of equations
Math 115
Finite Mathematics
2. Simple example
3. Examples: (Section 2.6) #2, 14, 32, 40, 46
Math 115
Finite Mathematics
Topic 2: Systems of Linear Equations — Session 4
Outline
I.
Economic Systems
A. The Leontief Closed Model
1. What is produced is consumed internally.
2. Example:
3. As a matrix A
Produced by (input)
F
C
T
.4 .2 .25
% Consumed by (output) C .3 .4 .3 
T 
.3 .4 .45

F
4. If X is the matrix giving the amount produced by each industry then a closed
economic system can be modeled by the equation: AX  X
B. The Leontief Open(Production) Model
1. Not all of what is produced in consumed within the system
2. Modify example
3. If D is the matrix giving the consumer demand for each industry then an open
economic system can be modeled by the equation: AX  X  D
II.
Examples: (Section 2.7) #2, 8, 18
III.
Review for Exam 1