Chapter 4.1 Linear Systems
1. State the definition of a system (in the context of
equations and inequalities) in your own words.
2. State the definition of a Solution set of a system
(in the context of equations and inequalities) in your
own words.
3. Write an example of a system of equations.
4. Write an example of a system of inequalities.
Chapter 4.2 Solutions of Linear Systems
1. What does it mean to say 2 systems of equations
are equivalent?
2. Give an example of 2 equivalent systems.
Example 1: Solve the system 2x+3y=8
5x4y=10
Example 2: Solve the system and graph 2x3y=10 2x-3y=14
Example 2a: Solve the system of equations 3x7y=5 9x-21y=15
3. What does inconsistent equations (or inconsistent
system) mean? Give an example.
4. What does dependent equations (or dependent
system) mean? Give an example.
5. What does independent equations (or independent
system) mean? Give an example
Chapter 4.3 Skip
4.7 Augmented Matrices
1. Solve the following system using the elimination
method.
3x+4y=8
5x-3y=23
2. Write the augmented matrix for the system in
question 1.
3. Make a list of the operations that are "legal" for
an augmented matrix?
4. Transform the matrix you wrote in question 3 to a
diagonal matrix.
5. What does it mean to diagonalize a matrix?
6. What would happen in this process if your system
was inconsistent? Dependent?
Chapter 4.4 f(x) Terminology and Systems as
Models
3x+10 y=-24
6x+7y=-9
1. What does the notation f ( x) mean? g(x)?
2. What is the common confusion with this notation?
Example 1:
Given f(x)= 2x-3 and g(x) = x2+4x Find
f(3), f(-4), g(2), and f(2)/g(2)
Example 2:
Suppose your family is going to buy a new
refrigerator. One brand costs $2,000 to buy and
$30 a month to operate. Another brand costs $3,000
to buy and $25 a month to operate. Let f(x) be the
cost for the cheaper unit for x months, and let g(x) be
the cost for x months for the more expensive unit.
Write the equations for f(x) and g(x)
Plot the graph of the system
Solve the system.
Chapter 4.5 Linear Equations with 3 or more
variables
1. Write an example of a system of equations with 3
equations and 3 unknowns (variables)
2. What is an n-tuple?
3. What is the standard form for the solution to a
system of 3 equations and 3 unknowns?
4. What does the graph of a 3 variable linear
equation look like?
5. What is an xy trace? yz trace? xz trace? For the
equation 3x+y+2z=6, write the equations of the 3
traces.
6. How do you write the x intercept of a 3 variable
linear equation? What does it look like on a graph? (
y intercept?, z intercept?)
Example:
For the following equation, graph the three traces.
3x+2y+z=12
(If possible, use a 3D graph program (or TI 89) to
graph the equation in 3D.)
4.6 Systems of Equations with 3(or more)
Variables
1. What does the solution to a system of three
equations with three variables look like on a graph?
Example:
Solve the following system using the elinination
method shown in the text.
3x+4y+2z=6
x+3y-5z+-7
5x+7y-3z=3
(Find a 3D grapher, and graph the system also.)
4.9 skip
4.8 Higher Order Augmented Matrices
1. Write an augmented matrix for the following
system.
3x-2y+4z=11
4x+3y-2z=-5
5x-7y-3z=1
2. What is the matrix equivalent of eliminating a
variable?
3. What are the "legal"operations for transforming
the matrix in #1 above? Give an example of an
illegal operation.
5. What is a row operation?
6. What is a good method for keeping track
(showing work) of the operations you have
performed on a matrix?
7. Diagonalize the matrix in question 1 and write the
solution set.
4.10 Systems of Linear Inequalities
1. Graph the equation y=3x+4
2. What role does the line you graphed play in the
graph of y<3x+4? Or y>3x+4?
3. What two forms can the line take, and when do
you use each form?
4. What does the shading indicate on an inequality
graph?
5. How do you decide which side of the boundary
line to shade?
6. Graph the following 2 inequalities on the same set
of axes.
3x-4y>9
1
y ≤ x−4
2
4.11 Linear Programming
1. Write the inequality sign for each of the phrases:
at most, at least, a maximum of, a minimum of, no
more than, no less than, greater than, less than, more
than, fewer than, bigger than, smaller than.
Example: You are in the business of manufacturing
MP3 players. You make two models of MP3 player,
model B and model G. Your factory machinery has
a capacity of 75 B per week or 48 G per week or a
mix of both models with a maximum total of 96
MP3 players per week. You have a maximum or
180 labor hours available per week in the assembly
department. Model B takes 1 labor hour each and
model G takes 3 labor hours each in the assembly
department. In the testing and quality control
department you have only one worker who works 40
hours per week. Testing model B takes .5 hours and
testing model G takes .25 hours. If model G sells
for $200 and model B sells for $300, how many of
each model should you make to earn the most
revenue?
a. Write the constraint equations for this problem
and graph them on the same axes (Put B on the x
axis and G on the y axis).
b. Describe the feasible region ( area of the graph
where all of the inequalities are satisfied) for this set
of inequalities.
c. Write a revenue function for your company.
Assume you want a profit of $5,000. (For graphing
purposes ignore temporarily the fact that you cannot
sell pieces of MP3 players)
d. What is the slope of this revenue function?
e. Assume you want a profit of $6,000. Write the
profit function and find the slope. What do you
notice about the two slopes you found? If you used
another dollar amount for the profit, what would the
slope of the profit function be?
f. What do you know about lines that have the same
slope?
g. Draw several different profit lines on the
feasible region. What is the greatest profit you can
make and still operate at a point in the feasible
region?