Download Algebra II - Chapter 2 Day #1 - Somerset Independent Schools

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Algebra II - Chapter 2 Day #1
Topic: Solving Systems of Equations Using Tables and Graphs
Standards/Goals:
../ A.Le.: I can solve systems of two linear equations using various methods, including
../
elimination, substitution, and graphing .
A.CED.2:
o
../
I can create equations in two or more variables to represent relationships between
quantities.
o I can graph equations on the coordinate plane with labels and scales.
A.REl.ll: I can use technology to graph equations.
Concepts:
1.
Systems of Equations - Is a set of two or more equations.
two or more related unknowns.
These can be used when you have
The goal of solving a system of equations is to find a set of
values that replace the variables in the equations and are able to make each equation true.
2. linear system - consists of linear equations.
3. Solution of a system - is a set of values for the variables that makes all the equations true. You
can solve a system of equations graphically or by using tables.
Examples:
#1. Find the solution to the following
system of equations.
Make use of technology to assist you in
#2. Find the solution to the following system of equations.
finding the solution.
Make use of technology to assist you in
finding the solution.
-3x + 2y = 8
x + 2y =-8
=
2x + y 4
x- Y = 2
#3. Word Problem: You bought a total of 6 pens and pencils for $4. If each pen costs $1 and each
pencil costs $0.50, how many pens and pencils did you buy?
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#4. STEM (Marine Biology):
The diagrams shown, display the birth lengths
and growth rates of two species of sharks.
a.
If the growth rates stay the same, at what age
would a Spiny Dogfish and a Greenland shark be the same length?
b.
If the growth rates continue, how long will each shark be when it
is 25 years old?
c.
Explain why the growth rates for these sharks may not
continue indefinitely.
#5. STEM (Statistics/ Linear Regression):
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The table below shows the populations of the New York Cit~ and Los Angeles metropolitan
the U.S. Census data reports from 1950 through the year 2QlOO.
YEAR
NYC population
L~ population
1950
1960
1970
1980
1990
2000
12,911,994
14,759,429
16,178,700
16,121,297
18,087,291
21,199,865
4,367,911
6,742,696
7,032,075
11,487,568
14,531,529
16,373,645
. T
a.
Assuming these linear trends continue, when will the population
b.
What will that population
be?
regions from
of these regions be equal?
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HOMEWORK - Chapter 2 Day #1
I.
Solving Systems of Equations Using Graphs/Tables.
Solve each system by graphing or using a table. Check your answers
#1.
y =x- 2
#2.
Y = -x + 3
#3.
Y = -2x + 7
II.
Y = 3/2 x - 2
2x + 4y = 12
x+y=2
Writing System of Equations from Word Problems
Write and solve a system of equations for each situation. Check your answers.
1. A store sells small notebooks for $8 and large notebooks for $10. If you buy 6
notebooks and spend $56, how many of each size notebook did you buy?
2. A shop has one pound bags of peanuts for $2 and three-pound bags of peanuts for
$5.50. If you buy 5 bags and spend $17, how many of each size bag did you buy?
III.
STEM PROBLEMS(Statistics & Regression)
1. Metropolitan PopuLations: The table shown below displays the populations of the San
Diego and Detroit metropolitan regions.
Year
San Diego Population
Detroit Population
1950
334,387
1,849,568
1960
573,224
1970
696,769
1,511,482
1980
875,538
1,203,339
1,223,400
951,270
1990
2000
~~--
-~-'~~-
1,670,144
-~'-'~--1,027,974
1,110,549
a. When were the populations of these regions equal?
b. What was that population?
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2.
Life Expectancy: The table below, shows the life expectancy of U.S. men and women,
according to the U.S. Census reporting from 1970 through 2000. Find a linear model for
the set of data. In what year will the two quantities be equal?
Year
1970
1975
1980
1985
:1:990
1995
2000
3. Sports:
Men (years)
f
67.1
Women (years)
I
74.7
68.8
70.0
71.1
76.8
77.4
78.2
71.8
72.5
78.8
78.9
74.3
79.7
You can choose two tennis courts at two university campuses to learn how to
play tennis. One campus charges $25 per hour. The other campus charges $20 per hour
PLUSa one-time registration fee of $10.
a. Write a system of linear equations to represent the cost 'c' for 'h' hours of court
use at each campus.
b. Using a graphing calculator, find the number of hours for which the costs are the
same.
c. If you want to practice for a total of 10 hours, which university campus should
you choose? Explain.
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Algebra II - Chapter 2 Day #2
Topic: Solving Systems of Equations Algebraically
Standards/Goals:
../
A.REI.6: I can solve systems of linear equations for both an exact answer and approximately
../
A.1.e.: I can solve systems of two linear equations using various methods, including
elimination, substitution, and graphing •
./'
A.CEO.3.: I can interpret
•
solutions to systems of equations as being viable or non-viable.
Concepts:
1. Systems of Equations - Is a set of two or more equations.
These can be used when you
have two or more related unknowns. Thegoal of solving a system of equations is to find
a set of values that replace the variables in the equations and are able to make each
equation true.
Let's first look at different ways to solve a system of equations. Standard A.l.a/A.REI.6.
Examples:
#1. What is the solution of the system of equations?
(Use the substitution
method).
5x - 3y =-1
x+y=3
#2. What is the solution of the system of equations?
2x-3y= 14
4x + 3y = 46
(Use the elimination
method).
#3. What is the solution of the system of equations? (Use the elimination with multiplication
method).
7x + Sy
=2
8x - 9y
=
17
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Another concept we will look at today deals with classifying a system of two linear equations based
on the number of solutions that it has:
-/ A.CEO.3.: I can interpret solutions to systems of equations as being viable or non-viable.
Here is a summary.
---- -------
- - - --- -- - - --- -- --- --------- ------------Graphical Solutions of Linear Systems
-----:--------
Concept Summary
Parallel Lines
Coinciding Lines
Intersecting Lines
: , Yi
I
I
y!
'
1
l
I
,
i
J...
no solution
Inconsistent
infinitely many solutions
Consistent
i I Dependent
one solution
Consistent
Independent
Without graphing, is each system independent, dependent, or inconsistent?
#1.
-Bx + Y = 4
x -1/3
Y= 1
#2.
2x + 3y
=1
4x + Y = -3
#3.
Y = 2x - 3
6x - 3y
=9
Let's now take a look at some applications of systems of equations by working on some word
problems.
Word Problems:
1. Music: A music store offers piano lessons at a discount for customers buying new
pianos. The costs for lessons and a one-time for materials (including music books, CD's,
software, etc. are shown below. What is the cost of each lesson and the one-time fee
for materials?
6 lessons:
$300
12 lessons:
$480
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2. Labels: You are in charge of ordering labels for a small business. A company that makes
custom labels charges a yearly fee plus a cost per label. You paid $375 last year for 300
labels. This year, you ordered 1,000 labels and paid $725. What are the yearly fee and
cost per label, assuming the prices did not change?
3. Tests: A student took 60 minutes to answer a combination of 20 multiple choice and
extended response questions. She took 2 minutes to answer each multiple choice
question and 6 minutes to answer each extended response question.
a. Write a system of equations to model this relationship between the number of
multiple choice questions 'm' and the number of extended response questions 'r'.
b. How many of each type of questions was on the test?
4. STEM (Chemistry): A scientist wants to make 6 milliters of a 30% sulfuric acid solution.
The solution is to be made from a combination of a 20% solution sulfuric acid solution
and a 50% sulfuric acid solution. How many milliliters of each solution must be
combined to make the 30% solution?
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HOMEWORK - Chapter 2 Day #2
I.
#1.
Solve each system below by using substitution. Check your answers
y =x+ 1
#2.
x = Y- 2
2x + Y = 7
3x - Y = 6
#3.
4x= 8y
2x + 5y = 27
#5.
2y- 3x = 4
x =-4
II.
#1.
#4.
3x + 2y
x+y=3
=9
Solve each system by using elimination.
x + Y = 10
#2.
4x - 3y = -2
x- Y =2
#3.
#5.
4x + 5y = 14
x-y=o
x+y=2
#4.
5x + 4y = 2
-5x - 2y = 4
14x + 2y = 10
#6.
4x + 3y =-6
x - 5y = 11
5x - 6y
= -27
III.
Classifying Systems
Without graphing, classify each system of independent, dependent, or inconsistent.
#1. 3x - 2y = 8
#2.
2x + 8y 6
#3.
3m
5n + 4
4y 6x - 5
x -4y + 3
n - 6/5 -3/5 m
=
=
=
=-
=
IV.
Word Problems
#1. Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that
day, you bought six oranges and three grapefruits for a total of $4.80. What is the price of
each type offruit?
#2. There are a total of 15 apartments in two buildings. The difference of two times the
number of apartments in the first building and three times the number of apartments in the
second building is 5.
a. Write a system of equations to model the relationship between the number of
apartments in the first building T and the number of apartments in the second
building's'.
b. How many apartments are in each building?
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Algebra II - Chapter 2 Day #3
Topic: Solving Systems of Equations in the form of Word Problems
Standards/Goals:
*MAINGOAL
./ A.REI.6: I can solve systems of linear equations for both an exact answer and approximately
in the context of a word problem
*SECONDARY GOALS:
./ A.CED.2: I can create an equation in two or more variables to represent a real-life situation .
./
A.REl.ll:
I can use technology to graph equations that model real-life situations
./
A.CED.3: I can interpret
real-life scenarios
solutions to systems of equations as being viable or non-viable
in
Examples:
1. Baseball: last year, a baseball team paid $20 per bat and $12 per glove, spending a total of
$1040. This year, the prices went up to $25 per bat and $16 per glove. The team spent
$1350 to purchase the same amount of equipment as last year. How many bats and gloves
did the team purchase each year?
2.
Football: A high school football team is playing a game this Friday night. Tickets for
admission to the game cost $3 for students and $5 for adults. During one game, $2995 was
collected from the sale of 729 tickets. How many tickets were sold to students and how
many were sold to adults?
3. Coffee: A cafe sells a regular cup of coffee for $1 and a large cup of coffee for $1.50.
Melissa and her friends buy 5 cups of coffee and spend a total of $6. Write and solve a
system of equations to find the number of large cups of coffee they bought.
4.
Bills: A student has some $1 bills and $5 bills in his wallet. He has a total of 15 bills that are
worth $47. How many of each type of bill does he have?
5. Transportation: A church youth group with 26 members is going skiing. Each of the five
chaperones will drive a van or sedan. The vans can seat seven people, and the sedans can
seat five people. Assuming there are no empty seats, how many of each type of vehicle
could transport all 31 people to the ski area in one trip?
6.
Break-Even Point: Charley's Donut Shop sells cinnamon rolls $3.50 each. The electricity to
run the oven in Charley's Shop is $140 per day and the cost of making one of the delicious
rolls is $2.40. How many rolls need to be sold each day to break even?
7.
Shipping Expenses: Your business needs to ship a package to another store. FedEx charges
$3.50 per pound plus a $25 service charge. UPScharges $5.50 per pound without any
service charge. At what weight does it cost the same for both companies? If your package
weighs 18 pounds, should you use UPSor FedEx?
8. Geometry: If the perimeter of the SQUAREbelow is 64 inches, what are the values of x and
y?
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Multiple Choice Practice:
1. Which graph shows the solution of the following
system?
4x + y = 1
x + 4y =-11
a.
c.
b.
d.
Which equation represents a line with a slope of ~ and a v-intercept
a. y=~x-%
2.
3.
b.
c.
y=%x-~
y=Yzx+%
d.
y=%x+~
Which is the equation of a line that is perpendicular
of %?
to the line in the graph?
a. y = -3x + 2
b. y = 1/3 x + 5
c. Y = -1/3 x - 4
d. y = 3x-1
4.
Consider the equation:
solutions?
a. 2y 1.Sx - 2
b. 2y = 1.5 x - 1
3x - 4y = 2. Which equation below would form a system with NO
=
3x + 4y = 2
4y-3x=-2
c.
d.
Which graph shows the solution to the given system?
~x-y=1
S.
x=3
a.
y
~ •....
:=
b.
,
('.
..;
~-
Q --f--+-II-'f--+-j
- - 2~-t-t-+-+-+-+-I
- -f- -- -- -+-+-+-1
6.
-
,-
- .- -r-r-r-r-r-r-r-r-r-
y.
-- 2 ---
---- ... -- .--
~···6-----ic
~
x
c_
_.__.
-.-
I.., .••••
=
-- ......f'..
y
5
- °
is true about the given system?
=
The system has:
a.
Zero solutions
Exactly one solution
c.
d.
Two solutions
Infinitely many solutions
......
~i'--I-"-O =z:
tr'j-=.
-=l
b.
x
....•..•.••..
L-L-L...__,=,
Which of the following
-4y 12 - 8x
Y 2x - 3
c.
-)
- f-~-
~",
~
-- e-_._-- ,-
r-T---t--"i.°1""./+-4-'~~I-'-r--:t-+-+-I
~-f-,
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HOMEWORK - Chapter 2 Day #3
1. Exercise: Each morning you do a combination of aerobics, which burns about 12
calories per minute, and stretching, which burns about 4 calories per minute. Your goal
is to burn 416 calories during a 60 minute workout. How long should you spend on each
type of exercise to burn the 416 calories?
2. Class Trip: Suppose 28 members of your class went on a rafting trip. Class members
could either rent canoes for $16 each or rent kayaks for $19 each. The class spent a
total of $469. How many people rented canoes and how many people rented kayaks?
3. Shipping Costs: Your business needs to ship a package to another store. Company A
charges $2.50 per pound plus a $20 service charge. Company B charges $4.50 per
pound without any service charge.
a. At what weight does it cost the same for both companies?
b. If your package weighs 14 pounds, which shipping company should your business
use?
4. CONCEPT QUESTION: Is it possible for a linear system with infinitely many solutions to
contain two lines with DIFFERENTy-intercepts?
5. Tickets: Your school sells tickets for its winter choir concert. Student tickets are $5 and
adult tickets are $10. If your school sells 85 tickets and makes $600, how many of each
ticket did they sell?
6. Groceries: A grocery store has small bags of apples for $5 and large bags of apples for
$8. If you buy 6 bags and spend $45, how many of each size of bag did you buy?
7. Technology Question:
new business.
The table below shows the monthly income and expenses for a
Month
Income
Expenses
1
May
$1500
$21,400
2
June
$3500
$18,800
3
July
$5500
$16,200
4
August
$7500
$13,600
a.
Use your graphing calculator to find linear models for income and expenses as
functions of the number of the month.
b. In what month will income equal expenses?
8. Error Analysis: You and your friend are solving the system 4x - y = 5 and 4x + y = 3.
Your friend says there is no solution, and you say the solution is (1, -1). Who is correct?
Explain.
4
9. Baseball: Last year, a baseball team paid $20 per bat and $12 per glove, spending a total
of $1040. This year, the prices went up to $25 per bat and $16 per glove. The team
spent $1350 to purchase the same amount of equipment as last year. How many bats
and gloves did the team purchase each year?
10. Error Analysis: You and your friend are solving the system: y = 7x + 5 and 3y = 21x + 15.
You say that there are infinitely many solutions and your friend says the solution is (5/7, 0). Which of you is correct? What mistake was made?
11. Break-Even Point: Jenny's Bakery sells carrot muffins at $2 each. The electricity to run
the oven is $120 per day and the cost of making one carrot muffin is $1.40. How many
muffins need to be sold each day to break even?
12. Geometry:
and y?
If the perimeter of the SQUARE below is 72 inches, what are the values of x
3y
2x
13. Chemistry (STEM): MX Labs need to make 500 gallons of a 34% acid solution. The only
solutions available are a 25% acid solution and a 50% acid solution. How many gallons
of each solution should be mixed to make the 34% solution?
14. Online Music: An online music company offers 15 downloads for $19.75 and 40
downloads for $43.50. Each price includes the same one-time registration fee. What is
the cost of each download and the registration fee?
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Algebra II - Chapter 2 Day #4
Topic: Solving Systems of Inequalities
Standards/Goals:
./
D.2.a./A.REI.12:
o
I can graph a system of linear inequalities in two variables with and without
o
o
technology to find the solution set to the system.
I can graph the solutions to a linear inequality in two variables as a half-plane.
I can graph the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Today, we will learn how to solve a system of linear inequalities. We will use many of the same
skills that you would use to graph a linear equation, except that these are inequalities and it will
involve some additional work that graphing equations does NOT require. Just like graphing
linear equations, you can graph linear inequalities in more than just one way.
Solving a System by Using a Table:
Assume thatg and m are whole numbers. What is the solution of the system of inequalities?
g+m~6
{5g + 2m ~ 20
Solving a System by Graphing:
#1. What is the solution of the system of inequalities?
2X - Y~-3
,
{ Y~-~x+l
-
~
17
o
~ ~
!2
1
-,
--
2
#2. What is the solution of the system of inequalities?
X
+ 2y s 4
{y
~ -x-1
41~y
IT
.J
J
X
••
~
S
1
•
1
-'"
-r.r
,
-J
1T
Using a System of Inequalities:
#1. Your city's cultural center is sponsoring a concert to raise at least $30,000 for the city's
Youth Services. Tickets are $20 for balcony seats and $30 for orchestra seats. If the center has
500 orchestra seats, how many of each type of seat must they sell?
#2. Tickets for a dance are sold for $5 to seniors and $7 to juniors. The dance hall can hold 560
students. How many of each type of ticket must be sold to raise at least $3,500?
3
HOMEWORK - Chapter 2 Day #4
1. Assume that x and yare whole numbers. wha1 is the solution of the sys1em of
inequalities? (Hint: Use a table)
x+y:2::5
{ 2x + 3y ~ 15
2. For the following systems of inequalities, solve them by graphing. (Use the provided
graph to assist you).
y - 3x s -4
a. { y ~ x- 3
, ~y
x
•
•
X
b.
{
+ 2y :::;-6
y >
_ x +5
'~y
x
b
>x
< x +1
y
c.
{y
!l
6
••
4
..•..
)'
y <3
d. { y:::;
~X
+1
-
....
x
I
X
e.
[
+ 2y:::;
10
x + y_< 3
3. Concert Tickets: Tickets to a concert cost $70, or $20 with a student discount. Ticket
sales must EXCEED$500,000 for the group to perform. If 20,000 seats are available,
how many of each type must be sold? Show the solution on your calculator.
4. Pizza Parlor: A pizza parlor charges $1 for each vegetable topping and $2 for each meat
topping. You want at least five toppings on your pizza. You have $10 to spend on
toppings.
How many of each type of topping can you get on your pizza?
5. Football & TV: You spend no more than 3 hours each day watching TV and playing
football. You play football for at least 1 hour each day. What are the possible numbers
of hours you can spend on each activity in one day?
6. College Admissions:
mathematics section.
school of your choice
inequalities to model
graphing.
An entrance exam has two sections, a verbal section and a
You can score a maximum of 1600 points. For admission, the
requires a math score of at least 600. Write a system of
scores that meet the school's requirements. Then solve by
Algebra II - Chapter 2 Day #5
Topic: Solving Systems of Equations with 3 variables
I
Standards/Goals:
• o.l.c.: I can solve algebraically a system containing three variables.
Today, we will begin to focus our enerqies on systems of equations with 3 different variables.
The process will be very similar in nature to how we handled it with just 2 variables.
We will first focus our energy on using SUBSTITUTION.
1.
x - 2y + 3z = -4
y-z=3
z =-1
x + 2y + z = 1
2.
y-z=2
4z
3.
=8
2x + 3y - 2z =-1
x + 5y = 9
4z-5x
=4
We will also use elimination to solve system of equations with 3 variables:
4.
2x-y + z 4
x + 3y-z = 11
4x + y-z = 14
=
5.
x + Y + 2z = 3
2x + y + 3z = 7
-x - 2y + z = 10
6. STEM: Engineering/Manufacturing:
In a factory, there are 3 machines: A, B, & c.
When all three machines are working, they produce 287 bolts per hour. When only
machines A and C are working, they produce 197 bolts per hour. When only machines A
and B are working, they produce 202 bolts per hour. How many bolts can each machine
produce per hour?
7. STEM: Business/Economics: You manage a clothing store and budget $6000 to restock
200 shirts. You can buy T-shirts for $12 each, polo shirts for $24 each, and rugby shirts
for $36 each. If you want to have twice as many rugby shirts as polo shirts, how many of
each type of shirt should you buy?
8. {Continued from #7}: Suppose you want to have the same number of t-shirts as polo
shirts. Buying 200 shirts with a budget of $5400, how many of each shirt should you
buy?
HOMEWORK - Chapter 2 Day #5
Use substitution for #'51 - 3:
1.
x + 3V = 1
V + 2z
=5
z=3
2.
4x - V + 2z = 6
V + 4z = 2
2V=4
3.
x - 2V + z = -4
-4x + V - 2z
=1
2x + 2V - z
= 10
Use elimination for #'54 - 6:
4.
x - V + z =-1
+ V + 3z =-3
2x - V + 2z = 0
x
5.
x - 2V + 3z
= 12
2x -V-
2z
=5
z
=4
2x + 2V-
6.
3x + 3V + 6z = 9
2x + V + 3z = 7
x + 2V - z -10
7. STEM: Sports: A stadium has 49,000 seats. Seats sell for $25 in Section A, $20 in
Section B, and $15 in Section C. The number of seats in Section A equals the total
number of seats in Sections Band C. Suppose the stadium takes in $1,052,000 from
each sold-out event. How many seats does each section hold?
8. STEM: Business: A worker received a $10,000 bonus and decided to split it among
three different accounts. He placed part in a savings account paving 4.5% per year,
twice as much in government bonds paving 5% and the rest in a mutual fund that
returned 4%. His income from these investments after one year was $455. How much
did the worker place in each account?
Algebra II - Chapter 2 Day #5 Part II
Topic: Solving Systems of Equations with 3 variables
Standards/Goals:
•
D.l.c.: I can solve algebraically a system containing three variables.
Today, we will continue to work on systems of equations with 3 different variables.
1.
2.
=1
-z
x + 2y + z
=4
3x-y-z
=3
3x + y
x + Y + 2z = - 7
3x + y- 2z = 7
-x - 3y + z = - 9
3.
x - 2y + z = 1
2x + z = 9
-3x + Y = - 3
4. You are an office supply distributor and budget $7200 for 80 office chairs. You can buy
leather chairs for $125 each, mesh chairs for $100 each, and fabric chairs for $75 each.
If you want to have 3 times as many fabric chairs as leather chairs, how many of each
type should you buy?
5. The sum of three numbers is -2. The sum of three times the first number, twice the
second number, and the third number is 9. The difference between the second number
and half the third number is 10. Find the numbers.
6. Monica has $1, $5, and $10 bills in her wallet that are worth $96. If she had one more
$1 bill, she would have just as many $1 bills as $5 and $10 bills combined. She has 23
bills total. How many of each denomination does she have?
7.
Solve the following system:
2x - 3y + z = 5
2x - 3y +
z =-2
-4x + 6y - 2z
= 10
1//
HOME'f"ORK - Chapter 2 Day #5 Part II
I. Solve the following Systems
X+Y+Z=3
X+Y+Z=-l
1. 2x-y+2z=-5
!
2x - Y + 2z
2.
{
-x+2y-z=4
-x + y - 2z = 5
{
3x+2y-
=
6
3. x-2z =-3
{
z=13
2X - y + z = --4
X- y+2z=lO
4.
2X+ y =9
5. 3x + y - 2z
{
3x - 3y + 6z = -2
=
2X- y-z
{
3x- y=--4
7. 3x-y+z=4
{
5x+z =7
{
2x-2z
3x+ y+z =16
x+ y+4z =5
x+3y+z=O
8.
=4
-x+2y+z=1
6.
0
x-2y=1
x+y-z=O
2y+3z=15
9.
= 18
{
-2x+2z=3
3x+ y-2z
=0
10. You have 17 coins in pennies, nickels, and dimes in your pocket. The value of the coins is $0.47.
There are four times the number of pennies as nickels. How many of each type of coin do you
have?
11. For a party, you are cooking a large amount of stew that has meat, potatoes, and carrots. The
meat costs $6 per pound, the potatoes cost $3 per pound, and the carrots cost $1 per pound.
You spend $48.50 on 13.5 pounds of food. You buy twice as many carrots as potatoes.
a. Write a system of three equations that represent how much food you bought.
b. How much of each ingredient did you buy?
12. The first number plus the third number is equal to the second number. The sum of the first
number and the second number is six more than the third number. Three times the first number
minus two times the second number is equal to the third number. What is the sum of the three
numbers?
II. MULTIPLE CHOICE:
#1. Which of the following is a system with the solution (6, -2, -3)?
cs: {2X+
y- z=5
x+4y+2z=16
©
15x+6y-2z=12
®
z=31
x-2y+2z=8
y-4z=-13
#2. What is the value of
3X - 4 Y + 2z
{
4x+ y-
{ 4x-3y+
-3x+
{-2X-2Y+
z=16
3x-3y+2z=9
®
z=-20
{-X+2Yz=-7
3x+5y+2z=2
-2x+3y+4z=-30
z in the solution ofthe system?
= 20
x+ y-z=-4
6x - y+ 2z =23
A.
2
B. -1
C. 5
D. -5
Algebra II - Chapter 2 Day #6
Topic: Solving Systems of Equations Using Matrices
Today, we will learn how to use a matrix to solve a system of equations.
We will also utilize technology
in this process.
Standards/Goals:
./' 1.1.e.:I can solve systems of equations by using inverses of matrices and determinants .
./' A.REI.8: I can represent a system of linear equations as a single matrix equation .
./' N.VM.8: I can add, subtract and multiply matrices with appropriate dimensions.
We want to first re-introduce ourselves to the concept of a matrix.
Definition - Matrix: A matrix is a rectangular array of numbers. You will usually display the
array within brackets (as shown below). The dimensions of a matrix are the numbers of rows
and columns in the array. Each number that is inside of the matrix is known as a matrix
element. You can identify a matrix element by its row and column numbers.
4
5
A = [~
~]
We want to be able to determine
individual elements inside of a matrix.
1. What is m12 in matrix M?
M = [8205 7 4]
2. What is aZ3 in matrix A?
A= [6 ~9 1;]
-3
-2
10
We also want to know how to represent a system of equations as a matrix equation.
3. How do you represent the system of equations shown below with a matrix?
6x - y = 12
+ 10y = 1
f
t-x
4.
How do you represent the system of equations shown below with a matrix?
X
{
-
3y
+z
=6
x + 3z = 12
y = -5x + 1
EXAMPLES: Write matrix equations that correspond to the following systems:
SP - 3q + r = 5
#1. { 4p - 3r = 1
2q = 3r - 2
#2.
{2X _ Y = -1
x + 3y = 17
{ 3a + 2b
#3.
=5
4a = 3c + 7
6b - 6c = - 5
2
5. What linear system of equations does this matrix represent?
[=: ~~~]
5 1 0 8
We want to learn how to multiply matrices by their inverse.
EXAMPLES: What are the solutions to the following matrix equations?
#1. [~
~3]X
= [_~2]
#2..
[~
_\]
X
= [~]
We also want to learn how to solve a system of linear equations by using a matrix and by
utilizing technology.
#1. What is the solution of this system?
X
{
+ 2y = 16
3x + y = 8
#2. What is the solution of this system of equations? Use your calculator.
+ 2b - c = 7
2a - c = 5
{
a - 4b + c =-4
3a
#3. What is the solution ofthe system of equations?
f
2a + 3b - c = 1
-4a + 9b + 2c = 8
-2a + 2c = 3
3
We also want to be able to solve a real-world application
to solve a system of linear equations.
WORD PROBLEMS: Consider the following situation:
problem by using a matrix equation
#1.
A
~
it
..
fSome machines
I charge
a
! percentage for
I counting your
~.-~---
..•
,-,-
.,
:e=~==============::::::::;:';"""':';''';;;;'''';;;;''-----'---'---I
---')
#2. PIGGY BANK: There are 34 coins in your friend's piggy bank made up of nickels, dimes, and
quarters. The total value of the coins in the bank is $3. If the number of nickels is 6 more than
the number of dimes and the number of quarters put together, how many coins of each type
are in the bank?
#3. BASEBALL: A baseball field has 6200 seats in the lower three tiers. Seats sell for $120 in
section A, $100 in section B, and $75 in section C. If tickets are sold for all of the seats, the total
in sales is $604,000. The number of seats in section Cis 500 fewer than those in section B.
How many seats are in each section of the field?
#4. FRUIT STAND: You work at a fruit stand that sells apples for $2 per pound, oranges for $5
per pound, and bananas for $3 per pound. Yesterday you sold 60 pounds of fruit and made
$180. You sold 10 more pounds of apples than bananas.
a. Write a matrix equation to show the system of equations for this situation.
b. How many pounds of each kind of fruit did you sell yesterday?
c. What kind of fruit did you sell the most of?
4
HOMEWORK - Chapter 2 Day #6
Identify the indicated element.
6 3
[! ~I:]
A=
B=[~
1. A13
2.824
5.
6. A2l
B3l
21
5
1
13
2
°
-10
3.812
4.A22
S.A11
Write a matrix to represent each system.
9. {
3x+ Y =-4
10.
-2x+4y=7
{ -3x+4y=
Write the system of equations represented
120[-! ~ _~ l~ J
Solve each matrix equation.
15.
4X-
6x=11
130[~
y+2z=10
11. 5x+2y-3z=0
{
x-3y+
z=6
2
by each matrix.
:1-;]
140[
!-:~;-:~
J
If an equation cannot be solved, explain why.
0.25 -0.75]
_ [ 1.5]
X[ 3.5
2.25
-3.75
16. [3 -9] X-_ [12]
1 -6
°
~l
r ~l
l-l 2 2 l-2
1S.r ~ ~ -
X =
Solve each system.
4X+ y-
5X - 2y + z
= -
-x-
= 5
2z=3
19. 2y+ z=4
{
3x- 3y- z=9
22. Suppose the
20.
{
y-2z
3x+2y+2z=
2
3X+ 5z =-4
1
21.
{
-2x + Y - 3z = 9
-x-
2y + 9z=0
movie theater you work at sells popcorn in three different sizes. A small costs $2, a
medium costs $5, and a large costs $10. On your shift, you sold 250 total containers of popcorn and
brought in $1726. You sold twice as many large containers as small ones.
a. How many of each popcorn size did you sell?
b. How much money did you bring in from selling small size containers?
23. The following matrix shows the prices passengers on an airline flight paid for a recent ticket and
how many passengers were on that flight. Some passengers paid full price for their tickets, and some
bought their tickets during a half-price sale. How many passengers bought each price of ticket?
100]
[ 1201 24011 20,160
1
Algebra II - Chapter 2 Day #7
Topic: Linear Programming
Today, we will use our previously learned skills regarding the graphing of systems of linear inequalities
into a real-world
application,
known as linear programming.
J
Standards/Goals:
MAIN GOAL:
./ D.2.b.: I can solve linear programming problems by finding maximu
and minimum values of
a function over a region defined by linear inequalities.
SECONDARY GOAL:
./ D.2.a./ A.REI.12: I can graph a system of linear inequalities in two variables with and without
technology
Definition
to find the solution set to the system.
- Linear Programming
some quantity,
- is a method
for finding
a minimum
or maximum
value of
given a set of constraints.
Graph the following
inequalities
to determine
the feasible
region.
x;::::
2
y~3
{ y<6
x + y::; 10
F
.v
--
i-v-
c
I
<A
-
'/
0
-
----------
-+
From above, we see what the 'feasible region' is going to be. We also want to introduce ourselves to
the idea of an objective function. Oftentimes, this quantity will be either cost or profit. Suppose that
the objective function above is C = 2x + y.
The two points that represent the least and greatest values for the objective function are:
2
Let's focus some more on findi ng the minlrn um and maximum values of the objective
function for a given feasible regon.
The feasible region determined I ya system of constraints is given below. Find the minimum
and maximum values of the objective functio n for the given feasible reg;~>n.
#2. C x + 2y
#1. C=x-y
=
:.1_
--j
ty I
-
--
1-
--
2-
(0,0)
y
(5,0)
x
2
r-~---~-~+----+---
Graph each system of constrait ts. Nameall
maximize or minimize the obje ttlve function
#1. C = 3x-
+:
Then, find the vj'ues of x and y that
# 2. C = 2x + 4y
Constraints:
x~3
Y
Constraints:
x~O
y~O
2x + y::;; 6
x+y;::3
2x - 3y ~ - 9
Y
y
IO
IO
9
9
8
I
8
7
7
6
6
5
,
,
4
I
I
I
-9 -8 -7 -6 -5 -4 -3 -2 -1
5
4
3
3
2
2
1
0
1
_L
I
3
4
1
i
I
1 2
6
7 8
9 10
X
-9 -8 -7 -6 -5 -4 -3 -!2 -1
0
-1
-2
2
-3
-4
-3
-4
-5
-5
-6
-6
,
-7
-7
-8
-8
-9
-9
-
-
I
2
3 4
5
6
7
8
9 10
X
3
HOMEWORK - Cha~ter 2 Day #7
Find the value of x and y that mrXimizes or minimizes the objective function for each graph
~~
I
#1. Maximum for
P
#2 Minimum for
= 6x + 2y
P
#3. Maximum for
= 4x + Y
P=x y
·10
8
,6
·4
. 2 .•••.•.•,•..••••
o
#9. Minimum for
#4. Maximum for
P
= 2x + y
#6. Minimum for
P = x + 9y
P = 5x + lay
·Y
·6
6
4
·4
2
-2
=
o
Graph each system of
7.
+ 2y ::s; 6
x
> 2
{y
;:::1
X
10.
{
+Y
s 5
4x + Y :5: 8
x ;:::O,y ;::: 0
Minimum for
C=x+ 3y
-s.
-2
Use the graphs provided for each.
~X
+ Y ::s; 5
8. x + 2y :5:8
x ;:::0, y ;:::0
Minimum for
C=3x+4y
8(6#3)
4 - 6 ..
Name all vertices. Then find the values of x and y that maximize or
minimize the objective function.
X
.Q
I
Maximum for
~=x+3y
{X
+ Y :5: 6
9. 2x + Y :5:10
x ::s; 0, y ;:::Ii)
Maximum for
P=4x+y
I
*6
#7
y
y
I
10
I.
i I
I
\0
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
I
1
1
-9 -8
7 --6
5 -41 -31 -2 -I
0
1
2
3
4
1
5
6
7
8
9 10
-1
I
I
9
X
8 -7 -6 -5 -4 -3 -2 -1 0
I
1
2
3 4
5
6
7
8
9 \0
-I
-2
-2
-3
-4
-3
-4
-5
--6
-5
--6
-7
-7
-8
-8
-9
-9
X
C-
.~
o
y
y
10
\0
9
9
,
8
7
6
6
5
5
4
4
,
3
I
8
7
3
2
I
1
!
1
-9 -8 -7 -6 -51 -4 -3 -2 -1
0
1
2
3
-1
1
I
4
6
I
I
7
8
9 \0
X
9 -8 . -7 -6 -5 -4 -3 -11 -I
2
I
0
-2
-3
-4
-3
-4
--6
I
I
-7
-8
-9
1 2
3
4
5
6
7
8 9 10
-1
-2
-5
1
I
-5
--6
-7
-8
I
I
-9
TheMathWorksheetSite.com
X
Algebra II - Chapter 2 Day #8
Topic: Linear Programming
Today, we will continue learning how to use linear programming
J
I
in real-life situations.
Standards/Goals:
MAIN GOAL:
./ D.2.b.: I can solve linear p ,ogramming problems by finding maximum
a function over a region djfined
SECONDARY GOAL:
D.2.a./ A.REI.12: I can graph a system of linear inequalities
technology to find the solution se~ to the system.
values of
in two variables with and without
I
Examples:
and minimum
by linear inequalities.
I
Construct the constraints and g[aph the feasible regions for the following situations.
1. A plane carrying relief f~od and water can carry a maximum of 50,000 pounds, and is
limited in space to carrYIng no more than 6000 cubic feet. Each container of water
weighs 60 pounds and takes up 1 cubic foot, and each container of food weighs 50
pounds and takes up 10 fubiC feet. What is the region of constraint for the number of
containers of food and 'f.ater that the plane can carry?
2. A furniture company makes two kinds of sofas, the Standard model and the Deluxe
model. The Standard mbdel requires 40 hours of labor to build, and the Deluxe model
requires 60 hours of labbr to build. The finish of the Deluxe model, however uses both
teak and fabric, while th!e Standard uses only fabric, with the result that each Deluxe
sofa requires 5 square yards of fabric and each Standard sofa requires 8 square yards of
fabric. Given that the companv can use 200 hours of labor and 25 square yards of fabric
per week building sofas] what is the region of constraint for the number of Deluxe and
Standard sofas the company can make per week?
Use linear programming
3.
to! solve
the problem:
Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no
I
more than a total of 50m cans and packages and spend no more than $600. How can
I
you maximize your profit? How much is the maximum profit?
Mixdd Nuts
Roasted Peanuts
12 cans per case
You pay: 1$24per case
Sell at: $3.50 per can
$18 pro fit per case
20 packages per case
You pay: $15 per case
Sell at: $1.50 per package
$15 profit per case
4. STEM: Air Quality: A city wants to plant maple and spruce trees to absorb carbon
dioxide. It has $2100 tJ spend on planting spruce and maple trees. The city has 45,000
square feet available f+ planting.
a. Use the data from the table. Write the constraints for the situation.
b. Write the objective function.
I
1
Planting Cost
Area Re . uired
Carbon Dioxide Absorption
Spruce
Maple
$40
$30
ft2
600
650lbfyear
1
900 ft2/
300lbfy'ear
!!n?~
~h~?!~ts:n~~.+~!~i!
~~!!~e
following
Situ~tions:
r
1. Sarah is looking through
clothing catalog and she is willing to sgend up to $80 on
clothes and $10 for shipping. Shirts cost $12 each plus $2 shipping, and a pair of pants
cost $32 plus $3 shipping. What is the region of constraint for the number of shirts and
pairs of pants Sarah can buy?
Solve the following using Ii~ear programming
2. Teams chosen from 30 f~rest rangers and 16 trainees are planting trees. An experienced
team consisting of two rangers can plant 500 trees per week. A training team consisting
of one ranger and two trrinees can plant 200 trees per week.
I
~rienCed
Teams
I
Numberof Teams
Numberof
I
Rangers
Numberof Trainees
Numberof Trees Planted
I
Training
Teams
Total
x
y
x+y
2x
y
30
0
2y
16
500x
200y
500x+ 200y
a. Write an objectiv e function and constraints for a linear
program that models the problem.
b.
How many of eJh type of team should be formed to
maximize the nur,ber of trees planted? How many trainees
are used in this solution? How many trees are planted in a
week?
3. You are going to make and sell baked goods. A loaf of Irish soda bread is made with 2 c
flour and ~ c sugar. Kugelhopf cake is made with 4 c flour and 1 c sugar. You will make a
profit of $1.50 on each loaf of Irish soda bread and a profit of $4 on each Kugelhopf
I
cake. You have 16 c floJr and 3 c sugar.
a. How many of eafh kind of baked goods should you make to maximize the profit?
b. What is the malimum profit?
4. Suppose you make and Isell skin lotion. A quart of regular skin lotion contains 2 c oil and
1 c cocoa butter. A quart of extra-rich skin lotion contains 1 c oil and 2 c cocoa butter.
You will make a profit ~f $lO/qt on regular lotion and a profit of $8/qt on extra-rich
lotion. You have 24 c oil and 18 c cocoa butter.
a. How many quais of each type of lotion should you make to maximize your
profit?
b. What is the maximum profit?
I
Algebra II - chapter 2 Day #9
I
Topic: Linear Programming
Today, we will continue lear~ing how to use linear programming
I
in rea/T/ife situations.
I
Standards/Goals:
MAIN GOAL:
./ D.2.b.: I can solve line~r programming problems by finding maximum and minimum values
of a function over a region defined by linear inequalities.
I
i
SECONDARYGOAL:
.,/ D.2.a./A.REI.12: I can graph a system of linear inequalities in two variables with and without
technology to find the solution set to the system.
I
EXAMPLE: Cooking BakingItray
of corn muffins takes 4 cups of milk and 3 cups of wheat
flour. Baking a tray of bran
uffins takes 2 cups of milk and 3 cups of wheat flour. A baker
has 16 cups of milk and 15 c ps of wheat flour. He makes $3 profit per tray of corn muffins
and $2 profit per tray of br~ muffins. How many trays of each type of muffin should the
baker make to maximize his profit?
Understanding
the Proble
1. Organize the information
lin a table.
I Com Muffin
Bran Muffin
Trays, x
Trays,y
Milk (cups)
Flour (cups)
Profit
2.W hat are the constraints
and the objective
Objective Function:
Constraints:
Planning the Solution
3. Graph the constraints
function?
-----
on the grid at the right.
4. Label the vertices of the
easible region on your graph.
Getting an Answer
S. What is the value of the objective
function
6. At which vertex is the 0 jective function
7. How can you interprettte
at each vertex?
maximized?
solution in the context of the problem?
Total
Example:
Construct the constraints an d graph the feasible regions for the folioing
Your school band is selling
situations.
ca lendars as a fundraiser. Wall calendars casIt $48 per case of 24.
You sell them at $7 per calerdar.
Pocket calendars cost $30 per case of 40. You sell them at
$3 per calendar. You make a profit of $120 per case of wall calendars a~d $90 per case of
pocket calendars. If the barn can buy no more than 1000 total calenda s and spend no more
than $1200, how can you m, ximize your profit if you sell every Calendar? What is the
maximum profit?
Wal Calendars
Pocket Calendars
x
y
Number of Units
24x
40y
Cost
48x
30y
1~Ox
gOy
Number of Cases
Profit
Total
1000
1200
120x + gOy
I
HOMEWORK - Chapter 2 Day #9
1. STEM: Air Quality: A cit~ wants to plant maple and spruce trees to absorb carbon
dioxide. It has $2100 to spend on planting spruce and maple trees. The city has 45,000
square
a.
b.
c.
d.
feet available for/Planting.
/
Use the data froT the table. Write the constraints for the situation.
Write the objectire function.
Graph the feasib,e region and find the vertices.
How many of each tree should the city plant to maximize carbon dioxide
abso rption?
I
I
Planting Cost
Area Requi(~d
Carbon Dioxide Absorption
Spruce
$30
600 ft2
650lb/year
Maple
$40
900 ft2
300lbfyear
2. A doctor all ots 15 minut PS for routine office visits and 45 minutes for full physicals. The
doctor cannot do more than 10 physicals per day. The doctor has 9 available hours for
appointments each day./ A routine office visit costs $60 and a full physical costs $100. How
many routine office visi,s and full physicals should the doctor schedule to maximize her
income for the day? Wh1at is the maximum income?
3. A caterer must make at least 50 gal of potato soup and at least 120 gal of
tomato soup. One chef fan make 5 gal of potato soup and 6 gal of tomato soup
in 1 h. Another chef ca~ make 4 gal of potato soup and 12 gal of tomato soup in
1 h. The first chef earn~I$20/h. The second chef earns $22/h. How many hours
should the company ask each chef to work to minimize the cost?
y
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