1. Unit 3: Systems of Equations

2. Lesson 1 Day 1 – 09/29/15 Solving Equations by Graphing Objective: SWBAT use graphing to solve simple linear equations and systems of linear equations

3. Warmup: Solve the following equations
1. 3𝑥−2=7 2. 2𝑥+3=4 𝑥−6 −2 3. 𝑥 2 −3= 𝑥 −2= ln 𝑥 +3=sin⁡(𝑥)

4. Graphing Calculators… UNITE!
Let’s explore how to

5. HOMEWORK – Due Thursday (via email)
1. Finish the guided notes exploration. 2. READ pages and take notes: system of equations, linear system, solution of a system, consistent solution, independent system, dependent system, inconsistent solution, three graphical solutions of linear systems. 3. Try Example 3 on your graphing calculator and do the Got It? Question #.3

6. Lesson 2 Day 1 – 10/05/15 Solving Systems of Equations by Substitution Objective: SWBAT Solve systems of equations using the substitution method

7. Graphing Packet (THURS) Homework worksheet (FRI)
Review from Sub days
Graphing Packet (THURS)
Homework worksheet (FRI)
Test Corrections (Due THURS)

8. The Substitution Method
For any system of equations in two variables, you can find the solution (x,y) by….
Steps:
1. Solve for one of the variables in either of the equations (Eq. 1).
2. Substitute the expression that is equivalent to the solved-for variable from step 1 into the OTHER equation (Eq. 2), and SOLVE that equation.
3. Substitute the value of the solved variable into either of the original equations and solve for the 2nd variable.
4. CHECK your solution by plugging in to either equation.

9. Example Problems 6𝑥−3𝑦=12 4𝑥+𝑦=14 Solution: (x,y)=(3,2) 2. 𝑦= 𝑥 2 +3
3𝑦+5 𝑥 2 =17
Solution: (x,y) = (1,4), (-1,4)
Step 1
Step 2
Step 3
STEP 4!

10. Practice! (-7,5) (2,-3) (-5,4) System 1: 3x + 7y = 14 2x + 7y = 21
Answers:
(-7,5)
(2,-3)
(-5,4)
System 1:
3x + 7y = 14
2x + 7y = 21
System 2:
4x + 9y = –19
-4x – 7y = 13
System 3:
-3x + 2y = 23
5x + 2y = –17

11. Homework
Finish Worksheet #3.2 by using the Substitution method
Research (online) solving systems of equations in THREE variables
Test Corrections – Due THURSDAY

12. Lesson 2 Day 2 – 10/06/15 Solving Systems of Equations by Substitution Objective: SWBAT Solve systems of equations using the substitution method KEEP HOMEWORK OUT

13. Bellwork −4𝑥+8𝑦−12𝑦−16𝑥 6𝑦+4𝑥−2𝑦+9𝑥 10𝑥+3𝑦−5𝑥+𝑦−𝑥 12𝑥−4𝑦+5𝑡−10𝑥+2𝑡
Simplify the following expressions
−4𝑥+8𝑦−12𝑦−16𝑥
6𝑦+4𝑥−2𝑦+9𝑥
10𝑥+3𝑦−5𝑥+𝑦−𝑥
12𝑥−4𝑦+5𝑡−10𝑥+2𝑡
10𝑥+9𝑦−8𝑧−7𝑥+6𝑦−5𝑧+4𝑥−3𝑦−2𝑧−𝑥+𝑦
Answers:
−20𝑥−4𝑦
13𝑥+4𝑦
4𝑥+4𝑦
2𝑥−4𝑦+7𝑡
6𝑥+13𝑦−15𝑧

14. Writing Solutions to a system of equations
Because a solution to a system of equations is the intersection point on a graph, we write the solution to a system as a coordinate pair (x,y).
Examples:
If the solution to a system is x=4 and y=-2, then…
We write that as 𝒙,𝒚 = 𝟒,−𝟐
2. If the solution to a system is x=76 and y=100.2, then…
We write that as 𝒙,𝒚 = 𝟕𝟔,𝟏𝟎𝟎.𝟐

15. Write the following solutions!
x = 5 and y = 10
x = -3 and y = -8
y = 4 and x = 1.3
x = and y = 1 2
𝑥=100 𝑎𝑛𝑑 𝑦=−12 𝑎𝑛𝑑 𝑧=−34.5
𝑥=−2 𝑎𝑛𝑑 𝑦=−6 𝑎𝑛𝑑 𝑧=13
𝑥=1 𝑎𝑛𝑑 𝑦=1 𝑎𝑛𝑑 𝑧=1
𝑥=2 𝑎𝑛𝑑 𝑦=90 𝑎𝑛𝑑 𝑧=1 𝑎𝑛𝑑 𝑎=4
Answers:
(x,y)=(5, 10)
(x,y)=(-3, -8)
(x,y)=(1.3, 4)
(x,y)=(-12.3, 1 2 )
(x,y,z)=(100, -12, -34.5)
(x,y,z)=(-2, -6, 13)
(x,y,z)=(1, 1, 1)
(x,y,z,a)=(2, 90, 1, 4)

16. Solving Systems of Equations using Substitution – Three Variables
Suppose we are given a system of equations with three different variables x, y, and z. How do we go about solving those?
Using substitution, we use the same process as with two variables, but… it’s a few extra steps.
Our solution to a system with the variables x, y, and z will look like:
𝑥,𝑦,𝑧 =(#,#,#)
Example problems:
Per. 2, 4, 6:
𝑥+2𝑦+5𝑧=−10
3𝑥−6𝑧=21
3𝑦=6
Per. 1, 3:
𝑥+2𝑦+5𝑧=−10
3𝑥−4𝑦+6𝑧=−23
2𝑥+𝑦−2𝑧=10

17. Practice! Per. 1, 3 −𝑥−5𝑦−5𝑧=2 4𝑥−5𝑦+4𝑧=19 𝑥+5𝑦−𝑧=−20 2. −4𝑥−5𝑦−𝑧=18
2𝑥+7𝑦−6𝑧=14
−3𝑦+8𝑧=−18
3𝑦=−6
𝑥−14𝑦+3𝑧=13
7𝑥−5𝑧=14
4𝑥=−12
𝑥+12𝑦−8𝑧=9
2𝑥−7𝑧=−14
5𝑧=20
4. 8𝑥+7𝑦+14𝑧=19
3𝑦+8𝑧=11
7𝑧=−14
Per. 1, 3
−𝑥−5𝑦−5𝑧=2
4𝑥−5𝑦+4𝑧=19
𝑥+5𝑦−𝑧=−20
2. −4𝑥−5𝑦−𝑧=18
−2𝑥−5𝑦−2𝑧=12
−2𝑥+5𝑦+2𝑧=4
3. −𝑥−5𝑦+𝑧=17
−5𝑥−5𝑦+5𝑧=5
2𝑥+5𝑦−3𝑧=−10
Sol’ns:
(-2,-3,3)
(-4,0,-2)
(-1,-4,-4)
Sol’ns:
(5,-2,-3)
(-3,-5,-7)
(7,-3,4)
(-2,9,-2)

18. Homework
Finish Worksheet #3.3 by using the Substitution method
Test Corrections – Due THURSDAY

19. Lesson 2 Day 3 – 10/08/15 Solving Systems of Equations by Substitution Objective: SWBAT Solve systems of equations using the substitution method Turn in Test corrections to the tray please! Keep HW #3.3 out.

20. PSAT PREP Problems #4 and 5 Take three minutes and select an answer choice. These should be short and quick!

21. Group Practice – solving systems
Get into groups of 4
Each member of the group must choose a letter A, B, C, D
We will work on systems of equations problems and I will call on a student at random to give me the solution to the problem. You will be assessed on whether you answer correctly or not.
Answer (x,y) or (x,y,z) goes on the whiteboard.

22. Let’s Practice! 𝒙+𝒚+𝒛=−𝟏 𝟐𝒙−𝒚+𝟐𝒛=−𝟓 −𝒙+𝟐𝒚−𝒛=𝟒 𝒙+𝒚=𝟏𝟎 𝒙+𝒚+𝒛=𝟑 𝒙−𝒚=𝟐
Alg 2 Honors
𝒙+𝒚+𝒛=−𝟏
𝟐𝒙−𝒚+𝟐𝒛=−𝟓
−𝒙+𝟐𝒚−𝒛=𝟒
𝒙+𝒚+𝒛=𝟑
𝟐𝒙−𝒚+𝟐𝒛=𝟔
𝟑𝒙+𝟐𝒚−𝒛=𝟏𝟑
𝟐𝒙+𝒚=𝟗
𝒙−𝟐𝒛=−𝟑
𝟐𝒚+𝟑𝒛=𝟏𝟓
𝟐𝒙−𝒚+𝒛=−𝟒
𝟑𝒙+𝒚−𝟐𝒛=𝟎
𝟑𝒙−𝒚=−𝟒
𝟐𝒙−𝒚−𝒛=𝟒
−𝒙+𝟐𝒚+𝒛=𝟏
𝟑𝒙+𝒚+𝒛=𝟏𝟔
Algebra 2 Periods 2, 4 ,6
𝒙+𝒚=𝟏𝟎
𝒙−𝒚=𝟐
−𝒙+𝟑𝒚=−𝟏
𝒙−𝟐𝒚=𝟐
𝒙+𝒚=𝟕
𝒙+𝟑𝒚=𝟏𝟏
𝟑𝒙−𝒚+𝒛=𝟐𝟎
𝟐𝒙+𝒚=𝟖
𝟏𝟐𝒛=𝟑𝟔
𝒙+𝟑𝒚−𝟒𝒛=𝟒
𝒙+𝒚+𝟐𝒛=−𝟒
𝟖𝒛=−𝟏𝟔

23. Clear your desks, except for a pencil NO CALCULATOR ALLOWED
QUIZ time!
Clear your desks, except for a pencil
NO CALCULATOR ALLOWED

24. Homework
READ pages and take notes on Elimination Method of Solving Systems of Equations, Try "Got It? #3, #4 a,b, and #5 a,b”

25. Lesson 3 Day 1 – 10/09/15 Solving Systems of Equations by Elimination Objective: SWBAT Solve systems of Two- and Three- Variable equations using the Elimination method Keep HW notes in notebook

26. PSAT PREP Problem #11 Take two minutes and select an answer choice
PSAT PREP Problem #11 Take two minutes and select an answer choice. These should be short and quick!

27. The Elimination Method
For any system of equations in two variables, you can find the solution (x,y) using elimination by….
Steps:
1. “Match up” (or “vertically align”) the x and y terms in similar places in both equations.
2. Use an equivalent system to make the coefficient of one of the “matched up” terms be the opposite of the coefficient of its counterpart
3. Add the two equations vertically.
4. Solve for the variable in the resulting equation.
5. Substitute the value of that variable back in to one of the original equation to solve for the other variable.
Example: 𝑥+3𝑦=1
2𝑥=2𝑦+10
Example: 𝑥+3𝑦=  x + 3𝑦=1
2𝑥=2𝑦  2𝑥−2𝑦=10
Example: x + 3𝑦= ∙(−2)  −2𝑥−6𝑦=−2
2𝑥−2𝑦=  𝑥−2𝑦=10
Example: −2𝑥−6𝑦=−2
𝑥−2𝑦=10
0 −8𝑦=8
𝑦=−1
𝑥=4

28. What you do not finish: Becomes extra HW Afterwards: page 146 #20
Let’s Practice!
Work TOGETHER in pairs (one group of 3) on the problems on the sheet handed to you.
What you do not finish: Becomes extra HW
Afterwards: page 146 #20

29. Constructing Systems of Equations
Page 146 #20 in your textbook:
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.
Write a system of equations to model the relationship between the number of multiple choice questions m and extended response questions r.
𝒎+𝒓=𝟐𝟎
𝟐𝒎+𝟔𝒓=𝟔𝟎
How many of each type of question was on the test?
𝟏𝟓 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐞 𝐜𝐡𝐨𝐢𝐜𝐞, 𝟓 𝐞𝐱𝐭𝐞𝐧𝐝𝐞𝐝 𝐫𝐞𝐬𝐩𝐨𝐧𝐬𝐞

30. Homework for the weekend
Textbook problems:
Page 146 #13-19 odds,
#22-30 All
QUIZ TUESDAY on Elimination Method
Watch video on elimination method in three variables

31. Lesson 3 Day 2 – 10/12/15 Solving Systems of Equations by Elimination Objective: SWBAT Solve systems of Two- and Three- Variable equations using the Elimination method Keep HW Problems on desk – will review

32. PSAT Prep Warmup
Problem # Take three-four minutes and select an answer choice. These should be short and quick!

33. HW Questions?

34. Which Method is Better?
For the following problems, decide which method of solving the system would work better: graphing, substitution, or elimination?
2𝑥−3𝑦=10 𝑎𝑛𝑑 −2𝑥+4𝑦=12
𝑥−4𝑦=20 𝑎𝑛𝑑 𝑦=3𝑥−4
𝑥+𝑦=1 𝑎𝑛𝑑 𝑦−𝑥=−2
2 𝑥 2 −𝑦=12 𝑎𝑛𝑑 𝑦= 𝑥 −1
𝑦+2𝑥=1 𝑎𝑛𝑑 𝑦−3𝑥=9
2𝑥+2𝑦+𝑧=1 𝑎𝑛𝑑 −𝑥+𝑦−𝑧=2 𝑎𝑛𝑑 3𝑥+4𝑦=4

35. Elimination Method with Three Variables
Solve the system of equations below using the elimination method. This should look very similar to the video you watched over the weekend at:
𝑥+2𝑦+5𝑧=−10
3𝑥−4𝑦+6𝑧=−23
2𝑥−2𝑦−2𝑧=4
Steps (ESPN):
“Eliminate” one variable from two equations by adding/subtracting two pairs of original equations
Solve the resulting system of equations
Plug in those two values to an original equation to solve for the last variable.
Notify your brain that your answers are checked.

36. THE SOLUTION
𝑥+2𝑦+5𝑧=−10 3𝑥−4𝑦+6𝑧=−23 2𝑥−2𝑦−2𝑧=4 Solution:(𝒙,𝒚,𝒛)=(𝟏,𝟐,−𝟑)

37. Post-It Board (Exit Ticket)
There are three colors of post-its (yellow, orange, and purple)
If you prefer:
Graphing – grab a PURPLE post-it
Substitution – grab a YELLOW post-it
Elimination – grab an ORANGE post-it
You must write your name and the reason you prefer that method. Please use mathematical language in your reason.
There are no wrong answers! This is your opinion!

38. Homework for tonight
Worksheet 3-5 on the Elimination Method (choose 4 from each section + #22-23)
QUIZ TUESDAY on Elimination Method

39. Lesson 3 Day 3 – 10/13/15 Solving Systems of Equations by Elimination Objective: SWBAT Solve systems of Two- and Three- Variable equations using the Elimination method Keep HW Problems on desk – I will check them!

40. PSAT Prep Warmup
Problem # Take three-four minutes and select an answer choice. These should be short and quick!

41. HW Questions?

42. Secret Message for the PSAT
To find out my secret message, you must solve your systems of equations and write the letter that corresponds to each number listed on the blanks.
Secret Message Hint:
It is an Italian phrase meaning:
“GOOD LUCK!”

43. Only pencils on your desks NO CALCULATORS
QUIZ TIME!
Only pencils on your desks
NO CALCULATORS

44. None – Practice and Rest for the PSAT!
Homework
None – Practice and Rest for the PSAT!

45. PSAT PREP A. 2 B. 4 C. 8 D. 12 E. 16 Answer: B. 4
The volume, V, of the right circular cone with radius r and height h, shown below, can be found using the formula
𝑉 = 1 3 𝜋 𝑟2ℎ. A cone-shaped paper cup has a volume of 142 cubic centimeters and a height of 8.5 centimeters. What is the radius, to the nearest centimeter, of the paper cup?
A.   2
B.   4
C.   8
D. 12
E. 16
Answer: B. 4

46. Lesson 4 Day 1 – 10/15/15 Solving Systems of Inequalities Objective: SWBAT Solve systems of Two-Variable Inequalities By graphing

47. Warmup
1. Solve the system of equations: 2𝑥−𝑦=−3 𝑦=− 1 2 𝑥+1 2. Graph the system of equations above and circle the solution. 3. Graph 2𝑥−𝑦≥−3 4. Graph 𝑦≥− 1 2 𝑥+1

48. Systems of Inequalities
Instead of solving systems of equations, we might encounter systems of inequalities. In this case, let’s see how to solve it! Example 1: 2𝑥−𝑦≥−3 𝑦≥− 1 2 𝑥+1
𝟐𝒙−𝒚≥−𝟑 𝒚≥− 𝟏 𝟐 𝒙+𝟏 𝑻𝒉𝒆 𝒇𝒖𝒍𝒍 𝒔𝒚𝒔𝒕𝒆𝒎
(𝑩𝑶𝑻𝑯)

49. Solution Space to a System of Inequalities
The solution space to a system of inequalities is the area where the inequality solution spaces intersect.
Last Problem (Previous Slide)
A solution space consists of all of the coordinate pairs that are contained within that space.
Solution Space

50. 𝑦≥− 2 3 𝑥+1000 𝑦≤500 Let’s Do This! On your graphing calculators….
Solve the system:
𝑦≥− 2 3 𝑥+1000
𝑦≤500

51. More Practice
Where is the solution space? Label with an S on these pictures. S S S

52. More Practice
(1) 𝑦> 2 3 𝑥+3
𝑦>− 4 3 𝑥−3
(2)
4𝑥+𝑦<2
𝑦<−2
(3)
3𝑥+2𝑦≥−2
𝑥+2𝑦≤2

53. HOMEWORK Complete worksheet 3-6
Watch and take notes on video on Graphing Systems of Linear Inequalities:
Complete worksheet 3-6

54. Lesson 4 Day 2 – 10/19/15 Solving Systems of Inequalities Objective: SWBAT Solve systems of Two-Variable Inequalities By graphing Keep HW Worksheet on Desk

55. Warmup
1. Graph 𝑦≥− 2 3 𝑥+4 2. Graph −3x+𝑦≥−𝑥+1 3. Solve the following system of inequalities − 3 4 𝑥+𝑦>6 3𝑥−4𝑦>12

56. Modeling with Linear Inequalities
Next year’s Heritage Festival is promising to be a good one. We will change it into a fundraiser for the Spanish department. The department wants to raise $30,000! Tickets will be sold for the event at $20 for a bleachers seat and $30 for a reserved chair on the floor. If the school can set up a maximum of 500 chairs on the floor, how many tickets of each seat type must be sold to reach the fundraising goal?

57. Modeling Problem Questions to consider:
1. How are you raising money?
2. Are you guaranteed to sell every seat? What if there is another event happening in the city that night?
3. What are the maximum number of each type of seat you can sell? Do you know this information?
Determine how many of each seat type you could sell and show your calculations

58. You try! You try: “Got it?” Question #3 on page 151 in groups!
We actually just worked on the same problem as page 151 “Problem 3”.
You try: “Got it?” Question #3 on page 151 in groups!
Show your full solution and give examples of the different amounts of toppings you can have.

59. Possible amounts of toppings
Graph: 𝑚≥5−𝑣 𝑚≤− 1 2 𝑣+5 m is the dependent var; v is the independent var.
Possible amounts of toppings
Veggies
Meat 5
1 or 2 4
2, 3, or 4 3
3, 4, 5, or 6 2
4, 5, 6, 7, or 8 1
5, 6, 7, 8, 9, or 10

60. Exit Ticket – half sheet to turn into tray
Page 152 “Lesson Check” #1, 4

61. HOMEWORK
Due Friday–
Page 153 #11-21 ALL, #22-30 ALL

62. Lesson 4 Day 3 – 10/20/15 Solving Systems of Inequalities Objective: SWBAT Solve systems of Two-Variable Inequalities By graphing HW due on thursday

63. Warmup
1. Solve the following system of inequalities − 2 5 𝑥+𝑦>2 4𝑥−2𝑦>6 2. Solve the following system of inequalities 𝑦≥ 𝑥−2 +1 𝑦< − 𝑥− Solve the following system of inequalities 𝑦< 𝑥+3 −2 𝑦> 1 4 𝑥−1

64. Graph: y≥5−𝑥 𝑦≤− 1 2 𝑥+5 y is the meat topping; x is veggie topping.
Possible amounts of toppings
Veggies
Meat 5
1 or 2 4
2, 3, or 4 3
3, 4, 5, or 6 2
4, 5, 6, 7, or 8 1
5, 6, 7, 8, 9, or 10

65. Linear Programming 𝑥+𝑦<8 𝑥≥0 𝑦≥0
What if there are “constraints” on your possible solutions to a problem?
Example: Only positive values, total amount has a maximum of 18, etc.
Example:
𝑥+𝑦<8
𝑥≥0
𝑦≥0
Let’s try this on our graphing calculators!

66. Whiteboard Practice Graph the following Systems of Inequalities
2𝑦−4𝑥≤0
𝑥≥0
𝑦≥0
2. 𝑦≥−2𝑥+4
𝑥>−3
𝑦≥1
3. 𝑦≤ 2 3 𝑥+2
𝑦≥ 𝑥 +2

67. Whiteboard Practice Graph the following Systems of Inequalities
4. 𝑦<𝑥−1
𝑥≤8
𝑦>− 𝑥−2 +1
5. 2𝑥+𝑦≤3
𝑦> 𝑥+3 −2
𝑥≥−3
6. 𝑦< 𝑥−1 +2
𝑥≥0

68. - HONORS: READ pages 157-159 and take notes
HOMEWORK
Due Thursday –
Page 153 #11-21 ALL, #22-30 ALL,
Quiz tomorrow!
- HONORS: READ pages and take notes

69. Lesson 4 Day 4 – 10/22/15 Solving Systems of Inequalities Objective: SWBAT Solve systems of Two-Variable Inequalities By graphing Keep HW on Desk

70. HW Review For your HW problems page 153 #11-30 All,
Post one of the solutions and graphs on the board.
When you finish, walk around with your solutions to see other solutions.
If you find an error and somebody else agrees with you on that error, CHANGE IT!
Go back to your original problem and be sure that your solution is still what you originally thought!

71. Whiteboard Practice (cont’d)
Graph the following Systems of Inequalities
4. 𝑦<𝑥−1
𝑥≤8
𝑦>− 𝑥−2 +1
5. 2𝑥+𝑦≤3
𝑦> 𝑥+3 −2
𝑥≥−3
6. 𝑦< 𝑥−1 +2
𝑥≥0

72. In groups… Page 154 #58 parts a and/or b Try both parts a and b
Try one last problem!
Page 154 #58 parts a and/or b
Try both parts a and b

73. Clear your desks except pencils
Quiz time!
Clear your desks except pencils

74. Homework
Worksheet packet 3-7
Study for Test on Tuesday!
Study Guide Problems are good to use this weekend to help study!

75. HONORS Warmup
In groups of 2 or 3, do the “Getting Ready” problem on page 157
While you may “Guess and Check” to find your solution, how can you do this problem using equations/inequalities and systems?

76. HONORS: Linear Programming (Section 3-4 in text)
Important Terms:
Constraints – Limits on your problem. These are written as inequalities or equations in our problems
Linear Programming – A method for finding a minimum or maximum value of some quantity with the given constraints.
Feasible Region – The graph of the system of constraints that define your possible solutions.
Objective Function – Defines the quantity you are trying to minimize or maximize.
Example Problem: page 158 “Got it?” problem #1

77. HONORS: One more example – on you!
Do page 160 #10

78. Clear your desks except pencils
Quiz time!
Clear your desks except pencils

79. HONORS: Homework
Read through "Problem 2" on page 159 and do "Got it? #2" on page 159, and do page #17-22 All
Study for Test on Tuesday!
Study Guide Problems are good to use this weekend to help study!

80. Lesson 5 Day 1 – 10/26/15 Solving Systems of Equations and Inequalities Objective: SWBAT Solve systems of Two-Variable Equations using Multiple reresentations Keep HW on desk

81. Honors Warmup
You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?

82. Honors Warmup Solution
To organize our information:
10𝑥+20𝑦≤140 → 𝑦≤− 1 2 𝑥+7
6𝑥+8𝑦≤ → 𝑦≤− 3 4 𝑥+9
𝑥≥0, 𝑦≥0
Maximize V = 8X + 12Y
Quantity
Cabinet X
Cabinet Y
Total
Cost
10X
20Y
≤140
Floor Space (sq. ft) 6X 8Y
≤72
Storage Volume (cubic ft) 8X
12Y
V = 8X+12Y

83. Honors Warmup Solution
Maximize V = 8X + 12Y Potential Solutions (Vertices): (0,0)  8(0)+12(0)= 0 (0,7)  8(0)+12(7)= 84 (12,0) 8(12)+12(0)= 96 (8,3)  8(8)+12(3)=100 Since (8,3) gives the highest total volume of storage space, we should buy 8 of Cabinet X and 3 of Cabinet Y.

84. HW Questions. We will not do any full-length problems
HW Questions? We will not do any full-length problems. If you have need of full problems, come to the review session after school! Test Review Questions? ANY Questions about Unit 3? Please raise your hand.

85. This…. Is…. Review JEOPARDY!
Period 1:
Separated into teams of 3, NOT with others at your table.
Periods 2, 4, 6:
Separated into teams of a maximum of 4, NOT with others at your table.
Period 3:
Separated into teams of 4/5, NOT with others from your table.

86. Homework Due Thursday:
Complete the study guide for the test!
Due Thursday:
Periods 1/3 - READ pages and take notes, try pg. 179 #13-17, Odds and check answers in the back of the book
Periods 2/4/6 - Watch video on "Supply and Demand"

87. Lesson 6 Day 1 – 10/29/15 Systems of Equations Extension ObjectiveS: SWBAT Solve systems of Two-Variable Equations in supply-demand problems Honors: SWBAT Solve systems of multi-variable equations using matrices

88. Warmup What is the relationship between supply and demand?
How does price relate to the supply and the demand of a product?
What is the equation of the line containing the points (4,3) and (-6,-2)?

89. Supply and Demand Basics
Supply – the amount of a good your suppliers (or manufacturers) will sell you
Demand – the amount of a good that your buyers want to purchase, the “desire” for a product
Law of Supply and Demand - The law of supply and demand defines the effect that the availability of a particular product and the desire (or demand) for that product has on price.
Generally, if there is a low supply and a high demand, the price will be high.
In contrast, the greater the supply and the lower the demand, the lower the price will be.
Equilibrium- A condition or state in which economic forces are balanced. There is neither a shortage nor a surplus of goods.

90. Let’s see what a famous teacher has to say about these topics…

91. In your Jeopardy teams…
Work through the Activity Sheet 1: Senior Class Buttons
You should complete #1-10 on the activity sheet. Everyone must complete his/her own activity sheet.

92. Test Corrections are Due on Thursday of next week Format: 1. Problem
Homework
Test Corrections are Due on Thursday of next week
Format: 1. Problem
2. “What I did incorrectly was…”
3. Full, corrected solution

93. Honors Warmup Solve the system of equations below: 1 3 𝑥− 4 5 𝑦=10
7 4 𝑥+ 2 9 𝑦=−12

94. THE MATRIX

95. What is a Matrix? How do Matrices work?
A matrix is simply a “chart” or an extended list of numbers and other values
A matrix is defined by how many rows and how many columns there are.
For instance, a 2 x 3 matrix is one with 2 rows and 3 columns
In general, a matrix is defined by R x C horizontal rows and vertical columns

96. Example Dimensions
Example Example Example Example 4

97. Adding/Subtracting Matrices
To add/subtract matrices together, simply add the elements that are in the same corresponding positions in the two matrices.
Example 1:
Addition
Example 2:
Subtraction
Caution: You cannot add/subtract two matrices that have different dimensions

98. Multiplying by a scalar (or “dilating”)
Often, we will want to change matrix values altogether by multiplying by a scalar factor. It is very intuitive; just how you might imagine it.
This is similar to the “distributive property”

99. Multiplying Matrices
To Multiply matrices, you multiply each row of the first matrix by each column of the second.
When multiplying “rows” and “columns”, you multiply each element of the row by the element in the same corresponding order of the column. Sum together the products of the row x column.
Example:
=58, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑡ℎ𝑒 1𝑠𝑡 𝑟𝑜𝑤 𝑏𝑦 1𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛
or element 1,1 in the result 64
139
154

100. Example of Multiplying
Notice that the order in which we multiply matrices matters. 𝐀∙𝐁 𝐢𝐬 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐭 𝐟𝐫𝐨𝐦 𝐁∙𝐀.

101. Systems of Equations as Matrices
Given what we now know about matrix multiplication, we can define our systems of equations as matrices.
Example:
2𝑥+3𝑦=2
−𝑥+5𝑦=−14
Then 2𝑥+3𝑦 −𝑥+5𝑦 = 2 −14 can be written as
− 𝑥 𝑦 = 2 −14

102. Inverse of a matrix Solve the following for B 2 5 −1 3 ∙𝐵= 2 3
2 5 −1 3 ∙𝐵= 2 3
How do we “get rid of” a matrix?
What is the inverse of “matrixing”? Unfortunately, we cannot divide by matrices. We just can’t. End of story.
BUT…. We can find
the inverse of a matrix!

103. Inverse Matrices The inverse matrix of matrix A:
If A= −1 3 , then 𝐴 −1 = 1 2∙3− −1 ∙5 3 −5 1 2
𝐴 −1 = −5 1 2
𝐀 −𝟏 = 𝟑 𝟏𝟏 − 𝟓 𝟏𝟏 𝟏 𝟏𝟏 𝟐 𝟏𝟏

104. Find the Inverse of Matrices….

105. Solutions
𝐴 −1 = 7 −4 −5 3 𝐵 −1 = 𝐴 −1 = −3 − 𝑁𝑜 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑋

106. Honors Homework:
Try worksheet (ALL Problems) with matrix operations – solving equations

107. Lesson 6 Day 2 – 10/30/15 Systems of Equations Extension ObjectiveS: SWBAT Solve systems of Two-Variable Equations in supply-demand problems Honors: SWBAT Solve systems of multi-variable equations using matrices

108. Warmup Let’s review from yesterday…
If the set price is too high, then how will supply and demand react?
If the set price is too low, then how will supply and demand react?
What is the equation of the line containing the points (6,10) and (-6,14)?

109. Continue our study of Economics…
Try Activity Sheet 2!
ALL PROBLEMS
Same groups as yesterday

110. Homework
Write down the 11 Unit 4 vocabulary terms on page 192 and define the words (found on the page number next to the word).
If you cannot find a clear definition on that page, look elsewhere! This is for your benefit.
If the vocab term is something that involves a picture, you should include a picture!
Vocab Quiz – next Thursday, 11/05.

111. Honors Warmup Solve the system of equations below: 1 3 𝑥− 4 5 𝑦=10
7 4 𝑥+ 2 9 𝑦=−12

112. Inverse of a matrix Solve the following for B 2 5 −1 3 ∙𝐵= 2 3
2 5 −1 3 ∙𝐵= 2 3
How do we “get rid of” a matrix?
What is the inverse of “matrixing”? Unfortunately, we cannot divide by matrices. We just can’t. End of story.
BUT…. We can find
the inverse of a matrix!

113. Inverse Matrices The inverse matrix of matrix A:
If A= −1 3 , then 𝐴 −1 = 1 2∙3− −1 ∙5 3 −5 1 2
𝐴 −1 = −5 1 2
𝐀 −𝟏 = 𝟑 𝟏𝟏 − 𝟓 𝟏𝟏 𝟏 𝟏𝟏 𝟐 𝟏𝟏

114. Find the Inverse of Matrices….

115. Solutions
𝐴 −1 = 7 −4 −5 3 𝐵 −1 = 𝐴 −1 = −3 − 𝑁𝑜 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑋

116. Solving an equation with inverse matrices
Solve the following for B
2 5 −1 3 ∙𝐵=
𝐵= −
𝑩= − 𝟗 𝟏𝟏 𝟖 𝟏𝟏
𝟑 𝟏𝟏 − 𝟓 𝟏𝟏 𝟏 𝟏𝟏 𝟐 𝟏𝟏 ∙ 𝟑 𝟏𝟏 − 𝟓 𝟏𝟏 𝟏 𝟏𝟏 𝟐 𝟏𝟏 ∙

117. Solving Systems of Equations using Matrices
Example:
2𝑥+5𝑦=2
−𝑥+3𝑦=3
Set up an equation with matrices to solve:
2 5 −1 3 ∙ 𝑥 𝑦 = 2 3
Using the inverse Matrix 𝐴 −1 , we can solve for x and y.
(See last slide, same problem)
𝒙 𝒚 = − 𝟗 𝟏𝟏 𝟖 𝟏𝟏 , 𝑠𝑜 𝑥=− 𝑎𝑛𝑑 𝑦= 8 11

118. On the Calculator…
It’s easy! There is a “MATRIX” menu to use.
Hit “2nd”, “ 𝑥 −1 ” to go to it.
Notes on the TI-Emulator in class.

119. TEST Corrections due Thursday
Honors Homework
Pg. 179 #12-17 ALL, #24-29 ALL,
Pg. 180 #32-37 ALL,
Pg. 181 #45-47 ALL,
TEST Corrections due Thursday