UNIT 3: SYSTEMS OF EQUATIONS. LESSON 1 DAY 1 – 09/29/15 SOLVING EQUATIONS BY GRAPHING OBJECTIVE:...

UNIT 3: SYSTEMS OF EQUATIONS

LESSON 1 DAY 1 – 09/29/15SOLVING EQUATIONS BY GRAPHING

OBJECTIVE: SWBAT USE GRAPHING TO SOLVE SIMPLE LINEAR EQUATIONS AND SYSTEMS OF LINEAR EQUATIONS

Warmup: Solve the following equations

1.

2.

3.

4.

5.

Graphing Calculators… UNITE!• Let’s explore how to

HOMEWORK – Due Thursday (via email)

1. Finish the guided notes exploration.

2. READ pages 134-137 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

LESSON 2 DAY 1 – 10/05/15SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION

OBJECTIVE: SWBAT SOLVE SYSTEMS OF EQUATIONS USING THE SUBSTITUTION METHOD

Review from Sub days

1.Graphing Packet (THURS)

2.Homework worksheet (FRI)

3.Test Corrections (Due THURS)

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.

Example Problems

Solution: (x,y)=(3,2)

2.

Solution: (x,y) = (1,4), (-1,4)

Step 1

Step 2

Step 3

STEP 4!

Practice!• System 1:

3x + 7y = 14

2x + 7y = 21

• System 2:

4x + 9y = –19

-4x – 7y = 13

• System 3:

-3x + 2y = 23

5x + 2y = –17

Answers:

1.(-7,5)

2.(2,-3)

3.(-5,4)

Homework

•Finish Worksheet #3.2 by using the Substitution method

•Research (online) solving systems of equations in THREE variables

•Test Corrections – Due THURSDAY



KEEP HOMEWORK OUT

Bellwork • Simplify the following expressions

Answers:

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:

1. If the solution to a system is x=4 and y=-2, then…

We write that as

2. If the solution to a system is x=76 and y=100.2, then…

We write that as

Write the following solutions!

1. x = 5 and y = 10

2. x = -3 and y = -8

3. y = 4 and x = 1.3

4. x = -12.3 and y =

Answers:1. (x,y)=(5, 10)2. (x,y)=(-3, -8)3. (x,y)=(1.3, 4)4. (x,y)=(-12.3, )5. (x,y,z)=(100, -12, -34.5)6. (x,y,z)=(-2, -6, 13)7. (x,y,z)=(1, 1, 1)8. (x,y,z,a)=(2, 90, 1, 4)

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:

Example problems:Per. 2, 4, 6: Per. 1, 3:

Practice!• Per. 1, 3

2.

3.

• Per. 2, 4, 6

2.

3.

4.

Sol’ns:

1. (5,-2,-3)

2. (-3,-5,-7)

3. (7,-3,4)

4. (-2,9,-2)

Sol’ns:

1. (-2,-3,3)

2. (-4,0,-2)

3. (-1,-4,-4)

Homework

•Finish Worksheet #3.3 by using the Substitution method

•Test Corrections – Due THURSDAY



TURN IN TEST CORRECTIONS TO THE TRAY PLEASE! KEEP HW #3.3 OUT.

PSAT PREP

Problems #4 and 5

Take three minutes and select an answer choice. These should be short and quick!

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.

Let’s Practice!Algebra 2 Periods 2, 4 ,6

Alg 2 Honors

QUIZ time!

•Clear your desks, except for a pencil

•NO CALCULATOR ALLOWED

Homework

•READ pages 144-145 and take notes on Elimination Method of Solving Systems of Equations, Try "Got It? #3, #4 a,b, and #5 a,b”

LESSON 3 DAY 1 – 10/09/15SOLVING SYSTEMS OF EQUATIONS BY ELIMINATION

OBJECTIVE: SWBAT SOLVE SYSTEMS OF TWO- AND THREE- VARIABLE EQUATIONS USING THE ELIMINATION METHOD

KEEP HW NOTES IN NOTEBOOK

PSAT PREP

Problem #11

Take two minutes and select an answer choice. These should be short and quick!

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:

Example: x

Example:

Example: + 0

𝑦=−1

𝑥=4

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

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.

a. Write a system of equations to model the relationship between the number of multiple choice questions m and extended response questions r.

b. How many of each type of question was on the test?

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



KEEP HW PROBLEMS ON DESK – WILL REVIEW

PSAT Prep Warmup

•Problem #12-13

Take three-four minutes and select an answer choice. These should be short and quick!

HW Questions?

Which Method is Better?• For the following problems, decide which method of solving

the system would work better: graphing, substitution, or elimination?

Elimination Method with Three VariablesSolve the system of equations below using the elimination method. This should look very similar to the video you watched over the weekend at: https://www.khanacademy.org/math/algebra2/systems_eq_ineq/fancier_systems_precalc/v/systems-of-three-variables

Steps (ESPN):

1. “Eliminate” one variable from two equations by adding/subtracting two pairs of original equations

2. Solve the resulting system of equations

3. Plug in those two values to an original equation to solve for the last variable.

4. Notify your brain that your answers are checked.

THE SOLUTION

Solution:

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!

Homework for tonight

•Worksheet 3-5 on the Elimination Method (choose 4 from each section + #22-23)

•QUIZ TUESDAY on Elimination Method



KEEP HW PROBLEMS ON DESK – I WILL CHECK THEM!

PSAT Prep Warmup

•Problem #18-19

Take three-four minutes and select an answer choice. These should be short and quick!

HW Questions?

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!”

QUIZ TIME!

•Only pencils on your desks•NO CALCULATORS

Homework

•None – Practice and Rest for the PSAT!

PSAT PREPThe volume, V, of the right circular cone with radius r and height h, shown below, can be found using the formula . 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

http://www.actstudent.org/sampletest/math/math_02.html





LESSON 4 DAY 1 – 10/15/15SOLVING SYSTEMS OF INEQUALITIES

OBJECTIVE: SWBAT SOLVE SYSTEMS OF TWO-VARIABLE INEQUALITIES BY GRAPHING

Warmup1. Solve the system of equations:

2. Graph the system of equations above and circle the solution.

3. Graph

4. Graph

Systems of InequalitiesInstead of solving systems of equations, we might encounter systems of inequalities. In this case, let’s see how to solve it!

Example 1:

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

Let’s Do This!• On your graphing calculators….

Solve the system:

More Practice• Where is the solution space? Label with an S on these

pictures.

SS

S

More Practice

(2) (3)

(1)

HOMEWORK• Watch and take notes on video on Graphing Systems of

Linear Inequalities:

https://www.khanacademy.org/math/algebra/two-variable-linear-inequalities/graphing-inequalities/v/graphing-systems-of-inequalities-2

•Complete worksheet 3-6



KEEP HW WORKSHEET ON DESK

Warmup

1. Graph

2. Graph

3. Solve the following system of inequalities

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?

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

You try!• 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.

Graph:

m is the dependent var; v is the independent var.

Possible amounts of toppingsVeggies Meat

0 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 0

Exit Ticket – half sheet to turn into tray

•Page 152 “Lesson Check” #1, 4

HOMEWORK

•Due Friday–

Page 153 #11-21 ALL, #22-30 ALL



HW DUE ON THURSDAY

Warmup




Graph:

y is the meat topping; x is veggie topping.

Possible amounts of toppingsVeggies Meat

0 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 0

Linear Programming• 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:

• Let’s try this on our graphing calculators!

Whiteboard Practice• Graph the following Systems of Inequalities

2.

3.

Whiteboard Practice• Graph the following Systems of Inequalities

4.

5.

6.

HOMEWORK

•Due Thursday –

Page 153 #11-21 ALL, #22-30 ALL,

Quiz tomorrow!

- HONORS: READ pages 157-159 and take notes



KEEP HW ON DESK

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!

Whiteboard Practice (cont’d)• Graph the following Systems of Inequalities

4.

5.

6.

In groups…• Try one last problem!

•Page 154 #58 parts a and/or b•Try both parts a and b

Quiz time!

•Clear your desks except pencils

Homework

•Worksheet packet 3-7

•Study for Test on Tuesday!•Study Guide Problems are good to use this weekend to help study!

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?

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

HONORS:One more example – on you!

•Do page 160 #10

Quiz time!

•Clear your desks except pencils

HONORS: Homework

•Read through "Problem 2" on page 159 and do "Got it? #2" on page 159, and do page 160-161 #17-22 All

•Study for Test on Tuesday!•Study Guide Problems are good to use this weekend to help study!

LESSON 5 DAY 1 – 10/26/15SOLVING SYSTEMS OF EQUATIONS AND INEQUALITIES

OBJECTIVE: SWBAT SOLVE SYSTEMS OF TWO-VARIABLE EQUATIONS USING MULTIPLE RERESENTATIONS

KEEP HW ON DESK

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?

Honors Warmup Solution• To organize our information:

• Maximize V = 8X + 12Y

Quantity Cabinet X Cabinet Y Total

Cost 10X 20Y

Floor Space (sq. ft) 6X 8Y

Storage Volume (cubic ft) 8X 12Y V = 8X+12Y

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.

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.

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.

Homework• Complete the study guide for the test!

Due Thursday:• Periods 1/3 - READ pages 174-178 and take notes, try pg. 179 #13-17, 25-29 Odds and check answers in the back of the book

• Periods 2/4/6 - Watch video on "Supply and Demand"

http://www.snagfilms.com/show/we_the_economy/supply_and_dance_man

LESSON 6 DAY 1 – 10/29/15SYSTEMS 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

Warmup

1. What is the relationship between supply and demand?

2. How does price relate to the supply and the demand of a product?

3. What is the equation of the line containing the points (4,3) and (-6,-2)?

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.

http://www.investopedia.com/terms/l/lawofsupply.asp

http://www.investopedia.com/video/play/law-supply-and-demand/

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

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.

Homework

•Test Corrections are Due on Thursday of next week

•Format: 1. Problem 2. “What I did

incorrectly was…”

3. Full, corrected solution

Honors Warmup

1. Solve the system of equations below:

THE MATRIX

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

Example Dimensions

Example 1 Example 2 Example 3 Example 4

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

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”

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:

or element 1,1 in the result

64154139

Example of Multiplying

Notice that the order in which we multiply matrices matters.

Systems of Equations as Matrices• Given what we now know about matrix multiplication, we

can define our systems of equations as matrices.• Example:

Then can be written as

Inverse of a matrix• Solve the following for B

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!

Inverse Matrices• The inverse matrix of matrix A:

• If A= , then

Find the Inverse of Matrices….

Solutions

Honors Homework:

•Try worksheet (ALL Problems) with matrix operations – solving equations

LESSON 6 DAY 2 – 10/30/15SYSTEMS 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

Warmup• Let’s review from yesterday…

1. If the set price is too high, then how will supply and demand react?

2. If the set price is too low, then how will supply and demand react?

3. What is the equation of the line containing the points (6,10) and (-6,14)?

Continue our study of Economics…

•Try Activity Sheet 2!•ALL PROBLEMS•Same groups as yesterday

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.

Honors Warmup

1. Solve the system of equations below:

Inverse of a matrix• Solve the following for B

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!

Inverse Matrices• The inverse matrix of matrix A:

• If A= , then

Find the Inverse of Matrices….

Solutions

Solving an equation with inverse matrices

• Solve the following for B

Solving Systems of Equations using Matrices

• Example:

Set up an equation with matrices to solve:

Using the inverse Matrix , we can solve for x and y.(See last slide, same problem)

On the Calculator…

•It’s easy! There is a “MATRIX” menu to use.

•Hit “2nd”, “” to go to it.

•Notes on the TI-Emulator in class.

Honors Homework

•Pg. 179 #12-17 ALL, #24-29 ALL,

•Pg. 180 #32-37 ALL, •Pg. 181 #45-47 ALL, •TEST Corrections due Thursday

UNIT 3: SYSTEMS OF EQUATIONS. LESSON 1 DAY 1 – 09/29/15 SOLVING EQUATIONS BY GRAPHING OBJECTIVE:...

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