Unit 5 Teacher Notes - It har and not very fun to do as a thingy. PDF

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It har and not very fun to do as a thingy....

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Unit 5 Notes and Objectives: Systems of Equations

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Name: _______________________ Per:___ Day

Date

Assignment (Due the next class meeting) 5.1 Worksheet Solving Linear Systems by Graphing 5.2 Worksheet Solving Linear Systems by Substitution 5.3 Worksheet Solving Linear Systems by Elimination, Part I 5.4 Worksheet Solving Linear Systems by Elimination, Part II 5.5 Worksheet Solving Systems in Different Forms and Modeling 5.6 Worksheet Solving Systems of Linear Inequalities by Graphing Unit 5 Practice Test Unit 5 Test

NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit. Remember if you have no missing assignments there is a 2% bonus to your semester grade. HW reminders:  If you cannot solve a problem, get help before the assignment is due.  Need help? Try www.khanacademy.com or www.classzone.com  Mr. Schulewitch’s Class website https://www.washoeschools.net/Page/9908

5.1 Warm-Up: 1. Graph the line: y = 3x – 4

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2. Graph the line: 3x – 4y = -12

Unit 5 Notes and Objectives: Systems of Equations

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5.1: Solving Linear Systems by Graphing Essential Question: How do you approximate the solution of a system of linear equations by graphing? A ______________ of linear equations consists of __________ or more linear equations in the same variables. In order to _________________ a system of linear equations, we can graph the lines and find the point of _______________________________. Example 1) What is the solution to the system below

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Unit 5 Notes and Objectives: Systems of Equations Examples: Solve each system by graphing. Check your solution. 2)

3)

Check your solution by using substitution with each equation:

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6) y = -x + 3 3|Page

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Unit 5 Notes and Objectives: Systems of Equations

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y = 2x – 5

Reflect: Describe the graphs of y = -5 and x = -3. Explain how to solve the linear system by graphing. What would the graph look like? What is the solution of the linear system?

A system of linear equations can have … One Solution Infinitely Many Solutions

No Solution

8) A system of two linear equations has infinitely many solutions. What must be true about the equations? a) They are perpendicular. b) They are the same line. 4|Page

Unit 5 Notes and Objectives: Systems of Equations

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c) They have the same y-intercept. d) They are parallel. 9) Which of the following equations will have no solutions with 6x + 2y = 8? a) y = -3x + 4 b) y = 6x – 5 c) y = -3x – 1 d) y = 6x + 8

Example 10) Stella has five coins in her wallet, and they are only dimes and quarters. The coins have a total value of $0.95. Write a system of equations to model this situation.

Would you want to solve this system by graphing? Why or why not? Use trial and error to find out how many dimes and how many quarters Stella has in her wallet.

10) The functions and are graphed to the side. Approximate the value when?

Reflection: Describe some disadvantages to solving systems by graphing?

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Unit 5 Notes and Objectives: Systems of Equations

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Example 11: A coffee shop sells teas for $4 each and coffees for $5 each. If the coffee shop sold 9 drinks for a total of $40, how many of each type of drink were sold? 13) Write two equations to model this situation. 14) Solve the system by graphing. Teas = Coffees =

5.2 Warm-up: 1. Line m is shown on the graph below. Construct line b on the graph so that:  line m and line b represent a system of linear equations with a solution of (1, -3 )  line b has a slope between 0 and 1. m

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Unit 5 Notes and Objectives: Systems of Equations

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5.2 Notes: Solving Linear Systems by Substitution Essential Question: How do you use substitution to solve a system of linear equations? If you know the value of one variable in a system, you can find the value of the other variable by _________________________ the known value into one of the equations. Solve each system by using substitution. Check your solution 1)

Sometimes you don’t know a specific numerical value but you do have an equation with one variable ______________________. You can substitute the expression into the other equation. 2)

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Unit 5 Notes and Objectives: Systems of Equations

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5)

6)

7) For the system , what is the most efficient way to isolate one variable? Solve the system.

Word problems: Write a system of equations to model each situation, and then solve. 8) Sam and Troy are best friends. They are both doing track this year. Sam can run 2 feet per second, and Troy can run 3 feet per second. If Sam got a 3 foot head start, how many seconds would it take Troy to catch up? How many feet would they have to run?

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Unit 5 Notes and Objectives: Systems of Equations

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9) Lindsey and Gretchen work at two different hair salons and pay different amounts for their station. Lindsey pays $140 for rent, and $10 per customer that she works on that month. Gretchen only pays $100 for rent, but has to pay $18 per customer. How many customers would it take for them to pay the same amount?

10) Ally goes to the grocery store and buys pounds of peaches and pounds of kiwis. Peaches cost dollars per pound and kiwis cost dollars per pound. The total cost, , that Ally pays for the fruit is represented by the equation: What could represent in the equation above?

11) The distance (in yards) that a golfer hits their driver can be approximated using the formula D = ks, where s is the person’s swing speed in mph and k is a constant determined by the range of the golfers swing speed. Find the units for the coefficient k.

5.3 Warm-Up: 9|Page

Unit 5 Notes and Objectives: Systems of Equations

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1. Which of the following represents a function? (You may choose more than one) A. B. C. x y x y x y 1 1 2 7 2 4 2 2 3 7 3 6 3 3 4 7 3 8 4 4 5 7 5 10 2. Write an equation for #1.C in slope-intercept form.

3. What comes to mind when you hear the word “elimination?”

5.3: Solving Linear Systems by Elimination , Part I Essential Question: How do you solve a system of linear equations by adding? The elimination method eliminates (gets rid of) one of the variables when you ___________ the two equations together. Then you will have an equation that you can ______________.

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Unit 5 Notes and Objectives: Systems of Equations

Steps in solving an equation by elimination: 1) Put both equations in ____________________ form.

2) Find a variable that will be easiest to eliminate.

3) Add vertically.

4) Solve for remaining variable.

5) Substitute that value into an equation, to solve for other variable.

Solve the linear system by the elimination method. 2) 3)

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Example 1: Solve by elimination.

Unit 5 Notes and Objectives: Systems of Equations

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4)

5) In the system, , can you add the two equations to solve for one of the variables? If so, solve the system. If not, what could you do to make a system you could solve?

Sometimes the original system does not have opposite ___________________. You can change any equation by _________________ it by a negative one (or any other number) to make opposite _____________________.

6) 5x – 3y = 19 5x + 4y = 5

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7)

Unit 5 Notes and Objectives: Systems of Equations 8) x + y = 3 x+y=5

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9) 0.25x – 0.05y =1 0.25x + 0.1y = 2.5

10) The Spanish club sells food at the sporting events. At the football game they charge $3 for the popcorn and $1 for the sodas. They made $75 at the football game. At the track meet they sold the popcorn for $2 and the sodas for $1. They made $55 at the track meet. How many bags of popcorn and sodas did they sell, if they sold the same at both games?

Reflect: Explain how you know whether to use elimination versus substitution to solve a system of equations?

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Unit 5 Notes and Objectives: Systems of Equations

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Reflect: Determine which method of solving is easiest for each system? Write “Elimination” or “Substitution.” Do not solve the systems. a)

b)

5.4 Warm-Up: 1. Write the equation of the line in slope intercept form that passes through (0, 5) and (-2, 3).

2. Which line below is parallel to the line y = 3x – 4, and has no solution? A) 3x + y = 4 B) -3x + y = -4 C) -3x + y = 2 D) 3x + y = 2

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Unit 5 Notes and Objectives: Systems of Equations

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5.4: Solving Linear Systems by Elimination, Part II calculator ok Essential Question: How do you solve a system of linear equations by multiplying? Many times the equations in a system aren’t easy to graph, or aren’t easy to use the substitution method and won’t eliminate a variable if you just add the equations. We can use ________________________ with one or both equations so that one variable will eliminate. Steps: 1) Put each line in standard form. 2) Decide which variable you would like to eliminate. 3) Multiply one or both equations to create opposite terms for x and/or y. 4) Add vertically. 5) Solve for one variable. 6) Use substitution and solve for the other variable.

Example 1:

Reflect: In the example on the previous page, can the linear system be solved by elimination without multiplying? Explain your reasoning.

Solve the systems: 2)

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3)

Unit 5 Notes and Objectives: Systems of Equations

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5)

7)

Reflect: Describe the difference between systems with no solution and those with infinitely many solutions? How can you tell which is which?

Word Problems: Example 8: Josie owns a nail shop that charges $12 for a manicure and $20 for a pedicure. Her cousin owns a shop and charges $16 for a manicure and $30 for a pedicure. On Monday they compared how much they made. Josie made $520 and her cousin made $760. If they both sold the same amount of pedicures and manicures, how many pedicures and manicures did they each sell?

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Unit 5 Notes and Objectives: Systems of Equations

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Example 9: A store sells guitars and basses. In one day, a total of 5 instruments were sold. If guitars sell for $200 each and basses sell for $150 each, and the total cost was $900, how many of each type were sold?

5.5 Warm-Up: 1. Evaluate the expression when x = - 2, y = 3, and h = -5.

2. Josh is selling tickets to his band concert. He sold eight tickets, for a total of $14. Students tickets are $1.25 each, and adult tickets are $2.25 each. How many adult tickets did he sell?

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Unit 5 Notes and Objectives: Systems of Equations

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5.5: Solving Systems in Different Forms and Modeling Essential Question: Can you model and solve systems given linear situations? When you are asked to find the solution to a system of equations, what are you looking for?

Work with a partner to see if you can find the solution of the system below where the graph and equation represent two linear situations. Graph of Equation #1 Equation #2 y = 3x - 7

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Unit 5 Notes and Objectives: Systems of Equations

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Examples: 1. The equation and table of values are given to represent a system of linear equations. What is the solution to the system? Equation #1

Table for Equation #2 x y

y = -x +6

-1

9

2

3

3

1

2. The graph and table of values are given to represent a system of linear equations. What is the solution to the system? Graph for Equation #1

Table for Equation #2

Example 3: Can you solve situations with a system? 19 | P a g e

x

y

-2

-5

2

-3

4

-2

Unit 5 Notes and Objectives: Systems of Equations

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The equation and table of values are given to represent a system of linear equations. What is the solution to the system? Table for Equation #1 Equation #2 x y -1

-5

0

-3

2

1

y = 2x - 3

Example 4: Modeling Linear Situations: Try this with a partner! TV Repair by Tomas charges $77 for parts and $31 per hour of labor for a repair job. Tolliver's TV Shop charges a flat fee of $130 for the same job. a. Write a system of equations to represent the situation.

b. Graph the system of equations.

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c. Which shop is cheaper if the job takes 2 hours? 3 hours? 1 hour?

Unit 5 Notes and Objectives: Systems of Equations

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Reflect: What is one other way this problem could be solved?

Sequences review and alternate notation:

Arithmetic Sequences Common difference = d Explicit Recursive

Geometric Sequences Common ratio = r Explicit Recursive

d* *

*Reminder from FLEX day: Geometric Sequence: A list of numbers where you

multiply or divide to get from one term to the next.

Common Ratio: The constant that’s being multiplied or divided in a geometric sequence. Identifying Geometric Sequences: To determine whether a sequence is geometric or not, check for a common ratio. To do this, use division. Note that is the previous term. Function Notation Sequence Notation

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Unit 5 Notes and Objectives: Systems of Equations

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Example 5: Write a recursive and explicit formula for each sequence. (using sequence notation) A)

B)

C) Write an explicit formula for the geometric sequence given and . Assume the common ratio is positive.

D) Write an explicit formula for the geometric sequence given and . Assume the common ratio is positive.

5.6 Solving Systems of Linear Inequalities Warm-Up: 1. Write the equation of the line in (h,k) form that passes through the point (-4,3) and has a slope of 2.

2. Graph 2x – y < 5

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3. Solve:

Unit 5 Notes and Objectives: Systems of Equations

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Essential Question: How do you solve a system of linear inequalities? A _____________________ ___ __________________ consists of two or more linear inequalities that have the same variables. The _________________ of a system of linear inequalities are all the __________________ ___________ that make all the inequalities in the system true.

Example 1) Solve the system of inequalities by graphing. Check your answer.

Reflect: Are the following ordered pairs solutions to the previous system? How could you tell? (0, 0) (2, 3) (-4, 2) (-2, 4) (2, 0) (5, 0) 23 | P a g e

Unit 5 Notes and Objectives: Systems of Equations

More Examples: Graph each system of linear inequalities 2)

Describe the solutions:

3)

Describe the solutions:

4)

Describe the solutions:

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Unit 5 Notes and Objectives: Systems of Equations

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5)

6)

Example 7: Given the system of inequalities shown below, determine all the points that are solutions to this system of inequalities.

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Unit 5 Notes and Objectives: Systems of Equations

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Example 8: The solution set of a system of inequalities is shown in the graph below as a shaded region. The equations of the boundaries are and . Write a system of linear inequalities that could represent the solution.

9)

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Unit 5 Notes and Objectives: Systems of Equations

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10) Penelope is selling bracelets and earrings to make money for summer vacation. The bracelets cost $2 and earrings cost $3. She needs to make at least $600. Penelope knows that she will sell more than 50 bracelets. Use x for #bracelets and y for #earrings. Write a system of inequalities to represent this situation

Graph the two inequalities and shade the intersection

How many bracelets and earrings could Penelope sell?

11)

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