Module 11 & 12. Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities

Transcription

1 Module 11 & 12 Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities

2 What is a System of Equations? A system of linear equations consists of two or more linear equations that have the same variables. The solution to a system of linear equations is any order pair that satisfies all of the equations in the system.

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4 Lesson 11.1: Solving Systems by Graphing Essential Question: How can you find the solution of a system of linear equations by graphing?

5 How to Solve by Graphing 1. Graph line 1. Use the x and y intercept method Rearrange equation to slope intercept form 2. Graph line The solution is the point of intersection. 4. Write the solution as an ordered pair (x, y).

6 #1 Solve by Graphing: 2x + y = 6 -x + y = 3

7 # 2 Solve by Graphing: y = 2x 2 3y + 6x =18

8 #3: Solve by Graphing: y = - 2x 2 x + 2y = 2

9 #4: Solve by Graphing: y = 2x x + y = 6

10 Vocab: Solution Types Happens when the slopes are the same, but different y-intercepts. Happens when BOTH equations are exactly the same!

11 #5: Solve by Graphing and identify solution type: y = 2x - 2-2x + y = 4

12 #6: Solve by Graphing and identify solution type: y = -x - 2 x + y + 2= 0

13 #7: Solve by Graphing and identify solution type:

14 Rock and Bowl charges $2.75 per game plus $3 for shoe rental. Super Bowling charges $2.25 per game and $3.50 for shoe rental. For how many games will the cost to bowl be approximately the same at both places? What is that cost?

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16 Video club A charges $10 for membership and $4 per movie rental. Video club B charges $15 for membership and $3 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost? Write a system and solve by graphing.

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18 Lesson 11.2: Solving Linear Systems by Substitution Essential Question: How can you solve a system of linear equations by using substitution?

19 Theory of Substitution Method Substitution is used to reduce the system to one equation that has only one variable. Then you can solve this equation for the one variable and substitute again to find the other variable.

20 How to: Substitution Method Solving Systems of Equations by Substitution Step 1 Step 2 Step 3 Solve for one variable in at least one equation, if necessary. Substitute the resulting expression (from step 1) into the other equation. Solve that new equation to get the value of the first variable. Step 4 Step 5 Substitute the value (from step 3) into one of the original equations and solve for the other variable. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check.

21 1. Solve by Substitution: 3x + y = -3-2x + y = 7

22 2. Solve by Substitution: x - 3y = 9 x + 4y = 2

23 3. Solve by Substitution: 3x + y = 14 2x 6y= -24

24 4. LA Fitness has a $60 enrollment fee and costs $35 per month. 24Hour Fitness has no enrollment fee and costs $45 per month. Let t represent the total cost in dollars and m represent the number of months. Create and solve a system to determine in how many moths the gyms will cost the same. What will be the cost?

25 5. Solve by Substitution: x + y = 4 -x y= 6

26 6. ATT Internet has $100 setup fee and costs $65 per month. Comcast has a setup fee of $40 and costs $70 per month. Let t represent the total amount paid in dollars and m represent the number months. Create and solve a system to determine when (in how many months) the cost will be the same. What will that cost be?

27 7. Solve by Substitution: -3x + y = 12 6x 2y= 18

28 8. A boat travels at a rate of 18 kilometers per hour from its port. A second boat is 34 kilometers behind the first boat when it starts traveling in the same direction at a rate of 22 kilometers per hour to the same port. Let d represent the distance the boats are from the port in kilometers and t represent the amount of time in hours. Create and solve a system to determine how many hours it will take for the second boat to catch up with the first boar. How far will the boats be from the port?

29 Lesson 11.3 & 11.4: Solving Linear Systems by Elimination Essential Question: How can you solve a system of linear equations by using multiplication and elimination?

30 How to Solve by Elimination: 1. Arrange the equations in the same form (standard form or slope-intercept form). 2. Add or subtract to eliminate one variable. Simplify to solve. *****You might need to multiply to create elimination opportunities. 3. Plug in to solve for the other variable. 4. Write the answer as an ordered pair.

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32 1. Solve Using Elimination: 4x 2y = 12 x + 2y = 8

33 2. Solve Using Elimination: 2x + 6y = 6 2x - y = -8

34 3. Solve Using Elimination: 3x + 2y = 5 x + 2y = -1

35 4. Perfect Patios is building a rectangular deck for a customer. According the customer s specification, the perimeter should be 40 meters and the difference between twice the length and twice the width should be 4 meters. Create and solve a system to determine the length and width of the deck.

36 5. Solve Using Elimination: -4x - 2y = 4 4x + 2y = -4

37 6. Solve Using Elimination: 3x + 8y = 7 2x - 2y = -10

38 7. Redbox charges $2 for each video game rented, and $1 for each movie rented. One day last week, a total of 114 video games and moves were rented for a total of $177. Create and solve a system to find the number of video games and movies that were rented.

39 8. Solve Using Elimination: 2x + 3y = -1 5x - 2y = -12

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43 12.1: Creating Systems of Linear Equations Essential Question: How do you use systems of linear equations to model and solve real-world problems?

44 1. Video streaming service Atomic Stream charges $10 for membership and $1 for each download. Blitz Video charges $15 for a membership and $.50 per movie download. How many movies would you need to download for the services to have identical costs? What is that cost?

45 Set Up the System of Equations 2. One cable television provider has a $60 setup fee and charges $80 per month. Another cable provider has a $160 equipment fee and charges $70 per month.

46 Set Up the System of Equations 3. The Strauss family is deciding between two lawncare services. Green lawn charges a $49 startup fee plus $29 per month. Yard Guard charges $25 startup fee plus $37 per month.

47 4. Use the cost table below to create a system of linear equations. Then solve the system to determine when the costs of the two services will be equal.

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49 5. Set Up the System of Equations At how many hours do both companies charge the same?

50 6. Use the graph to make a linear model of each function. Then create the linear system, and state what the solution represents.

51 7. Use the graph to make a linear model of each function. Then create the linear system, and state what the solution represents.

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54 12.2: Graphing Systems of Linear Inequalities Essential Question: How do you solve a system of linear inequalities?

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56 Solving Systems of Linear Inequalities 1. Graph Line A 2. Graph Line B 3. Shading for Line A ***Plug in (0,0).if true shade where (0,0) is if NOT true shade where (0,0) is NOT*** 4. Shading for Line B 5. The solution is where the shading overlaps.

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63 Parallel Lines No Solution: Center Solution

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66 12.3: Modeling with Linear Systems Essential Question: How can you use systems of linear equations or inequalities to model and solve contextual problems?

67 1. Bobby will buy coffee and hot chocolate for his coworkers. Each cup of coffee costs $2.25 and each cup of hot chocolate costs $1.50. If he pays a total of $15.75 for 8 cups, how many of each did he buy?

68 2. A student is buying pens and markers for school. Packs of pens cost $2.75 each and packs of markers cost $3.25 each. If she bought a total of 6 packs and spent $17.50 how many of each did she buy?

69 3. A company has to buy computers and printers. Each computer costs $550 and each printer costs $390. If the company spends $8160 and buys a total 16 machines, how many of each did it buy?

70 4. John has to buy two different kinds of rope. Rope A costs $0.60 per foot and Rope B costs $0.90 per foot. John needs to buy at least 15 feet of rope, but he wants to spend no more than $18. Show and describe all combinations of the number of feet of each type of rope John can buy.

71 5. A student has to buy graph paper and printer paper. The printer paper costs $2 a pack, while the graphing paper costs $3 a pack. She wants to buy at least 6 packs of paper but wants to spend at most $27.

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