# 28 (Late Start) 7.2a Substitution. 7.1b Graphing with technology Feb 2. 4 (Late Start) Applications/ Choosing a method

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1 Unit 7: Systems of Linear Equations NAME: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make any necessary changes. All assignments are due the following class period unless indicated otherwise. This is a calculator based unit. Monday Tuesday Wednesday Thursday Friday 26 Unit 7 Launch/ Expectations/Pretest a Graphing 28 (Late Start) 7.2a Substitution b Practice b Graphing with technology Feb (Late Start) a Elimination 7.3b Elimination Continued 7.4 Applications/ Choosing a method Review Quiz a Special Cases 7.5b More Applications 6.5 Graphing Linear Inequalities 7.6a Graphing Systems 7.6b More graphing and Review Review Unit 7 UNIT 7 TEST Mid-Winter Break Section Page Assignment 7.1a p. 401 #11-13, 17-23, 25-27, 32, 35 [Graph paper for 8 graphs] 7.2a p. 408 #15-35 odd, 42, b --- Worksheet 7.1b --- Work in the Notes Packet 7.3a p. 414 #8-21, 25-27, all 7.3b --- Worksheet 7.4 p. 421 #13-17 (don t solve), odd, 44, 48, 49, 50 Quiz Review p. 417 & WS Pg 417 #1-10 and Review Worksheet 7.5a p. 429 #12-25, 28, 34, b --- Worksheet 6.5 p. 363 #18-38 even, all, 61-65, [Graph paper needed] 7.6a p. 435 #9-23, 38-39, b --- Worksheet Review

2 Lesson 7.1a Solving Linear Systems by Graphing (by hand) Essential Question: How can you solve a system of linear equations? Algebra 1. Your family opens a bed-and-breakfast. They spend \$600 preparing a bedroom to rent. The cost to your family for food and utilities is \$15 per night. They charge \$75 per night to rent the bedroom. a. Write an equation that represents the costs: b. Write an equation that represents the revenue: c. A set of two or more equations is called a system of linear equations. The two equations you wrote are a system of linear equations for this problem. 2. Use the cost and revenue equations from #1 to determine how many nights your family needs to rent the bedroom before recovering the cost of preparing the bedroom. This is the break even point. X nights C(dollars) R(revenue) a. In the same coordinate plane, graph the cost equation and the revenue equation. b. How many nights does your family need to rent the bedroom before breaking even? c. How does the answer to the question above relate to the graph? Explain. d. How can you solve a system of linear equations? How can you check your answer?

3 1. Graph and check to solve the linear system. x + y = -2 2x 3y = -9 How would you explain how to solve a system of equations by graphing to a friend who has never learned this before? 2. Use the graph and check method to solve the following systems of equations. (a) 2x + y = 2 (b) 5x + 2y = 4 x + y = -1 9x + 2y = You have a total of 18 math and science exercises for homework. You have six more math exercises than science exercises. How many exercises do you have in each subject?

4 Lesson 7.2 Solving Systems of Linear Equations by Substitution Algebra Essential Question: How can you use substitution to solve a system of linear equations? 1. Find the solution by graphing the system. (a) y 4x 2 y 2x 3 (b) What challenges did you encounter while trying to solve this system? USING THE SUBSTITUTION METHOD Given 2. two equations in Standard Form: (Equation 1) 5. (Equation 2) Given two equations in Slope-intercept Form: (Equation 1) (Equation 2)

5 2. Solving More Linear Systems by Substitution Solve the linear system by substitution Practice Problems: Solve each of the following linear systems by substitution. (a) 3x + 2y = -7 (b) x + y = -2 (c) 2x 1 4 y 1 x 3y = 5 2x 3y = -9 4x + y = Solving a System to Linear Models by Substitution At a Royal Oak Divisional Championship swim meet, the Ravens collected \$1590 from 321 people in attendance. If student admission is \$4 and adult admission is \$6, find the number of adults and students that were in attendance to watch Royal Oak s swim team win the championship? 4. Compare and contrast the substitution method and the graphing method for solving systems.

6 Lesson 7.1b Graphing with Technology Algebra 1 Warm-Up Exercise You are considering two payment plans for a Playstation 3 system that you are renting from your local electronics store. Plan A requires an initial fee of \$15, plus a weekly fee of \$3. Plan B requires an initial fee of \$4, plus a weekly fee of \$4. How many weeks would you have to rent the Playstation 3 in order for the two plans to cost the same? Use the graph to justify your answer. Solving Linear System Using a Graphing Utility Quite often, not every solution to a system of equations can be easily identified. Today we will learn how to utilize a graphing utility to deal with cases like these. Example: Consider the system of linear equations: 1. First, you must input your equations in slope-intercept form. (See Diagram 1) 1 2. Adjust the window by pressing ZOOM, and then selecting 6:Zstandard 3. The TRACE feature can give you a good approximation of where the solution is located. (See Diagram 2) 4. The Intersection feature, however, will give you the solution. 2 To use the Intersection feature: a. Go to the CALCULATE Menu by pressing 2nd, TRACE b. Select the 5:Intersect feature (See Diagram 3) c. For First Curve? press enter once your cursor is on Y 1, or line one d. For Second Curve? press enter once your cursor is on Y 2, or line two e. For Guess move your cursor to where you think the lines intersect, then press ENTER 3 - In the example, the solution to the system of linear equation is (, )

7 1. Solving Linear Systems Using a Graphing Utility Solve each system of linear equations using a graphing utility. (a) y x 6 y x 1 (b) y 3x 16 y 4x 9 (c) 3x 2y 4 x 3y 5 Solution: Solution: Solution: 0.25x y 2.25 (d) y 1.25 (e) 3x 9y 8 2x 6y 9 Solution: Solution: 2. Solving Systems of Linear Models In the beginning of the year, Mr. Milazzo and Mr. Hutton each created a Facebook TM profile. In January, the two decided to compare how many times each of their profiles is visited per month. Mr. Milazzo: There are currently 400 monthly visits and the visits are increasing at a rate of 25 visits per month. Mr. Hutton: There are currently 200 monthly visits and the visits are increasing at a rate of 50 visits per month. Is there a point in time where Mr. Milazzo and Mr. Hutton have the same number of visits? If so, which month?

8 Lesson 7.3 Solving Linear Systems Using Elimination (Linear Combination) Essential Question: How can you use elimination to solve a system of linear equations? 1. Given the following system of linear equations: -2x + 2y = -8 3x + 4y = -16 Do you think solving the system by either substitution or graphing would easy? 2. Add : Add the two equations. The use the result to solve the system. a) 3x y = 6 b) -2x y = -6 3x + y = 0 2x - y = 2 3. What about the next system? 2x + y = 7 x+ 5y = Let s Practice Elimination: (by adding the equations together) a) x + 6y = -8 b) 2y - x = 3 x + 6y = -16 x + 3y = 2 c) 2x + y = -7 d) x 2y = 8 -y + 2x = -1 3y = x - 5

9 Elimination (Linear Combination) Using Multiplication Example: -x + 3y = 6 3x + 6y = 12 Process: 5. Elimination ( sometimes called Linear Combination) With Multiplication Solve each of the following systems of linear equation using linear combination a) x + 2y = 3 b) -7x + 2y = 20 3x + 9y = 6-4x + y = 8 c) 3x 2y = -5 d) 2x + 4y = 3-2x + 5y = -4-5y = 3x A business with two locations buys seven large delivery trucks and five small delivery trucks. Location A receives three large trucks and two small trucks for a total cost of \$270,000. Location B receives four large trucks and three small trucks for a total cost of \$375, 000. What is the cost of each type of truck?

10 Lesson 7.4 Choosing the Best Method in Solving Systems of Equations Algebra Methods In Solving Systems of Linear Equations We have now learned three different ways in solving a system of linear equations: These methods are: Substitution: Elimination (Linear Combination): Graphing: *With the people in your table group. Solve the following problems using whatever method you prefer. Tell which method you choose and why. 1. Choosing a Method to Solve the System of Equations Choose the best method to solve the following systems of equations. x y 1 y 2x 3 (a) (b) (c) x y 5 3y 4x 21 2x 3y 7 3x y 5 Method Chosen: Method Chosen: Method Chosen: 2. More Choosing a Method to Solve a System of Equations Choose the best method to solve the following systems of equations. y 6x 18 x 2y 1 (a) (b) y 8x 24 4y 5x 23 Method Chosen: Method Chosen: (c) 1 2 x x y 5 1 y 0 3 Method Chosen:

11 Applications of Linear Systems Algebra Applications of Linear Systems Linear systems can be used in a variety of real-life situations. Use these steps as a guide when trying to setup and solve a system. Step 1 Define your variables. x =, y = State what x and y stand for. Step 2 Write a system (two equations) to represent the situation. Step 3 Solve the system using one of the methods learned in this chapter. 1. You buy 50 tickets at a local fair to ride some of the rides. You are going to ride the roller coaster and the Ferris wheel. If you ride 12 times, using 3 tickets for each Ferris wheel ride and 5 tickets for each roller coaster ride, then how many times did you go on each ride? Assume you used all 50 tickets. Step 1 Define your variables. Step 2 Write your system. Step 3 Solve the system and answer the question. 2. You have a choice between two different cell phone plans. Plan A charges \$25 per month plus \$0.15 per text message. Plan B charges \$50 per month plus \$0.05 per text message. How many text messages would you need to send per month in order for the total bills to be the same? What will be the final bill? 3. A bag contains 30 pieces of candy that includes both jolly-ranchers and tootsie rolls. A piece of candy is drawn at random. The number of tootsie rolls is equal to five times the number of jolly-ranchers. How many jolly-ranchers are in the bag? Follow-up: If you are currently enrolled under Plan A, and you average about 500 texts per month, then should you switch plans? If so, how much money would you save? Follow-up: What is the probability of selecting a jollyrancher from the bag without looking? 4. Terelle Pryor, The Ohio State University phenom quarterback, is returning to campus from his home in Pennsylvania. When he is in Pennsylvania, he drives at an average rate of 55 miles per hour. When he is driving through Ohio, he drives at an average rate of 65 miles per hour. The entire trip was 295 miles and takes 5 hours. How long (in hrs) does he drive in Pennsylvania, and how long does he drive in Ohio?

12 Lesson 7.5 Special Types of Linear Systems Algebra Essential Question: Can a system of linear equations have no solution or infinitely many solutions? Exploration 1: You invest \$450 for equipment to make skateboards. The materials for each skateboard cost \$20. You sell each skateboard for \$20. a) Write the cost and revenue equations. Then complete the table for your cost C and your revenue R. X (skateboards C(dollars) R(revenue) b) When will your company break even? Exploration 2: a) Solve each of the following systems of equations using the method of your choice. 3x 5y 1 x 3y 7 4y 5x 23 9y 3x 21 b) What do you notice about your solutions? c) How can you use the slope and y intercept to determine the number of solutions?

13 1. Solve each system using any method. Describe your solution set. 6y 30x 42 8x 12y 15 (a) (b) y 5x 7 3y 2 2x 2. (a) 14x 35y 49 5y 7 2x (b) 6x 5y 7 18x 15y Use only the slopes and y-intercepts of the graphs of the equations to determine whether the system of linear equations has one solution, no solution, or infinitely many solutions. Explain. a) x 3y 26 4x 42 12y b) 4x 7y 15 8x 15y 31 16y 18x 14 c) 9x 8y 7

14 Lesson 6.5: Graphing Linear Inequalities in a Coordinate Plane Algebra Essential Question How can you graph a linear inequality in a coordinate plane? Review/Warm-Up Exercises Solve the inequality. Graph your solution on a number line. (a) 2y 5 < 7 (b) 7 3x x x (c) Graph and label each of the following lines a. x = 4 b. y = -3 c. x 3 = -1 Exploration: a. Write an equation represented by the dashed line. b. The solutions of an inequality are represented by the shaded region. In words, describe the solutions of the inequality. c. Write an inequality represented by the graph. Which inequality symbol did you use? Explain your reasoning. A solution of a linear inequality in two variables is an ordered pair (x, y) that makes the inequality true. The graph of a linear inequality in two variables shows all the solutions of the inequality in a coordinate plane.

15 Checking solutions of linear inequalities 1. Substitute 2. Evaluate Example: Is (0, 2) a solution to 4x + 5y 12? 1. Checking Solutions of a Linear Inequality Check whether the ordered pairs are a solution to the inequality. (A) 3x y 2 Check (2,3) and (0,0) (B) 2x + y 3 Check (2,2) and (-2,2) Graphing a Linear Inequality In the XY-Plane 1. Graph the corresponding boundary equation. (a) When should a dashed line be used? (b) When should a solid line be used? 2. (a) Test a point (optional) (b) Shade the appropriate region Example: Graph the linear inequality x < -2 (a) What is the boundary equation? (b) Is the line solid or dashed? (c) What point could you test? (d) Did your point work? Which side will you shade? 2. Graphing a Linear Inequality with a horizontal or vertical line Graph the given inequality. (A) y < -3 (B) 6x 30 (C) y + 5 < 2y 1

16 Graphing Linear Inequalities Containing Two Variables A linear inequality is very similar to a linear equation. The only difference is that the equal sign of a linear equation is replaced by an inequality symbol ( >, <, <, > ) Given the following inequality: 2x 3y 6 (a) Rewrite the equation in slope-intercept form. (b) Graph the equation on the coordinate plane (c) Decide which side to shade. How do we know if our line should be dashed or solid? Practice: Graphing Linear Inequalities Graph each of the following linear inequalities below. Optional: Check your results using a test point. (a) x y 2 (b) 5x y 3 (c) 2x 3y 3

17 Application Problem: With two minutes left in a basketball game, the Miami Heat are 12 points behind. The Chosen One decides to take over. What are the combinations of 2-point and 3-point shots Lebron could score to earn at least 12 points? (a) What are the variables in this situation? (b) How can we express how many points Lebron will score? (c) Write a linear inequality for this situation. Which inequality symbol did you choose to use? (d) Graph the inequality then complete the table on the right to show three possible ways for Lebron to complete a comeback victory. 2pts 3pts Lesson 7.6: System of Linear Inequalities Algebra Essential Question How can you graph a system of linear inequalities? Exploration 1 : Match each linear inequality with its graph. Explain your reasoning. 2x y 4 Inequality 1 A. B. 2x y 0 Inequality 2 Exploration 2 : Consider the linear inequalities given in Exploration 1. 2x y 4 Inequality 1 2x y 0 Inequality 2 a. Use two different colors to graph the inequalities in the same coordinate plane. b. What is the result? c. What do you think the overlapping area represents? d. Do you think all systems of linear inequalities have a solution? Explain your reasoning.

18 Core Concepts Graphing a System of Linear Inequalities Step 1 Graph each inequality in the same coordinate plane. Step 2 Find the intersection of the half-planes that are solutions of the inequalities. This intersection is the graph of the system. In Exercises 1 2, tell whether the ordered pair is a solution of the system of linear inequalities. 1. (2, 3); y x 4 y 2x 4 2. (0, 4); y x 4 y 5x 3 3. Graphing Systems of Linear Inequalities Graph the system of linear inequalities (a) x 4 y 3 x y 4 (b) x 3 y 5 x 0 y 6 c) 2x + y > 3-6x + 3y > 9

19 4. Writing a System of Linear Inequalities Write a system of linear inequalities that defines the shaded region below (a) (b) (c) 5. During the summer you take two part-time jobs. The first pays \$5 an hour. The second pays \$8 an hour. You want to each at least \$150 a week and work 25 hours or less a week. (a) Write a system of inequalities that model the hours you can work at each job. (b) Graph the system. (c) List two possible ways to divide your time between the two jobs.

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