Algebra. Chapter 6: Systems of Equations and Inequalities. Name: Teacher: Pd:

Transcription

1 Algebra Chapter 6: Systems of Equations and Inequalities Name: Teacher: Pd:

2 Table of Contents Chapter 6-1: SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear equations in two variables by graphing Pgs: 1 5 HW: Pgs: 6 8 Chapter 6-2: SWBAT: solve systems of linear equations in two variables by substitution Pgs: 9-14 HW: Pgs: Lesson 6-3: SWBAT: solve systems of linear equations in two variables by elimination Pgs: HW: Pgs: Word Problems: SWBAT: Write and solve word problems whose solution requires solving systems of linear equations in two variables Pgs: HW: Pgs: Review Lesson 6-1 to 6-3: SWBAT: Demonstrate their knowledge of solving systems of linear equations in two variables Pgs: Lesson 6-5: SWBAT: graph and solve linear inequalities in two variables Pgs: HW: Pgs: Lesson 6-6: SWBAT: graph and solve systems of linear inequalities in two variables Pgs: HW: Pgs: Review CHAPTER 6 EXAM

3 Chapter 6 1 Solving Systems by Graphing SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear equations in two variables by graphing Warm Up Midterm Review #1 See attached Sheet If two or more equation are given, we call this a system of equations. The solution to a system of equations consists of the set of all ordered pairs, x, y, that satisfy (make true) all of the equations in the system. In today s lesson, we will investigate ways of finding this solution set for two linear equations. Practice: Use the graph below to estimate a solution to the system. Then check your solution algebraically. Solution: (, ) Check 1

4 Practice: Identifying Solutions of Systems Tell whether the ordered pair is a solution of the given system. x2y6 A) (4, 1); B) ( 1, 2); x y 3 2x5y8 3x2y5 x2y 6 xy 3 2x5y 8 3x2y 5 Example 2: Solving Systems of Linear Equations by Graphing All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection. Directions: Solve each system by graphing. Check your answer. C) Check: D) Check: 2

5 Practice: Solving Systems of Linear Equations by Graphing Directions: Solve each system by graphing. Check your answer. 3

6 4

7 Challenge / Regents Problem: Summary: y = 3x + 1 y = -x + 5 Exit Ticket: 5

8 Chapter 6-1 Solving Systems by Graphing HW Tell whether the ordered pair is a solution of the given system. 1) (3, 1); x 3y 6 4x 5y 7 2) (6, 2); 3x 2y 14 5x y 32 x 3y 6 4 x 5y 7 3x 2y 14 5x y 32 Solve each system by graphing. Check your answers. 3) y x 6 y 2x 9 Solution: 4) y x 6 y 3x 6 Solution: Check: Check: 6

9 5) x y 2 2x y Solution: 6) y 2x 6 y 3x 8 Solution: Check: Check: 7) 3x y 4 3x y 7 Solution: 8) Solution: Check: Check: 7

10

11 Chapter 6-2 Solving Systems by Substitution SWBAT: solve systems of linear equations in two variables by substitution Warm Up Midterm Review #2 See attached Sheet Solving Systems of Equations by Substitution Step 1 Solve for one variable in at least one equation, if necessary. Step 2 Substitute the resulting expression into the other equation. Step 3 Solve that equation to get the value of the first variable. Step 4 Substitute that value into one of the original equations and solve. Step 5 Write the values from Steps 3 and 4 as an ordered pair, (x, y). Step 6 Check! 9

12 Practice: Solve the system by substitution. 1) 10

13 Practice: Solve the system by substitution. 2) 11

14 Practice: Solve the system by substitution. 3) 12

15 Sometimes you substitute an expression for a variable that has a coefficient. When solving for the second variable in this situation, you can use the Distributive Property. Word Problems 13

16 Challenge : Solve for x and y, given ABCD is a rectangle. Summary: Exit Ticket: 14

17 Chapter 6-2 Solving Systems by Substitution HW Solve each system by substitution. Check your answers. y x 2 y x 4 1) Solution: 2) Solution: y 4x 1 y x 2 Check: Check: 3) y 3x 1 y 5x 3 Solution: 4) 2x y 6 x y 3 Solution: Check: Check: 15

18 5) 2x y 8 y x 7 Solution: 6) 2x 3y 0 x 2y 1 Solution: Check: Check: 7) Solution: 8) Solution: Check: Check: 16

19

20 Chapter 6-3 Solving Systems by Elimination SWBAT: solve systems of linear equations in two variables by elimination Warm Up Midterm Review #3 See attached Sheet Practice: Solve using Elimination by Addition. 18

21 In some cases, you will first need to multiply one or both of the equations by a number so that one variable has opposite coefficients. This will be the new Step 1. Practice: Solve using Elimination by Multiplication 2. 19

22 Word Problems At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $ Tori bought two pairs of gloves and two hats for $ What were the prices for the gloves and hats? 20

23 Closure 1. Write addition or Multiplication to tell which operation it would be easiest to use to eliminate a variable of the system. Explain your choice. 2. Tell how you can decide whether to use addition or multiplication to eliminate a variable in a system of equations. 21

24 Challenge Problem: Write the equation of a line that contains the point of intersection of the graphs 1 8x 3y = 7 and 10x + 4y = -1 and is perpendicular to the line y x 7. 3 Summary: Exit Ticket: 22

25 Chapter 6-3 Solving Systems by Elimination HW 23

26

27 Systems of Equations Word Problems SWBAT: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables Warm Up Midterm Review #4 See attached Sheet Example 1: Two health clubs offer different membership plans. The graph below represents the total cost of belonging to Club A and Club B for one year. Part A Write an equation to show the cost of each membership plan at Club A and Club B. Club A s Equation: Club B s Equation: Part B What is the number of the month when the total cost is the same for both clubs? Part C What is the total cost when both plans are the same? 25

28 Cost (dollars) 2. Angela is planning to go to an amusement park for the fourth of July. She can either go to Playworld or Fantastic Adventures Amusement Park Costs Playworld KEY 12 Fantastic Adventures Number of rides Part A Write an equation to show how the cost at each amusement park relates to the number of rides. Playworld: Fantastic Adventures: Part B For what number of rides would the cost be the same? rides 26

29 Example 3: At Connie s Couches, a person can rent a couch for $10 a month plus a one-time wear-and-tear fee of $50. At Harry s Homes, the charge is $20 a month and an additional charge of $20 for delivery with no wear-and-tear fee. Part A - If c equals the cost, write an equation representing the cost of the rental for m months at Connie s Couches. Equation: Part B - If c equals the cost, write an equation representing the cost of the rental for m months at Harry s Homes. Equation: Part C - On the accompanying grid, graph and label each equation. Part D - From your graph, determine in which month Harry s cost will equal Connie s cost. 27

30 Cost (dollars) 4. Angela wants to go to an amusement park for the 4 th of July. She can go to Playworld for $12 a day plus $2 per ride or Great Action Park for $2 a day plus $ 4 per ride. PART A Write an equation that represents the cost (c) of going to each amusement park in relationship to the number of rides (r). Playworld: Great Action Park: PART B: On the accompanying grid, graph each equation. 28 Amusement Park Costs Number of rides PART C For what number of rides would the cost be the same? PART D For what range of rides would you visit Playworld? For what range of months would you visit Great Action Park? 28

31 Word Problems 5. The total attendance at a school play was 850. The tickets for senior citizens were $1.50 each, and the regular tickets were $2.00 each. If the total receipts were $1, , how many tickets of each kind were sold? 6. Troy has 25 coins in dimes and nickels. The value of his coins is $1.60. How many dimes and nickels does he have? 29

32 Challenge Problem: Summary: Exit Ticket: 30

33 Cost (dollars) Homework Word Problems 1. The accompanying diagram represents the monthly cost of exercising at two local sports clubs. Monthly Sports Club Costs (x KEY NY Sports World Platinum Gym Number of months Part A Write an equation to show the cost of each membership plan at NY Sports World and Platinum Gym. NY Sports World s Equation: Platinum Gym s Equation: Part B What is the number of the month when the total cost is the same for both gyms? Part C In what month will the total cost for both gyms be the same? 31

34 Cost (dollars) 2. Taylor s Department Store sells CDs for $15.00 each. Buyer s Warehouse sells each CD for $10 each and has a $25 membership fee. Part A: Write an equation that represents the cost (c) of going to each store in relationship to the number of CDs bought. Taylor s: Buyer s: Part B: Make a graph that shows the cost of purchasing several different quantities of CDs at each store. 105 CD Costs Number of CDs 6 7 Part C: How many CDs would Dee have to purchase so that the cost is the same for Buyer s Warehouse and Taylor s Department Store? Answer: CDs 32

35 3. The Town Recreation Department ordered a total of 100 balls and bats for the summer baseball camp. Balls cost $4.50 each and bats cost $20.00 each. The total purchase was $ How many of each item were ordered? 4. In a store a total of 70 hammers and wrenches were sold. Hammers sold for $10.00 and wrenches sold for $5.00. A total of $ were sold. How many hammers and wrenches were sold? 5. Juan has 11 coins in dimes and quarters. The value of his coins is $2.15. How many dimes and quarters does he have? 33

36 Chapter 6-1 to 6-3 Review SWBAT: Demonstrate their knowledge of solving systems of linear equations in two variables 34

37 35

38 36

39 Word Problems 21. A baseball manager bought four bats and nine balls for $ On another day, he bought three bats and twelve balls at the same prices and paid $ How much did he pay for each bat and each ball? 22. Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have? 37

40 23. At Ron s Rental, a person can rent a big-screen television for $10 a month plus a one-time wear-and-tear fee of $100. At Josie s Rental, the charge is $20 a month and an additional charge of $20 for delivery with no wear-and-tear fee. a) If c equals the cost, write one equation representing the cost of the rental for m months at Ron s Rental and one equation representing the cost of the rental for m months at Josie s Rental. b) On the accompanying grid, graph and label each equation. c) From your graph, determine in which month Josie s cost will equal Ron s cost. 38

41 Chapter Graphing Linear Inequalities SWBAT: graph and solve linear inequalities in two variables Warm Up Directions: Graph each inequality. Solve for y: 6x + 2y = 4 What is a Linear Inequality? Which of the following points is a solution to the inequality above? (2, 1) (-1, -5) (-3, 2) (0, -2) Example 1: Identify Solutions of Inequalities Determine if the ordered pair is a solution of the inequality. A) (7, 3); y x 1 B) (4,5) y 3x 2 y x 1 Practice: Identify Solutions of Inequalities 39

42 A linear inequality describes a region of a coordinate plane called a half-plane. All points in the region are solutions of the linear inequality. The boundary line of the region is the graph of the related equation. Graphing Linear Inequalities Step 1: Solve the inequality for y (slope-intercept form). Step 2: Graph the boundary line. Use a solid line for or. Use a dashed line for < or >. Pick a point and plug it into the inequality to determine what area needs to be shaded. Step 3: Shade the region above the line for y > or. Shade the region below the line for y < or. Step 4: Check your answer. Example 2: Graphing Linear Inequalities in Two Variables C) Graph the solutions of each linear inequality. y 3x 4 Step 1: Solve for y. Step 2: Graph the boundary line. (Solid or dashed) Step 3: Shade the half-plane. Step 4: Check by plugging in a point in the shaded region. 40

43 Practice: Graphing Linear Inequalities in Two Variables Graph the solutions of each linear inequality. 4) y 2x 1 3 5) y x 2 5 6) y 3 41

44 Example 3: Writing Linear Inequalities from a Graph y D) E) y x x Practice: Writing Linear Inequalities from a Graph Challenge / Regents Problem: Graph the inequality below. Challenge Problem 42

45 Summary: Exit Ticket: 43

46 Chapter Graphing Linear Inequalities Homework Tell whether the ordered pair is a solution of the given inequality. 1) (1, 6); y x 6 2) ( 3, 12); y 2x 5 3) (5, 3); y x 2 Graph the solution of each linear inequality. Check your answer. 4) y x 4 5) 2x y 2 Check: Check: 44

47 6) x y 1 0 7) 2y 3x 6 Check: Check: Write an inequality to represent each graph. 8) 9) 10) 45

48 Chapter 6-6 Solving Systems of Linear Inequalities SWBAT: graph and solve systems of linear inequalities in two variables Warm Up Graph the solution of 4x 3y 12 What is a System of Inequalities?? How do we know if a point is a solution to the inequality?? Which of the following points is a solution to the system above? (2, 7) (-1, -5) (2, 1) (-5, -2) 46

49 Practice: Identify Solutions of Systems of Linear Inequalities To show all the solutions of a system of linear inequalities, graph the solutions of each inequality. The solutions of the system are represented by the overlapping shaded regions. Below are graphs of Examples 1 and 2. Example #1 Example #2 Ex ample 2: Solving a System of Linear Inequalities by Graphing Graph the systems of inequalities. Give two ordered pairs that (a) are solutions (b) are not solutions. y 2x 4 y x 1 C) D) 47

50 48 Practice: Solving a System of Linear Inequalities by Graphing Graph the systems of inequalities. Give two ordered pairs that (a) are solutions (b) are not solutions. 2a) 2b) Example 3: Solving a System of Linear Inequalities by Graphing Graph the systems of inequalities. Give two ordered pairs that (a) are solutions (b) are not solutions. E) y x x y F) x y y x G) H) y x x y x y x y 2 1

51 Challenge / Regents Problem: Summary: Exit Ticket: 49

52 50 Graphing Linear Inequalities Systems Homework Tell whether the ordered pair is a solution of the given inequality. 1) (2, 2); 3 1 y x y x 2) (2, 5); 2 2 x y x y 3) (1, 3); x y x y Graph the system of linear inequalities. a) Give two ordered pairs that are solutions. b) Give two ordered pairs that are NOT solutions. 4) x y x y 2 4 5) y x x y a) a) b) b)

53

54