Systems of Equations and Inequalities

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1 1

2 Systems of Equations and Inequalities

3 Table of Contents Solving Systems by Graphing Solving Systems by Substitution Solve Systems by Elimination Choosing your Strategy Solving Systems of Non linear Equations Writing Systems to Model Situations Solving Systems of Inequalities Linear Programming Systems of Equations with 3 Variables 3

4 Solving Systems of Equations Goals and Objectives Students will be able to solve systems of equations using graphing, substitution and elimination. The systems include both linear and non linear equations. 4

5 Solving Systems of Equations Why do we need this? Solving systems of equations is found everywhere. From finding the appropriate ticket prices to maximize profit, to working with different doses of medication, solving systems is a crucial topic in many different fields. 5

6 Solving Systems of Equations Definition: A system is two or more equations or inequalities. The solution to a system of equations are the points where the equations intersect. There could be no solution, one solution or many solutions. We use ordered pairs to denote the solutions. 6

7 Solving Systems of Equations To start, we will review working with linear systems. 7

8 Solving Systems of Equations by Graphing Solving Systems by Graphing Return to Table of Contents 8

9 Solving Systems of Equations by Graphing Consider this... Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? 9

10 Solving Systems of Equations by Graphing First, make a table to represent the problem. 10

11 Solving Systems using Graphing Next, plot the points on a graph Blocks Time (min.) 15 11

12 Solving Systems using Graphing The point where they intersect is the solution to the system. Write the answer in a complete sentence. 12

13 Solving Systems using Graphing Solve the system of equations graphically. y = 2x 3 y = x 1 13

14 Solving Systems using Graphing Solve the system of equations graphically. 2x + y = 3 x 2y = 4 14

15 Solving Systems using Graphing Solve the system of equations graphically. 3x + y = 11 x 2y = 6 15

16 Solving Systems using Graphing Find the equations for the lines: What is the point of intersection? 16

17 Solving Systems using Graphing Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines. y = 3x 1 ( 1, 2) y = 4x

18 Solving Systems using Graphing Consider these lines: y = 2x + 6 and y = 2x 3 What is the solution? 18

19 Solving Systems using Graphing Now, consider 2x + y = 5 and 2y = 4x To graph them, solve both for y. 19

20 Solving Systems using Graphing 1 Solve the system by graphing. y = x + 4 y = 2x +1 20

21 Solving Systems using Graphing 2 Solve the system by graphing. y = 0.5x 1 y = 0.5x 1 21

22 Solving Systems using Graphing 3 Solve the system by graphing. y = 3x + 4 4y = 12x

23 Solving Systems using Graphing 4 Solve the system by graphing. 2x + y = 3 x 2y = 4 23

24 Solving Systems using Graphing 5 Solve the system by graphing. y = 3x + 3 y = 3x 3 24

25 Solving Systems using Substitution Solving Systems using Substitution Return to Table of Contents 25

26 Solving Systems using Substitution Solve the system of equations graphically. y = x y = 2x

27 Solving Systems using Substitution Graphing can be inefficient or approximate. Another way to solve a system is to use substitution. Substitution allows you to create a one variable equation which you can use to find both of the coordinates, x and y. 27

28 Solving Systems using Substitution Now try solving the same system using substitution. y = x y = 2x

29 Solving Systems using Substitution Solve the system using substitution. Which variable will you substitute? y = 2x y + 3x = 21 29

30 Solving Systems using Substitution Solve the system using substitution. x = 5y 39 x = y 3 30

31 Solving Systems using Substitution You will choose which variable to substitute based on how the system is set up. Select which variable you would substitute in each system below. Explain why. y = 4x 9.6 y = 2x + 9 y = 4x + 1 x = 4y + 1 y = 3x 7x y = 42 31

32 Solving Systems using Substitution 6 Examine the system of equations. Which variable would you substitute? 2x + y = 5 2y = 10 4x A B x y 32

33 Solving Systems using Substitution 7 Examine the system of equations. Which variable would you substitute? 2y 8 = x y + 2x = 4 A B x y 33

34 Solving Systems using Substitution 8 Examine the system of equations. Which variable would you substitute? x y = 20 2x + 3y = 0 A B x y 34

35 Solving Systems using Substitution Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: 3x y = 5 2x + 5y = 8 35

36 Solving Systems using Substitution Let's apply this... Your class of 22 is going on a trip. You need to rent two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip? Write your answer in a complete sentence. Let v = the number of vans and c = the number of cars 36

37 Solving Systems using Substitution Solve this system using substitution. x + y = 6 5x + 5y = 10 37

38 Solving Systems using Substitution Solve the system: x + 4y = 3 2x + 8y = 6 38

39 Solving Systems using Substitution How can you quickly decide the number of solutions a system has? One Solution: No Solution: Infinitely Many Solutions: 39

40 Solving Systems using Substitution 9 Decide if this system has one solution, no solutions or infinitely many solutions. 3x y = 2 y = 3x + 2 A B C One solution No solution Infinitely many solutions 40

41 Solving Systems using Substitution 10 Decide if this system has one solution, no solutions or infinitely many solutions. 3x + 3y = 8 y = x A One Solution B No Solution C Infinitely Many Solutions 41

42 Solving Systems using Substitution 11 Decide if this system has one solution, no solutions or infinitely many solutions. y = 4x 2x 0.5y = 0 A One Solution B No Solution C Infinitely Many Solutions 42

43 Solving Systems using Substitution 12 Decide if this system has one solution, no solutions or infinitely many solutions. 3x + y = 5 6x + 2y = 1 A One Solution B No Solution C Infinitely Many Solutions 43

44 Solving Systems using Substitution 13 Decide if this system has one solution, no solutions or infinitely many solutions. y = 2x 7 y = 3x + 8 A One Solution B No Solution C Infinitely Many Solutions 44

45 Solving Systems using Substitution 14 Solve the system using substitution. y = x 3 y = x + 5 Choices for x: Choices for y: A 4 B 4 C 1 D 1 E 8 F 4 G 4 H 1 I 1 J 8 45

46 Solving Systems using Substitution 15 Solve the system using substitution y = x 6 y = 4 Choices for x: Choices for y: A 4 B 4 C 10 D 10 E 2 F 2 G 4 H 4 I 10 J No solution 46

47 Solving Systems using Substitution 16 Solve the system using substitution. y +2x = 14 y = 2x + 18 Choices for x: Choices for y: A 2 B 2 C 4 D 8 E 8 F 2 G 2 H 4 I 8 J 8 47

48 Solving Systems using Substitution 17 Solve the system using substitution. 4x = 5y + 50 x = 2y 7 Choices for x: Choices for y: A 2 B 5 C 6 D 6.5 E 10 F 2 G 5 H 6 I 6.5 J 10 48

49 Solving Systems using Substitution 18 Solve the system using substitution. y = 3x + 23 y + 4x = 19 Choices for x: Choices for y: A 6 B 5 C 5 D 4 E 3 F 3 G 3 H 4 I 5 J 5 49

50 Solving Systems using Elimination Solving Systems using Elimination Return to Table of Contents 50

51 Solving Systems using Elimination When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination. Elimination involves adding or subtracting the equations to eliminate a variable. 51

52 Solving Systems using Elimination You can eliminate either variable. The quickest method is to look for variables that have the same coefficients either positive or negative. 4x 5y = 10 4x y = 0 3x + 2y = 7 5x + 2y = 4 52

53 Solving Systems using Elimination When the coefficients are the same, but opposite signs of each other, you can just add them to eliminate a variable. 4x 5y = 10 4x y = 0 53

54 Solving Systems using Elimination When the coefficients are exactly the same, you can subtract them to eliminate a variable. To avoid minor mistakes, you can also multiply one equation by 1 and add. 3x + 2y = 7 5x + 2y = 4 54

55 Solving Systems using Elimination Solve using elimination. 5x + y = 44 4x y = 34 55

56 Solving Systems using Elimination Solve using elimination. 3x + y = 15 3x 3y = 21 56

57 Solving Systems using Elimination Solve using elimination. 5x + y = 17 2x + y = 4 57

58 Solving Systems using Elimination Solve the system by elimination. 4x + 3y = 16 2x 3y = 8 58

59 Solving Systems using Substitution 19 Solve the system using elimination: x + y = 6 x y = 4 Choices for x: Choices for y: A 5 F 5 B 5 G 5 C 1 H 1 D 1 I 1 E 10 J No Solution 59

60 Solving Systems using Substitution 20 Solve the system using elimination. 2x + y = 5 2x y = 3 Choices for x: Choices for y: A 1 B 1 C 2 D 2 E 8 F 1 G 1 H 2 I 2 J Infinitely many solutions 60

61 Solving Systems using Substitution 21 Solve the system using elimination. 2x + y = 6 3x + y = 10 Choices for x: Choices for y: A 2 F 2 B 2 G 2 C 4 H 4 D 4 I 4 E 8 J 8 61

62 Solving Systems using Substitution 22 Solve the system using elimination. 4x y = 5 x y = 7 Choices for x: Choices for y: A 2 F 2 B 2 G 2 C 4 H 4 D 4 I 4 E 11 J 11 62

63 Solving Systems using Substitution 23 Solve the system using elimination. 3x + 6y = 48 5x + 6y = 32 Choices for x: Choices for y: A 2 F 2 B 2 G 2 C 7 H 7 D 7 I 7 E 10 J 10 63

64 Solving Systems using Elimination If none of the coefficients match, you will need to multiply one, or both, equations by a number in order to create a common coefficient. 2x + 5y = 1 x + 2y = 0 64

65 Solving Systems using Elimination Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? 2x + 3y = 1 3x + 8y = 81 3x + 6y = 6 3x + 6y = 0 5x 6y = 39 2x 3y = 4 65

66 Solving Systems using Elimination Solve this system by eliminating the y's. 3x + 4y = 10 5x 2y = 18 66

67 Solving Systems using Elimination Now, solve the same system by eliminating the x's. 3x + 4y = 10 5x 2y = 18 67

68 Solving Systems using Elimination 24 Which variable can you eliminate with the least amount of work? 9x + 6y = 15 A x 4x + y = 3 B y 68

69 Solving Systems using Elimination 25 Which variable can you eliminate with the least amount of work? 3x 7y = 2 A x 6x + 15y = 9 B y 69

70 Solving Systems using Elimination 26 Which variable can you eliminate with the least amount of work? x 3y = 7 A x 2x + 6y = 34 B y 70

71 Solving Systems using Elimination 27 What will you multiply the first equation by in order to solve this system using elimination? 2x + 5y = 20 3x 10y = 37 After you have answered, solve it. 71

72 Solving Systems using Elimination 28 What will you multiply the first equation by in order to solve this system using elimination? 3x + 2y = 19 x 12y = 19 After you have answered, solve it. 72

73 Solving Systems using Elimination 29 What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 3x + 4y = 2 After you have answered, solve it. 73

74 Choose your strategy Choose Your Strategy Return to Table of Contents 74

75 Choose your strategy Good math strategies make work go quickly. If you take the time to choose the method best suited for the system, you will be efficient. Give pros and cons of each method. Graphing Substitution Elimination Pros Cons Pros Cons Pros Cons 75

76 Choose your strategy 30 What method would you choose to solve the system? 4s 3t = 8 t = 2s 1 A B C graphing substitution elimination 76

77 Choose your strategy 31 Now, solve the system. 4s 3y = 8 t = 2s 1 Choices for x: Choices for y: A 1/2 B 1/2 C 2 D 2 E 9 F 1/2 G 1/2 H 2 I 2 J 9 77

78 Choose your strategy 32 What method would you choose to solve the system? A graphing y = 3x 1 y = 4x B C substitution elimination 78

79 Choose your strategy 33 Now, solve the system. y = 3x 1 y = 4x Choices for x: Choices for y: A 1/4 B 1 C 1 D 4 E 4 F 1/4 G 1 H 1 I 4 J 4 79

80 Choose your strategy 34 What method would you choose to solve the system? A B C graphing substitution elimination 3m 4n = 1 3m 2n = 1 80

81 Choose your strategy 35 What method would you choose to solve the system? A B graphing substitution y = 2x y = 0.5x + 3 C elimination 81

82 Choose your strategy 36 What method would you choose to solve the system? A B graphing substitution 2x y = 4 x + 3y = 16 C elimination 82

83 Choose your strategy 37 What method would you choose to solve the system? A B graphing substitution u = 4v 3u 3v = 7 C elimination 83

84 Choose your strategy Solving Systems of Non linear Equations Return to Table of Contents 84

85 Solving Systems of Non linear Equations Not everything we graph or use is a linear equation. In fact, a lot is non linear. We will look at solving these situations both graphically and algebraically. 85

86 Systems of Non linear Equations Consider the system: y = x y = x + 3 Solve it Graphically 86

87 Systems of Non linear Equations Solve it Algebraically. To do so, use substitution. y = x y = x

88 Systems of Non linear Equations Solve the system Algebraically: y = x 3 + 5x y = x

89 Systems of Non linear Equations Now try graphically: y = x 3 + 5x y = x How do you find the solutions? 89

90 Systems of Non linear Equations What are the pro's and con's of solving non linear systems graphically and algebraically? When would you use one over the other? Graphically Algebraically Pros Cons Pros Cons 90

91 Systems of Non linear Equations Solve the system Algebraically. Use substitution. Solve. Then, check with a graph. 91

92 Systems of Non linear Equations Check Graphically: 92

93 Systems of Non linear Equations Solve Algebraically: 93

94 Systems of Non linear Equations Here is the graph. What do you notice about the solution? 94

95 Systems of Non linear Equations Remember to check for extraneous solutions!! 95

96 Systems of Non linear Equations Solve. Can it be done Algebraically? y = (x 3) y = x 1 96

97 Systems of Non linear Equations 38 Solve: y = x 2 3x 4 y = x 2 + 2x + 3 Choices for x: Choices for y: A 0 F 0 B 1 G 1 C 7/2 H 7/2 D 9/4 I 9/4 E 5/6 J 5/6 97

98 Systems of Non linear Equations 39 Solve: y = 2x 3 x y = x + 1 Choices for x: Choices for y: A 0.5 B 0.5 C 0 D 1 E 2 F 0.5 G 0.5 H 0 I 1 J 2 98

99 Systems of Non linear Equations 40 Solve: Choices for x: Choices for y: A 1 B 1 C 2 D 2 E 3 F 1 G 1 H 2 I 2 J 6 99

100 Systems of Non linear Equations 41 Solve: y = ln(x + 2) y = x 2 1 Choices for x: Choices for y: A 1 B 0 C 1 D 1.25 E 1.5 F 1 G 0 H 1 I 1.25 J

101 Systems of Non linear Equations 42 Solve the following system for EXACT values of x: A x = 2 and x = 4 B x = 0.5 and 2 C x = D x = E No solution 101

102 Modeling Situations Modeling Situations Return to Table of Contents 102

103 Modeling Situations Altogether, 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470. Let a = adults s = students Create a system and solve it. Which method (graphing, substitution or elimination) are you going to use? 103

104 Modeling Situations A tennis ball is shot out of a small cannon at the same time that an apple is shot out of an air gun. The tennis ball and the apple travel according to the system below. How many seconds after they are shot will they hit? Tennis Ball Apple 104

105 Modeling Situations A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. At the same time, a piece of copper with an initial temperature of 0 F is heated at a rate of 2.5 F/min. Let m = the number of minutes and T = the temperature in F. Write a system that models the given information. At what point will their temperatures match? 105

106 Modeling Situations This graph represents the system below. If it is a model of two objects and their travel in the air, with y = 0 representing the ground, at what point do they hit each other? 106

107 Modeling Situations 43 Your class receives $1084 for selling packages of greeting cards and gift wrap. You sold the packages of cards for $4 each and gift wrap for $6 each. There was a total of 205 items sold. Set up a system to model this situation. How many packages of cards were sold? 107

108 Modeling Situations 44 You have 15 coins in your pocket that are either quarters or nickels. They total $.95. Set up a system to solve. How many quarters do you have? 108

109 Modeling Situations 45 A group of friends is standing on a porch 5 feet above the ground. You launch a water balloon at them that travels on the function below. How many seconds after you launch the water balloon will it hit the porch? water balloon y = x 2 + 6x 109

110 Solving Systems of Inequalities Solving Systems of Inequalities Return to Table of Contents 110

111 Solving Systems of Inequalities Two or more inequalities form a system of inequalities. Since inequalities have more than one solution, answers are best shown on graphs. 111

112 Solving Systems of Inequalities Graphing a Systems of Inequalities 1. Graph the boundary lines for each inequality. (Remember use a dashed line for < and > and a solid line for < and >) 2. Shade the associated section for each inequality. 3. The intersection of the shaded areas is the solution. 112

113 Solving Systems of Inequalities Solve the system of inequalities. x + 2y < 6 x + y < 0 113

114 Solving Systems of Inequalities Solve the system of inequalities. 2x + y > 4 x 2y < 4 114

115 Solving Systems of Inequalities Solve: y 5 y x 2 + 6x 115

116 Solving Systems of Inequalities Solve: 116

117 Solving Systems of Inequalities Solve: 117

118 Solving Systems of Inequalities 46 Solve the system of linear inequalities. y > 2x + 1 y < x + 2 A B C 118

119 Solving Systems of Inequalities 47 Solve the system of linear inequalities. x > 2 y < 5 A B C 119

120 Solving Systems of Inequalities 48 Solve the system of linear inequalities. 2x 2y < 4 y 2x > 1 A B C 120

121 Solving Systems of Inequalities 49 Solve the system of Inequalities. y = x 2 2x + 1 y = x 2 + 2x + 1 A B C 121

122 Solving Systems of Inequalities 50 Solve the system of Inequalities: A B y x 2 y > 0.5x C 122

123 g Systems of Inequalities 51 Solve the system of Inequalities: y x 3 y 2x A B C 123

124 Linear Programming Linear Programming Return to Table of Contents 124

125 Linear Programming Solve the system of inequalities y 1 x

126 Linear Programming Linear programming takes the solution of the system of inequalities and uses the vertices of the region to minimize or maximize an equation. }system } Equation to be maximized or minimized 126

127 Linear Programming Put coordinates of the vertices in this column Find P(x, y) here 127

128 Linear Programming Graph the following system of inequalities. Use the vertices of the system to find the maximum and minimum values for the graph. 0 < y < 5 1 < x < 4 C(x, y) = 3x + 2y 128

129 Linear Programming 52 Given the following vertices of a system, which choice will maximize the function? f(x, y) = 3x 4y A (1, 0) B (3, 0) C (4, 5) D (2, 4) 129

130 Linear Programming 53 Given the following vertices of a system, which choice will minimize the function? f(x, y) = 3x 4y A (1, 0) B (3, 0) C (4, 5) D (2, 4) 130

131 Linear Programming 54 Given the following function, which choice will maximize the function? f(x, y) = 2xy A (1, 0) B (3, 0) C (4, 5) D (2, 4) 131

132 Linear Programming 55 Given the following function, which choice will minimize the function? f(x, y) = 2xy A (1, 0) B (3, 0) C (4, 5) D (2, 4) 132

133 Linear Programming Graph the following system of inequalities. Use the vertices of the system to find the maximum and minimum values for the graph. 0 < y 0 < x 4x + 6y < 36 C(x, y) = 4x 2y 133

134 Linear Programming Graph the following system of inequalities. Use the vertices of the system to find the maximum and minimum values for the graph. 0 < y 0 < x 3x + 6y 18 2x + 2y < 10 C(x, y) = 10x + 8y 134

135 Linear Programming Write a system of inequalities and a profit equation to model the given situation. Answer the question using linear programming. A very small company makes decorative vases and pots for plants. Since they are hand made, during one shift the workers can only produce at most 9 vases and at most 8 pots. If they sell for the same price of $100, what is the maximum profit the company can expect from one shift? Use x = vases and y = pots. 135

136 Linear Programming A store that wants to try selling the decorative vases and pots wants to make a display. They also sell other plants and supplies, so they want to keep their display of vases and pots to a total of 120 square feet. The vases take up 2 square feet, and the pots take up 4 square feet. The store sells the vases for $400 and the pots for $300. How many of each should the store purchase and display to maximize their profit? They are going to buy at least 4 vases and 4 pots. Write a system and a profit equation that models the given situation. Solve. 136

137 Linear Programming 56 Given the graph of the feasible region and P(x,y), which choice will maximize the function? y D P(x,y) = 10x + 7y A 3 2 C 1 B x 137

138 Linear Programming 57 Given the graph of the feasible region and P(x,y), which choice will maximize the function? D P(x,y) = 4x 2y C A B 138

139 Linear Programming 58 Given the graph of the feasible region and P(x,y), which choice will minimize the function? D P(x,y) = 4x 2y C A B 139

140 Systems with 3 variables Systems Of Equations with 3 Variables Return to Table of Contents 140

141 Systems with 3 variables The methods used to solve a system with 3 variables are the same as those used to solve 2 variable systems. The goal is to use either substitution or elimination to combine two equations to eliminate a variable, the difference is the process may have to be used a few times to find the solution. 141

142 Systems with 3 variables 142

143 Systems with 3 variables Tips: You can combine any of the equations. If done correctly, will get the same answer. There are many different ways to solve these systems. Don't get frustrated. The problems can be long. If a solution works out to be a true statement, such as 0 = 0, then there are infinitely many solutions. If a solution works out to be a false statement, such as 1 = 0, then there is no solution to the system. 143

144 Systems with 3 variables 144

145 Systems with 3 variables 145

146 Systems with 3 variables 146

147 Systems with 3 variables 147

148 Systems with 3 variables 148

149 Systems with 3 variables 149

150 Systems with 3 variables 150

151 This is the end of Systems of Equations. 151

152 152

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