Systems of Equations and Inequalities
|
|
- Phebe Fowler
- 5 years ago
- Views:
Transcription
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
Algebra I. Systems of Linear Equations and Inequalities. Slide 1 / 179. Slide 2 / 179. Slide 3 / 179. Table of Contents
Slide 1 / 179 Algebra I Slide 2 / 179 Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Table of Contents Slide 3 / 179 Click on the topic to go to that section 8th Grade Review of
More informationAlgebra I. Systems of Linear Equations and Inequalities. 8th Grade Review. Slide 1 / 179 Slide 2 / 179. Slide 4 / 179. Slide 3 / 179.
Slide 1 / 179 Slide 2 / 179 lgebra I Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Slide 3 / 179 Table of Contents Click on the topic to go to that section 8th Grade Review of Systems
More informationAlgebra I System of Linear Equations
1 Algebra I System of Linear Equations 2015-11-12 www.njctl.org 2 Table of Contents Click on the topic to go to that section Solving Systems by Graphing Solving Systems by Substitution Solving Systems
More informationAlgebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.
Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems
More informationName Class Date. What is the solution to the system? Solve by graphing. Check. x + y = 4. You have a second point (4, 0), which is the x-intercept.
6-1 Reteaching Graphing is useful for solving a system of equations. Graph both equations and look for a point of intersection, which is the solution of that system. If there is no point of intersection,
More informationUnit 12: Systems of Equations
Section 12.1: Systems of Linear Equations Section 12.2: The Substitution Method Section 12.3: The Addition (Elimination) Method Section 12.4: Applications KEY TERMS AND CONCEPTS Look for the following
More informationChapter 1-2 Add and Subtract Integers
Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More information3.1 Solving Linear Systems by Graphing 1. Graph and solve systems of linear equations in two variables. Solution of a system of linear equations
3.1 Solving Linear Systems by Graphing Objectives 1. Graph and solve systems of linear equations in two variables. Key Terms System of linear equations Solution of a system of linear equations Check whether
More informationDue for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
MTH 209 Week 1 Due for this week Homework 1 (on MyMathLab via the Materials Link) Monday night at 6pm. Read Chapter 6.1-6.4, 7.1-7.4,10.1-10.3,10.6 Do the MyMathLab Self-Check for week 1. Learning team
More informationConsistent and Dependent
Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;
More informationSOLVING SYSTEMS OF EQUATIONS
SOLVING SYSTEMS OF EQUATIONS 5.1.1 5.1.4 Students have been solving equations since Algebra 1. Now they focus on what a solution means, both algebraically and graphically. By understanding the nature of
More informationAlgebra Quadratics Applications HW#54
Algebra Quadratics Applications HW#54 1: A science class designed a ball launcher and tested it by shooting a tennis ball up and off the top of a 15-story building. They determined that the motion of the
More informationALGEBRA 1 FINAL EXAM TOPICS
ALGEBRA 1 FINAL EXAM TOPICS Chapter 2 2-1 Writing Equations 2-2 Solving One Step Equations 2-3 Solving Multi-Step Equations 2-4 Solving Equations with the Variable on Each Side 2-5 Solving Equations Involving
More informationx 2 + x + x 2 x 3 b. x 7 Factor the GCF from each expression Not all may be possible. 1. Find two numbers that sum to 8 and have a product of 12
Factor the GCF from each expression 4 5 1. 15x 3x. 16x 4 Name: a. b. 4 7 3 6 5 3. 18x y 36x y 4x y 5 4. 3x x 3 x 3 c. d. Not all may be possible. 1. Find two numbers that sum to 8 and have a product of
More informationLesson 12: Systems of Linear Equations
Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the
More information7.1 Solving Systems of Equations
Date: Precalculus Notes: Unit 7 Systems of Equations and Matrices 7.1 Solving Systems of Equations Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities.
More informationNOTES. [Type the document subtitle] Math 0310
NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.
More information3.1 NOTES Solving Systems of Linear Equations Graphically
3.1 NOTES Solving Systems of Linear Equations Graphically A system of two linear equations in two variables x and y consist of two equations of the following form: Ax + By = C Equation 1 Dx + Ey = F Equation
More informationUnit 7 Systems and Linear Programming
Unit 7 Systems and Linear Programming PREREQUISITE SKILLS: students should be able to solve linear equations students should be able to graph linear equations students should be able to create linear equations
More informationFSA Algebra I End-of-Course Review Packet. Algebra and Modeling
FSA Algebra I End-of-Course Review Packet Algebra and Modeling Table of Contents MAFS.912.A-APR.1.1 EOC Practice... 3 MAFS.912.A-CE1.1 EOC Practice... 5 MAFS.912.A-REI.2.3 EOC Practice... 7 MAFS.912.A-CE1.4
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationMath 8 Notes Units 1B: One-Step Equations and Inequalities
Math 8 Notes Units 1B: One-Step Equations and Inequalities Solving Equations Syllabus Objective: (1.10) The student will use order of operations to solve equations in the real number system. Equation a
More informationRELATIONS AND FUNCTIONS
RELATIONS AND FUNCTIONS Definitions A RELATION is any set of ordered pairs. A FUNCTION is a relation in which every input value is paired with exactly one output value. Example 1: Table of Values One way
More informationAlgebra. Chapter 6: Systems of Equations and Inequalities. Name: Teacher: Pd:
Algebra Chapter 6: Systems of Equations and Inequalities Name: Teacher: Pd: Table of Contents Chapter 6-1: SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear
More informationGraphing Linear Inequalities
Graphing Linear Inequalities Linear Inequalities in Two Variables: A linear inequality in two variables is an inequality that can be written in the general form Ax + By < C, where A, B, and C are real
More information7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems. Main Ideas:
7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems Toolkit: graphing lines rearranging equations substitution Main Ideas: Definitions: Linear
More informationMath 3 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More information6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities
6 th Grade - TNREADY REVIEW Q3 Expressions, Equations, Functions, and Inequalities INSTRUCTIONS: Read through the following notes. Fill in shaded areas and highlight important reminders. Then complete
More informationUnit Test Linear equations and Inequalities
Unit Test Linear equations and Inequalities Name: Date: Directions: Select the best answer for the following questions. (2 points each) 7L 1. The steps for solving are: 1) Read the problem and label variables,
More information2x + 5 = x = x = 4
98 CHAPTER 3 Algebra Textbook Reference Section 5.1 3.3 LINEAR EQUATIONS AND INEQUALITIES Student CD Section.5 CLAST OBJECTIVES Solve linear equations and inequalities Solve a system of two linear equations
More informationChapter 1. Worked-Out Solutions. Chapter 1 Maintaining Mathematical Proficiency (p. 1)
Chapter Maintaining Mathematical Proficiency (p. ). + ( ) = 7. 0 + ( ) =. 6 + = 8. 9 ( ) = 9 + =. 6 = + ( 6) = 7 6. ( 7) = + 7 = 7. 7 + = 8. 8 + ( ) = 9. = + ( ) = 0. (8) =. 7 ( 9) = 6. ( 7) = 8. ( 6)
More informationSY14-15 Algebra Exit Exam - PRACTICE Version
Student Name: Directions: Solve each problem. You have a total of 90 minutes. Choose the best answer and fill in your answer document accordingly. For questions requiring a written response, write your
More informationChapter 9 Solving Systems of Linear Equations Algebraically
Name: Chapter 9 Solving Systems of Linear Equations Algebraically 9.1 Solving Systems of Linear Equations by Substitution Outcomes: 1. Interpret algebraic reasoning through the study of relations 9. Solve
More informationEvaluate and Simplify Algebraic Expressions
TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So
More information4.1 Solving Systems of Equations Graphically. Draw pictures to represent the possible number of solutions that a linear-quadratic system can have:
4.1 Solving Systems of Equations Graphically Linear- Quadratic A Linear-Quadratic System of Equations is a linear equation and a quadratic equation involving the same two variables. The solution(s) to
More informationTopic 1. Solving Equations and Inequalities 1. Solve the following equation
Topic 1. Solving Equations and Inequalities 1. Solve the following equation Algebraically 2( x 3) = 12 Graphically 2( x 3) = 12 2. Solve the following equations algebraically a. 5w 15 2w = 2(w 5) b. 1
More informationSystems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1
Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you
More informationMATH 1710 College Algebra Final Exam Review
MATH 1710 College Algebra Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) There were 480 people at a play.
More informationLinear Functions, Equations, and Inequalities
CHAPTER Linear Functions, Equations, and Inequalities Inventory is the list of items that businesses stock in stores and warehouses to supply customers. Businesses in the United States keep about.5 trillion
More informationWarm Up. Unit #1: Basics of Algebra
1) Write an equation of the given points ( 3, 4) & (5, 6) Warm Up 2) Which of the following choices is the Associative Property 1) 4(x + 2) = 4x + 8 2) 4 + 5 = 5 + 4 3) 5 + ( 5) = 0 4) 4 + (3 + 1) = (4
More informationInvestigating Inequalities:
Investigating Inequalities: Choose roles: Record each group member s name next to their role: Anuncer: Recorder: Walker A: Walker B: Set-up: use the number cards to construct a number line on the floor.
More informationMath 3 Variable Manipulation Part 1 Algebraic Systems
Math 3 Variable Manipulation Part 1 Algebraic Systems 1 PRE ALGEBRA REVIEW OF INTEGERS (NEGATIVE NUMBERS) Concept Example Adding positive numbers is just simple addition 2 + 3 = 5 Subtracting positive
More information3-1 Solving Systems of Equations. Solve each system of equations by using a table. 1. ANSWER: (3, 5) ANSWER: (2, 7)
Solve each system of equations by using a table. 1. 9. CCSS MODELING Refer to the table below. (3, 5) 2. (2, 7) Solve each system of equations by graphing. 3. a. Write equations that represent the cost
More informationName: Systems 2.1. Ready Topic: Determine if given value is a solution and solve systems of equations
Name: Systems 2.1 Ready, Set, Go! Ready Topic: Determine if given value is a solution and solve systems of equations TE-16 1. Graph both equations on the same axes. Then determine which ordered pair is
More informationMath 2 Variable Manipulation Part 6 System of Equations
Name: Date: 1 Math 2 Variable Manipulation Part 6 System of Equations SYSTEM OF EQUATIONS INTRODUCTION A "system" of equations is a set or collection of equations that you deal with all together at once.
More informationChapter 4. Inequalities
Chapter 4 Inequalities Vannevar Bush, Internet Pioneer 4.1 Inequalities 4. Absolute Value 4.3 Graphing Inequalities with Two Variables Chapter Review Chapter Test 64 Section 4.1 Inequalities Unlike equations,
More informationUNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES. Solving Equations and Inequalities in One Variable
UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES This unit investigates linear equations and inequalities. Students create linear equations and inequalities and use them to solve problems. They
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationAlgebra 1 End-of-Course Assessment Practice Test with Solutions
Algebra 1 End-of-Course Assessment Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit
More information5.1 Inequalities and Compound Sentences
5.1 Inequalities and Compound Sentences 1. When is a mathematical sentence called an inequality? Advanced Algebra Chapter 5 - Note Taking Guidelines 2. What is an open sentence? 3. How do we graph inequalities
More informationFoundations of Math. Chapter 3 Packet. Table of Contents
Foundations of Math Chapter 3 Packet Name: Table of Contents Notes #43 Solving Systems by Graphing Pg. 1-4 Notes #44 Solving Systems by Substitution Pg. 5-6 Notes #45 Solving by Graphing & Substitution
More informationALGEBRA I SEMESTER EXAMS PRACTICE MATERIALS SEMESTER Use the diagram below. 9.3 cm. A = (9.3 cm) (6.2 cm) = cm 2. 6.
1. Use the diagram below. 9.3 cm A = (9.3 cm) (6.2 cm) = 57.66 cm 2 6.2 cm A rectangle s sides are measured to be 6.2 cm and 9.3 cm. What is the rectangle s area rounded to the correct number of significant
More informationAddition and Subtraction of real numbers (1.3 & 1.4)
Math 051 lecture notes Professor Jason Samuels Addition and Subtraction of real numbers (1.3 & 1.4) ex) 3 + 5 = ex) 42 + 29 = ex) 12-4 = ex) 7-9 = ex) -3-4 = ex) 6 - (-2) = ex) -5 - (-3) = ex) 7 + (-2)
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
ALGEBRA 1 Standard 1 Operations with Real Numbers Students simplify and compare expressions. They use rational exponents, and simplify square roots. A1.1.1 A1.1.2 A1.1.3 A1.1.4 A1.1.5 Compare real number
More informationLesson 7: Literal Equations, Inequalities, and Absolute Value
, and Absolute Value In this lesson, we first look at literal equations, which are equations that have more than one variable. Many of the formulas we use in everyday life are literal equations. We then
More informationPre-Algebra Chapter 2 Solving One-Step Equations and Inequalities
Pre-Algebra Chapter 2 Solving One-Step Equations and Inequalities SOME NUMBERED QUESTIONS HAVE BEEN DELETED OR REMOVED. YOU WILL NOT BE USING A CALCULATOR FOR PART I MULTIPLE-CHOICE QUESTIONS, AND THEREFORE
More informationMath 1101 Chapter 2 Review Solve the equation. 1) (y - 7) - (y + 2) = 4y A) B) D) C) ) 2 5 x x = 5
Math 1101 Chapter 2 Review Solve the equation. 1) (y - 7) - (y + 2) = 4y A) - 1 2 B) - 9 C) - 9 7 D) - 9 4 2) 2 x - 1 3 x = A) -10 B) 7 C) -7 D) 10 Find the zero of f(x). 3) f(x) = 6x + 12 A) -12 B) -2
More informationAlgebra I Chapter 6 Practice Test
Name: Class: Date: ID: A Algebra I Chapter 6 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. Find a solution of the system of linear inequalities.
More informationChapter 4: Systems of Equations and Inequalities
Chapter 4: Systems of Equations and Inequalities 4.1 Systems of Equations A system of two linear equations in two variables x and y consist of two equations of the following form: Equation 1: ax + by =
More informationLooking Ahead to Chapter 10
Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.
More information5-7 Solving Quadratic Inequalities. Holt Algebra 2
Example 1: Graphing Quadratic Inequalities in Two Variables Graph f(x) x 2 7x + 10. Step 1 Graph the parabola f(x) = x 2 7x + 10 with a solid curve. x f(x) 0 10 1 3 2 0 3-2 3.5-2.25 4-2 5 0 6 4 7 10 Example
More informationFall IM I Exam B
Fall 2011-2012 IM I Exam B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following equations is linear? a. y = 2x - 3 c. 2. What is the
More information10-1: Composite and Inverse Functions
Math 95 10-1: Composite and Inverse Functions Functions are a key component in the applications of algebra. When working with functions, we can perform many useful operations such as addition and multiplication
More information3. Find the slope of the tangent line to the curve given by 3x y e x+y = 1 + ln x at (1, 1).
1. Find the derivative of each of the following: (a) f(x) = 3 2x 1 (b) f(x) = log 4 (x 2 x) 2. Find the slope of the tangent line to f(x) = ln 2 ln x at x = e. 3. Find the slope of the tangent line to
More informationUnit 4: Inequalities. Inequality Symbols. Algebraic Inequality. Compound Inequality. Interval Notation
Section 4.1: Linear Inequalities Section 4.2: Solving Linear Inequalities Section 4.3: Solving Inequalities Applications Section 4.4: Compound Inequalities Section 4.5: Absolute Value Equations and Inequalities
More information6 which of the following equations would give you a system of equations with the same line and infinitely many solutions?
Algebra 1 4 1 Worksheet Name: Per: Part I: Solve each system of equations using the graphing method. 1) y = x 5 ) -x + y = 6 y = x + 1 y = -x 3) y = 1 x 3 4) 4x y = 8 y = 1 x + 1 y = x + 3 5) x + y = 6
More informationMath Analysis Notes Mrs. Atkinson 1
Name: Math Analysis Chapter 7 Notes Day 6: Section 7-1 Solving Systems of Equations with Two Variables; Sections 7-1: Solving Systems of Equations with Two Variables Solving Systems of equations with two
More informationy in both equations.
Syllabus Objective: 3.1 The student will solve systems of linear equations in two or three variables using graphing, substitution, and linear combinations. System of Two Linear Equations: a set of two
More informationMFM2P Foundations of Mathematics Unit 3 Lesson 11
The Line Lesson MFMP Foundations of Mathematics Unit Lesson Lesson Eleven Concepts Introduction to the line Using standard form of an equation Using y-intercept form of an equation x and y intercept Recognizing
More informationUnit 5 SIMULTANEOUS LINEAR EQUATIONS
MATH 8 Unit 5 SIMULTANEOUS LINEAR EQUATIONS By the end of this unit, students should be able to: 1. Solve simultaneous linear equations by graphing. 2. Understand what it means to solve a system of equations.
More informationLESSON 13.1 NONLINEAR EQUATIONS
LESSON. NONLINEAR EQUATIONS LESSON. NONLINEAR EQUATIONS 58 OVERVIEW Here's what you'll learn in this lesson: Solving Equations a. Solving polynomial equations by factoring b. Solving quadratic type equations
More informationFoundations of Algebra. Learning Goal 3.1 Algebraic Expressions. a. Identify the: Variables: Coefficients:
Learning Goal 3.1 Algebraic Expressions What you need to know & be able to do 1. Identifying Parts of Algebraic Expressions 3.1 Test Things to remember Identify Parts of an expression Variable Constant
More informationUnit 6 Systems of Equations
1 Unit 6 Systems of Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations Specific Outcomes: 6.1 Solve problems that involve systems of linear equations in
More informationCORE. Chapter 3: Interacting Linear Functions, Linear Systems. Algebra Assessments
CORE Algebra Assessments Chapter 3: Interacting Linear Functions, Linear Systems 97 98 Bears Band Booster Club The Bears Band Booster Club has decided to sell calendars to the band members and their parents.
More informationYou solved systems of equations algebraically and represented data using matrices. (Lessons 0-5 and 0-6)
You solved systems of equations algebraically and represented data using matrices. (Lessons 0-5 and 0-6) Solve systems of linear equations using matrices and Gaussian elimination. Solve systems of linear
More informationALGEBRA MIDTERM REVIEW SHEET
Name Date Part 1 (Multiple Choice): Please show ALL work! ALGEBRA MIDTERM REVIEW SHEET 1) The equations 5x 2y 48 and 3x 2y 32 represent the money collected from school concert ticket sales during two class
More informationPower Packet. Algebra 1 Unit 5. Name
Power Packet Algebra 1 Unit 5 Name This packet may be used on your test and will be collected when you turn your test in. Write and graph linear inequalities / 1 6 Solve one-step linear inequalities /
More informationBETHLEHEM CATHOLIC HIGH SCHOOL
BETHLEHEM CATHOLIC HIGH SCHOOL ALGEBRA SUMMER ASSIGNMENT NAME: - Variables and Expressions For Exercises, choose the correct letter.. The word minus corresponds to which symbol? A. B. C. D.. The phrase
More informationChapter 7 Quadratic Equations
Chapter 7 Quadratic Equations We have worked with trinomials of the form ax 2 + bx + c. Now we are going to work with equations of this form ax 2 + bx + c = 0 quadratic equations. When we write a quadratic
More informationWRITING EQUATIONS through 6.1.3
WRITING EQUATIONS 6.1.1 through 6.1.3 An equation is a mathematical sentence that conveys information to the reader. It uses variables and operation symbols (like +, -, /, =) to represent relationships
More informationd. Predict the number of photos this photographer will keep if she takes 200 photos.
1 Solve. Regression Practice Curve Fitting with Linear Models 1. Vern created a website about his school s sports teams. He has a hit counter on his site that lets him know how many people have visited
More informationChapter 4. Systems of Linear Equations; Matrices. Opening Example. Section 1 Review: Systems of Linear Equations in Two Variables
Chapter 4 Systems of Linear Equations; Matrices Section 1 Review: Systems of Linear Equations in Two Variables Opening Example A restaurant serves two types of fish dinners- small for $5.99 and large for
More informationPractice Ace Problems
Unit 5: Moving Straight Ahead Investigation 3: Solving Equations using tables and Graphs Practice Ace Problems Directions: Please complete the necessary problems to earn a maximum of 16 points according
More informationEssential Question How can you use substitution to solve a system of linear equations?
5.2 Solving Systems of Linear Equations by Substitution Essential Question How can you use substitution to solve a system of linear equations? Using Substitution to Solve Systems Work with a partner. Solve
More informationQuadratic function and equations Quadratic function/equations, supply, demand, market equilibrium
Exercises 8 Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Objectives - know and understand the relation between a quadratic function and a quadratic
More informationMath 4: Advanced Algebra Ms. Sheppard-Brick B Quiz Review Learning Targets
5B Quiz Review Learning Targets 4.6 5.9 Key Facts We learned two ways to solve a system of equations using algebra: o The substitution method! Pick one equation and solve for either x or y! Take that result
More information4-A5: Mid-Chapter 4 Review
-A: Mid-Chapter Review Alg H Write the equations for the horizontal and vertical lines that pass through the given point.. (, 0) Horiz. Vert.. (0, 8) Horiz. Vert. Use the slope formula to find the slope
More informationName Period Date DRAFT
Name Period Date Equations and Inequalities Student Packet 4: Inequalities EQ4.1 EQ4.2 EQ4.3 Linear Inequalities in One Variable Add, subtract, multiply, and divide integers. Write expressions, equations,
More informationSystems of Linear Equations in Two Variables. Break Even. Example. 240x x This is when total cost equals total revenue.
Systems of Linear Equations in Two Variables 1 Break Even This is when total cost equals total revenue C(x) = R(x) A company breaks even when the profit is zero P(x) = R(x) C(x) = 0 2 R x 565x C x 6000
More informationMAC 2233, Survey of Calculus, Exam 3 Review This exam covers lectures 21 29,
MAC 2233, Survey of Calculus, Exam 3 Review This exam covers lectures 21 29, This review includes typical exam problems. It is not designed to be comprehensive, but to be representative of topics covered
More informationRelationships Between Quantities
Algebra 1 Relationships Between Quantities Relationships Between Quantities Everyone loves math until there are letters (known as variables) in problems!! Do students complain about reading when they come
More informationSystems of Nonlinear Equations and Inequalities: Two Variables
Systems of Nonlinear Equations and Inequalities: Two Variables By: OpenStaxCollege Halley s Comet ([link]) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse.
More informationIntroduction to Systems of Equations
Systems of Equations 1 Introduction to Systems of Equations Remember, we are finding a point of intersection x 2y 5 2x y 4 1. A golfer scored only 4 s and 5 s in a round of 18 holes. His score was 80.
More informationOTHER METHODS FOR SOLVING SYSTEMS
Topic 18: Other methods for solving systems 175 OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method 18.1 OPENER 1. Evaluate ab + 2c when a = 2, b = 3, and c = 5. 2. Following is a set
More informationCCGPS Coordinate Algebra. EOCT Review Units 1 and 2
CCGPS Coordinate Algebra EOCT Review Units 1 and 2 Unit 1: Relationships Among Quantities Key Ideas Unit Conversions A quantity is a an exact amount or measurement. A quantity can be exact or approximate
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular
More informationSolving real-world problems using systems of equations
May 15, 2013 Solving real-world problems using systems of equations page 1 Solving real-world problems using systems of equations Directions: Each of these problems can be represented using a system of
More informationCreate your own system of equations: 1. Prove (2, 5) is a solution for the following system: 2. Is (-2, 0) a solution for the following system?
5.1 Explain Solving Systems of Linear Equations by Graphing - Notes Main Ideas/ Questions What You Will Learn What is a system of linear equations? Essential Question: How can you solve a system of linear
More informationCC Math I UNIT 7 Systems of Equations and Inequalities
CC Math I UNIT 7 Systems of Equations and Inequalities Name Teacher Estimated Test Date MAIN CONCEPTS Page(s) Study Guide 1 2 Equations of Circles & Midpoint 3 5 Parallel and Perpendicular Lines 6 8 Systems
More informationQuarter 2 400, , , , , , ,000 50,000
Algebra 2 Quarter 2 Quadratic Functions Introduction to Polynomial Functions Hybrid Electric Vehicles Since 1999, there has been a growing trend in the sales of hybrid electric vehicles. These data show
More information