Solving Systems of Linear Equations
|
|
- Sophia Amberly Warner
- 6 years ago
- Views:
Transcription
1 Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes Your definition Ordered Pair Your definition New Terms to Learn: System of Linear Equations (SLE) Your definition Formal definition Example READING ASSIGNMENT 2.3 Sections 4.1, 4.2 and 4.3 READING AND SELF-DISCOVERY QUESTIONS What is a system of equations 83
2 Chapter 2 Equations 2. What is a solution to a system of equations 3. How do you solve a system of equations in two variables graphically 4. What is the elimination method for solving a system of linear equations 5. What is the substitution method for solving a system of linear equations KEY CONCEPTS 2.3 Types of Solutions to Systems of Two Linear Equations in Two Variables Systems of two linear equations in two variables can have one solution, no solution, or an infinite number of solutions. The same types of solutions exist for any system of linear equations. Limitation: There are other types of systems of equations that are non-linear. 84
3 METHODOLOGY 2.3 GRAPHING THE SOLUTION TO A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES This means of solving a system of linear equations can give insight into what it means to be a solution to such systems of equations. Limitation/Caution: This method may only provide approximate results, based on the precision of the graph. Solve by graphing: Example 1 Example 2 + 3y = 1 2x + 2y = 6 Solve by graphing: 4x 2y = 6 3x + y = 2 Steps Discussion 1 Graph both lines Graph the lines represented by each equation. Feel free to rewrite the equations in the slope-intercept form if that makes it easier to graph them. Note that the slope-intercept format is an equation where the variable y has been isolated. Example y x y 3 2 Example x
4 Chapter 2 Equations Steps Discussion 2 Identify the point of intersection Identify the ordered pair that represents the point where the two graphed lines intersect. 4 3 y (-2, 1) 2 Example x Example 2 (Identify it on your previous graph) 3 Present the solution Present the solution using the ordered pair in terms of the variables in the equations. ( 2, 1) or x = 2 and y = 1 4 Validate Validate your work algebraically by substituting the solution into each given equation and verifying that each is a true expression. Example 1 Let x = 2, y = 1 x + 3y = 1 2x + 2y = (1) = 1 2( 2) + 2(1) = 6 1 = = 6 6 = 6 Example 2 86
5 METHODOLOGY 2.3 SOLVING A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES BY SUBSTITUTION The purpose of this methodology is to solve a system of two equations in two variables by using the Substitution Principle twice. Limitation/Caution: When you solve one of the equations for one variable in terms of the other, you should not substitute the result back into that same equation. Example 1 Example 2 Solve the system: Solve the system: + 2y x x + y = x + 5y 7 Steps Discussion 1 Select a variable and an equation Select a variable for the first substitution and an equation to use to determine an expression for the selected variable + 2y 3x Select x and the first equation (we choose this equation because x has no coefficient so we won't have to divide it away). 2 Solve one of the equations Solve the selected literal equation for the selected variable x + 2y x 2y 3 Substitute expression Substitute the expression for the variable solved for in Step 2 in the other equation. 3x 3( 3 2 y) 87
6 Chapter 2 Equations Steps Discussion 4 Solve for first variable Solve the resulting equation in one variable for that variable 9 6y 6y y = 22 y = 2 5 Substitute first variable Substitute the value obtained in Step 4 in the equation you created in Step 2 x 2y x 2( 2) 6 Solve for second variable Solve the equation in Step 5 for the other variable x = x = 1 7 Present solution Write the solution for each variable. (1, 2) or x = 1 and y = 2 8 Validate Validate your work by substituting the values for the variables in both original equations x + 2y 1 + 2( 2) = 3 1 4= 3 3 and 3x 3(1) 5( 2)
7 METHODOLOGY 2.3 SOLVING A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES BY ELIMINATION The purpose of this methodology is to solve a system of two equations in two variables by using the Addition Property of Equations to create a third equation in one variable that can be solved. Limitation/Caution: A system of linear equations may have no solution. Example 1 Example 2 Solve the system: Solve the system: + 2y 2x + 3y = 2 3x 5x 2 Steps Discussion 1 Select the variable Select the variable to eliminate between the two equations + 2y 3x Eliminate y 2 Multiply the equations by the appropriate numbers Multiply the two equations by numbers such that the coefficients of the selected variable differ only by a sign ( x + y = ) ( x y = ) x + 1y = 15 6x 1y = 26 3 Add the two equations Add the two equations to eliminate the selected variable 11x = 11 89
8 Chapter 2 Equations Steps Discussion 4 Solve the equation Solve the equation from Step 3 for the variable x = 1 5 Substitute first variable Substitute the value of the variable you solved for in Step 4 in either of the original equations x + 2y (1) + 2y 6 Solve for second variable Solve the equation you obtained in Step 5 for the remaining variable 2y 1 2y = 4 y = 2 7 Present solution Write the solution for each variable. (1, 2) or x = 1 and y = 2 8 Validate Validate your work by substituting the values for the variables in both original equations x + 2y 1 + 2( 2) = 3 1 4= 3 3 and 3x 3(1) 5( 2)
9 CRITICAL THINKING QUESTIONS What does a solution to a system of linear equations in two variables represent 2. What is the central idea behind the substitution method 3. What is the central idea behind the elimination method 4. If a system of linear equations has no solutions, how are the slopes of the equations related 5. If a system of linear equations has multiple solutions, how are the graphs of the equations related 6. Should graphical solutions to a system of linear equations have the same solutions as algebraic solutions Explain your answer. 7. How do you validate the solution to a system of linear equations 91
10 Chapter 2 Equations DEMONSTRATE YOUR UNDERSTANDING 2.3 Solve each system of equations, using the indicated method. 1. 5x 3y = 2 5x 4y = 1 (elimination method) y = 3 2x + 6y = 6 (by graphing) 4 y x
11 3. 3x + 4y = 23 5x 4y = 15 (elimination method) 4. 3x 6x + 2y = 1 (substitution method) 93
12 Chapter 2 Equations IDENTIFY AND CORRECT THE ERRORS 2.3 In the second column, identify the error you find in each of the following worked solutions and describe the error made. Solve the problem correctly in the third column. 1. Solve by elimination: Problem Describe Error Correct Process 2x 6x + y = 6 Worked Solution (What is wrong here) 2x 6x + y = 6 4x = 4 x = 1 The solution is x = Solve by elimination: 2x + 4y = 12 2x + 3y = 2 Worked Solution (What is wrong here) 2x + 4y = 12 2x + 3y = 2 7x = 14 x = 2 2(2) + 4y = y = 12 4y = 16 y = 4 The solution is (2, 4) or x = 2 and y = 4. 94
13 Problem Describe Error Correct Process 3. Solve by substitution: 2x 6x + y = 6 Worked Solution (What is wrong here) 2x 6x + y = 6 2x y = 2 + 2x 2 x ( x) = 2 2x + 2 2x = 2 = The solution set is all real numbers. 4. Solve by substitution: 5x = 6 Worked Solution (What is wrong here) 5x = 6 ( x y ) 5 = 2 5x = 6 5x + 5y = 1 5x = 6 = 16 The solution set is all real numbers. 95
Solving and Graphing a Linear Inequality of a Single Variable
Chapter 3 Graphing Fundamentals Section 3.1 Solving and Graphing a Linear Inequality of a Single Variable TERMINOLOGY 3.1 Previously Used: Isolate a Variable Simplifying Expressions Prerequisite Terms:
More informationSolving Quadratic and Other Polynomial Equations
Section 4.3 Solving Quadratic and Other Polynomial Equations TERMINOLOGY 4.3 Previously Used: This is a review of terms with which you should already be familiar. Formula New Terms to Learn: Discriminant
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationWhy? 2.2. What Do You Already Know? 2.2. Goals 2.2. Building Mathematical Language 2.2. Key Concepts 2.2
Section. Solving Basic Equations Why. You can solve some equations that arise in the real world by isolating a variable. You can use this method to solve the equation 1 400 + 1 (10) x = 460 to determine
More informationGraphical Solutions of Linear Systems
Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations
More informationSections 8.1 & 8.2 Systems of Linear Equations in Two Variables
Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems
More informationOne Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationGraphing Linear Systems
Graphing Linear Systems Goal Estimate the solution of a system of linear equations by graphing. VOCABULARY System of linear equations A system of linear equations is two or more linear equations in the
More information1.5 F15 O Brien. 1.5: Linear Equations and Inequalities
1.5: Linear Equations and Inequalities I. Basic Terminology A. An equation is a statement that two expressions are equal. B. To solve an equation means to find all of the values of the variable that make
More informationx y = 2 x + 2y = 14 x = 2, y = 0 x = 3, y = 1 x = 4, y = 2 x = 5, y = 3 x = 6, y = 4 x = 7, y = 5 x = 0, y = 7 x = 2, y = 6 x = 4, y = 5
List six positive integer solutions for each of these equations and comment on your results. Two have been done for you. x y = x + y = 4 x =, y = 0 x = 3, y = x = 4, y = x = 5, y = 3 x = 6, y = 4 x = 7,
More informationPut the following equations to slope-intercept form then use 2 points to graph
Tuesday September 23, 2014 Warm-up: Put the following equations to slope-intercept form then use 2 points to graph 1. 4x - 3y = 8 8 x 6y = 16 2. 2x + y = 4 2x + y = 1 Tuesday September 23, 2014 Warm-up:
More informationStudy Guide: Systems of Linear Equations
Study Guide: Systems of Linear Equations Systems of Linear Equations A system of linear equations is when two or more linear equations are involved in the same problem. The solution for a system of linear
More informationAlgebra I+ Pacing Guide. Days Units Notes Chapter 1 ( , )
Algebra I+ Pacing Guide Days Units Notes Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
More information6-4 Solving Special Systems
Warm Up Solve each equation. 1. 2x + 3 = 2x + 4 2. 2(x + 1) = 2x + 2 3. Solve 2y 6x = 10 for y Solve by using any method. 4. y = 3x + 2 2x + y = 7 5. x y = 8 x + y = 4 Know: Solve special systems of linear
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationGeometry Summer Assignment 2018
Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each
More information6.3. MULTIVARIABLE LINEAR SYSTEMS
6.3. MULTIVARIABLE LINEAR SYSTEMS What You Should Learn Use back-substitution to solve linear systems in row-echelon form. Use Gaussian elimination to solve systems of linear equations. Solve nonsquare
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More informationAnswers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets.
Name Geometry SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Geometry. We will use these concepts on a regular basis throughout
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationPair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables Linear equation in two variables x and y is of the form ax + by + c= 0, where a, b, and c are real numbers, such that both a and b are not zero. Example: 6x +
More informationQuantile Textbook Report
Quantile Textbook Report Algebra 1 Author Charles, Randall I., et al StateEdition West Virginia Grade Algebra 1 1 Foundations for Algebra 1.1 Variables and Expressions 750Q 1.2 Order of Operations and
More informationMATHEMATICS CONTENT ASSESSED ON THE ALGEBRA 1 EOC ASSESSMENT AND ITEM TYPES BY BENCHMARK
Body of Knowledge Algebra Standard 1 Real and Complex Number System Expand and deepen understanding of real and complex numbers by comparing expressions and performing arithmetic computations, especially
More information6-4 Solving Special Systems
6-4 Solving Special Systems Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 6-4 Solve the equation. 1. 2(x + 1) = 2x + 2 3 pts Solve by using any method. 2. y = 3x + 2 2x + y = 7 5 pts possible
More informationDefinition: A "system" of equations is a set or collection of equations that you deal with all together at once.
System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems
More informationSection 1.6. Functions
Section 1.6 Functions Definitions Relation, Domain, Range, and Function The table describes a relationship between the variables x and y. This relationship is also described graphically. x y 3 2 4 1 5
More informationWhat can I tell from a survey?
CCA Ch 10: Solving Comple Equations Name Team # 10.1.1 What can I tell from a survey? Association in Two-Way Tables 10-1. a. c. d. d. 10-. a. Complete the following two-way table: Laptop No Laptop TOTAL
More informationMath 1201 Unit 7: Systems of Linear Equations. Ch. 7 Notes
Math 20 Unit 7: Systems of Linear Equations Read Building On, Big Ideas, and New Vocabulary, p. 392 text. Ch. 7 Notes 7. Developing Systems of Linear Equations ( class) Read Lesson Focus p. 394 text. Outcomes.
More informationSNAP Centre Workshop. Solving Systems of Equations
SNAP Centre Workshop Solving Systems of Equations 35 Introduction When presented with an equation containing one variable, finding a solution is usually done using basic algebraic manipulation. Example
More informationare topics that you have not covered yet. Just do the best you can.
Summer assignment for Honors Algebra II 1 Honors Algebra II 010 Summer Assignment Dear student, Welcome to Honors Algebra II! You have signed up for a rigorous course that will challenge your minds, get
More information8.EE The Intersection of Two
8.EE The Intersection of Two Lines Alignments to Content Standards: 8.EE.C.8.a Task a. Draw the two lines that intersect only at the point. One of the lines should pass (0, ) through the point. b. Write
More informationMatrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.
Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced
More informationSystems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities Alex Moore February 4, 017 1 What is a system? Now that we have studied linear equations and linear inequalities, it is time to consider the question, What
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationInstruction. Student Activities Overview and Answer Key
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving systems of linear equations using substitution Common Core Standards Analyze and solve linear equations
More informationAlgebra 2 Honors Unit 1 Review of Algebra 1
Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine
More informationSolving and Graphing Inequalities
Solving and Graphing Inequalities Graphing Simple Inequalities: x > 3 When finding the solution for an equation we get one answer for x. (There is only one number that satisfies the equation.) For 3x 5
More informationevaluate functions, expressed in function notation, given one or more elements in their domains
Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates
More informationLesson 23: The Defining Equation of a Line
Student Outcomes Students know that two equations in the form of and graph as the same line when and at least one of or is nonzero. Students know that the graph of a linear equation, where,, and are constants
More informationUNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together
More informationSolving Systems of Linear and Quadratic Equations
9.5 Solving Systems of Linear and Quadratic Equations How can you solve a system of two equations when one is linear and the other is quadratic? ACTIVITY: Solving a System of Equations Work with a partner.
More informationStudent Guide: Chapter 1
Student Guide: Chapter 1 1.1 1.1.1 I can solve puzzles in teams 1-4 to 1-8 1.1 1.1.2 1.1 1.1.3 I can investigate the growth of patterns 1-13 to 1-17 1-18 to 1-22 I can investigate the graphs of quadratic
More informationSystems of Linear Equations with the System Solver
Systems of Linear Equations with the System Solver In these activities, you explore the steps involved in solving systems of linear equations. You ll make observations about the effects of those operations
More informationYOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More informationSystems and inequalites review
Name: Class: Date: Systems and inequalites review Multiple Choice Identify the choice that best completes the statement or answers the question, 1. The approximate solutions to the system of equations
More informationSection 5.3 (Solving Systems of Two Linear Equations in Two Unknowns Algebraically)
Systems of equations can be solved in a variety of ways and without a graphing calculator, I typically do not turn to graphical means of solution. This is because in real world problems the answers don
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 informationPoint of intersection
Name: Date: Period: Exploring Systems of Linear Equations, Part 1 Learning Goals Define a system of linear equations and a solution to a system of linear equations. Identify whether a system of linear
More informationGeneral Methodology for Solving Equations
Section. Pre-Activity Preparation General Methodology for Solving Equations Catering A catering company charges $6.9 per guest with an additional set up fee of $00. How many guests can be invited if the
More informationEquations With Two or More Variables
! Equations With Two or More Variables You have done a lot of work with relationships involving two related variables. However, many real-world relationships involve three or more variables. For example,
More informationPERT Practice Test #2
Class: Date: PERT Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. Ê 1. What is the quotient of 6y 6 9y 4 + 12y 2 ˆ Ê 3y 2 ˆ? a. 2y 4 + 3y
More informationSYSTEMS Solving Linear Systems Common Core Standards
I Systems, Lesson 1, Solving Linear Systems (r. 2018) SYSTEMS Solving Linear Systems Common Core Standards A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation
More informationLesson 3-1: Solving Linear Systems by Graphing
For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we
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 informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationAlgebra Performance Level Descriptors
Limited A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Algebra. A student at this level has an emerging ability to A student whose performance
More informationANSWER KEY. Checklist Do I understand? Test Date: A-Day 2/7. Algebra 1 Benchmark Review study guide is due on test day!
ANSWER KEY Name: Algebra 1 Benchmark Review study guide is due on test day! Test Date: A-Day 2/7 Checklist Do I understand? Unit 1 Solving Equations and Inequalities Two Step Equations Properties of Real
More informationSection 1.4. Meaning of Slope for Equations, Graphs, and Tables
Section 1.4 Meaning of Slope for Equations, Graphs, and Tables Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using
More informationPacing Guide Algebra 1
Pacing Guide Algebra Chapter : Equations and Inequalities (one variable) Section Section Title Learning Target(s) I can. Evaluate and Simplify Algebraic Expressions. Evaluate and simplify numeric and algebraic
More informationMath 10. Lesson 4 7 General Form of Linear Equation
Math 10 Lesson 7 General Form of Linear Equation I. Lesson Objectives: 1) To learn to use the general form of a linear equation. 2) To practice our algebra skills in order to manipulate equations. II.
More informationGraphing Systems of Linear Equations
Graphing Systems of Linear Equations Groups of equations, called systems, serve as a model for a wide variety of applications in science and business. In these notes, we will be concerned only with groups
More informationP.5 Solving Equations Graphically, Numerically, and Algebraically
0 CHAPTER P Prerequisites What you ll learn about Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically
More informationALGEBRA 2. Background Knowledge/Prior Skills Knows what operation properties hold for operations with matrices
ALGEBRA 2 Numbers and Operations Standard: 1 Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, and number
More informationChapter 6. Functions. 01/2017 LSowatsky 1
Chapter 6 Functions 01/2017 LSowatsky 1 6.1A Constant Rate of Change I can Identify proportional and nonproportional linear relationships by finding a constant rate of change CCSS 8.EE.5, 8.F.4 LSowatsky
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
More informationWest Windsor-Plainsboro Regional School District Algebra Grade 8
West Windsor-Plainsboro Regional School District Algebra Grade 8 Content Area: Mathematics Unit 1: Foundations of Algebra This unit involves the study of real numbers and the language of algebra. Using
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 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 informationGeometry 263 Prerequisites
Name Geometry 6 Prerequisites Dear Incoming Geometry Student, Listed below are fifteen skills that we will use throughout Geometry 6. You have likely learned and practiced these skills in previous math
More informationThe Method of Substitution. Linear and Nonlinear Systems of Equations. The Method of Substitution. The Method of Substitution. Example 2.
The Method of Substitution Linear and Nonlinear Systems of Equations Precalculus 7.1 Here is an example of a system of two equations in two unknowns. Equation 1 x + y = 5 Equation 3x y = 4 A solution of
More information5.2 Start Thinking Sample answer: x the cost of an incandescent light bulb, y the cost of a CFL, 30x 2 3 3, t 25; t 6F
5.1 Cumulative Review Warm Up. y 5 5 8 1. y 1. y 1 4 8 4. y 5 4 5 5. y 5 4 6. y 5 7. y 1 8 8. y 7 5.1 Practice A 4 5 1. yes. no., 0 4., 4 5., 6 6., 5 7. 8, 8. 0.67,.5 9. 16 bracelets, 1 necklaces 10. y
More informationPLC Papers Created For:
PLC Papers Created For: Josh Angles and linear graphs Graphs of Linear Functions 1 Grade 4 Objective: Recognise, sketch and interpret graphs of linear functions. Question 1 Sketch the graph of each function,
More informationSolve Systems of Equations Algebraically
Part 1: Introduction Solve Systems of Equations Algebraically Develop Skills and Strategies CCSS 8.EE.C.8b You know that solutions to systems of linear equations can be shown in graphs. Now you will learn
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 informationGoing from graphic solutions to algebraic
Going from graphic solutions to algebraic 2 variables: Graph constraints Identify corner points of feasible area Find which corner point has best objective value More variables: Think about constraints
More informationDeveloped in Consultation with Virginia Educators
Developed in Consultation with Virginia Educators Table of Contents Virginia Standards of Learning Correlation Chart.............. 6 Chapter 1 Expressions and Operations.................... Lesson 1 Square
More informationMath-1010 Lesson 1-6. Textbook 1-11 (Systems of Linear Equations)
Math-1010 Lesson 1-6 Textbook 1-11 (Systems of Linear Equations) College Finals are over. You re moving back home for the summer. You need to rent a truck to move your possessions from the dorm back to
More informationB.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
More informationLIVE Online Math Algebra Scope and Sequence
LIVE Online Math Algebra Scope and Sequence The course is broken down into units. The units, and lessons that make up each unit, are below. Note: If there is a specific concept/technique that is not listed,
More informationTopic: Solving systems of equations with linear and quadratic inequalities
Subject & Grade: Mathematics, 9 th Grade Topic: Solving systems of equations with linear and quadratic inequalities Aim: How would you find the solution set of a linear and quadratic inequality? Materials:.
More informationConcept: Solving Absolute Value Equations
Concept: Solving Absolute Value Equations Warm Up Name: 1. Determine what values of x make each inequality true. Graph each answer. (a) 9 x - 2 7 x + 8 9 x - 2 7 x + 8-7x) 2 x - 2 8 +2) 2 x 10 2) x 5 Remember:
More informationMathematics Department. Summer Course Work. Geometry
Decatur City Schools Decatur, Alabama Mathematics Department Summer Course Work In preparation for Geometry Completion of this summer work is required on the first c l a s s day of the 2018-2019 school
More informationPartial Fraction Decomposition
Partial Fraction Decomposition As algebra students we have learned how to add and subtract fractions such as the one show below, but we probably have not been taught how to break the answer back apart
More informationMath 4: Advanced Algebra Ms. Sheppard-Brick A Quiz Review Learning Targets
5A Quiz Review Learning Targets 4.4 5.5 Key Facts Graphing one-variable inequalities (ex. x < 4 ) o Perform algebra steps to get x alone! If you multiply or divide by a negative number, you must flip the
More informationPart 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2
Solving Systems of Equations Algebraically Teacher Notes Comment: As students solve equations throughout this task, have them continue to explain each step using properties of operations or properties
More informationAlgebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella
1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial
More informationMATHS WORKSHOPS Simultaneous Equations and Inequalities. Business School
MATHS WORKSHOPS Simultaneous Equations and Inequalities Business School Outline Recap of Algebra, Linear and Quadratic Functions Simultaneous Equations Inequalities Applications in Business Summary and
More informationP1 Chapter 3 :: Equations and Inequalities
P1 Chapter 3 :: Equations and Inequalities jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 26 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationSolving Systems of Linear Equations with Linear Combinations (The Elimination Method)
Student Handout Name Solving Systems of Linear Equations with Linear Combinations (The Elimination Method) Problem 1 Launch the Activity (Multiplying an Equation by a Constant) When asked to graph the
More informationBooker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.
BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website
More informationAlgebra 2 Summer Review Packet
Algebra Summer Review Packet Welcome to Algebra! Attached you will find the learning targets your teacher thinks you should know BEFORE you come to class in the fall and problems to help you practice these
More informationx y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational
More informationUSING THE QUADRATIC FORMULA and 9.1.3
Chapter 9 USING THE QUADRATIC FORMULA 9.1.2 and 9.1.3 When a quadratic equation is not factorable, another method is needed to solve for x. The Quadratic Formula can be used to calculate the roots of a
More informationAlgebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target
Algebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target SB Activity Activity 1 Investigating Patterns 1-1 Learning Targets: Identify patterns in data. Use tables, graphs,
More information