# Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Save this PDF as:

Size: px
Start display at page:

Download "Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:"

## Transcription

1 Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system: Example 1: Use the graph to solve the system. Then check your solution algebraically. 2x + y = 4 Equation 1 3x 5y = 6 Equation 2

2 Example 2: Solve the linear systems by graphing. y = 3x 2 Equation 1 y = -2x + 3 Equation 2 x 3y = 2 Equation 1 5x + y = 4 Equation 2

3 Name Date LESSON 6.1 Practice A For use with the lesson Solve Linear Systems by Graphing Tell whether the ordered pair is a solution of the linear system. 1. (0, 4); x + y = 4 x 5y = (3, 3); x + 2y = 9 4x y = (1, 2); 2x 3y = 8 3x + 2y = 1 Solve the linear system by graphing. Check your solution. 4. y = x y = -2x + 1 y = x 2 y = x - 2

4 Lesson 6.2 Solve Linear Systems by Substitution Example 1: Solve the linear system using substitution. y = 2x 3 Equation 1 x + 3y = 5 Equation 2 Example 2: Solve the linear system using substitution. 2x + 5y = 5 Equation 1 x 4y = 9 Equation 2

5

6 Name Date LESSON 6.2 Practice A For use with the lesson Solve Linear Systems by Substitution Solve for the indicated variable. 1. 9x + y = 7; y 2. 3x y = 10; y 3. x 4y = 1; x Solve the linear system by using substitution. 4. x = 1 y y = 2x 2 5. x = 4y + 14 y = 3x + 3

7 6. y = 3x 1 4x + 3y = 2 7. y = 2x + 4 5y 2x = x 2y = 14 x = 10 6y

8 opyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. Name CHAPTER 6 Practice Quiz 1 For use after the lessons Solve Linear Systems by Graphing and Solve Linear Systems by Substitution Solve the linear system by graphing. 1. x y = 2 2. x + 2y = 6 x + y = 4 3x + y = x + 3y = 9 x + 2y = 6

9 Solve the linear system using substitution. 4. 4x 2y = 14 x = 10 6y 5. x = 1 y y = 2x 2 6. x + 2y = 6 7x + 3y = 8

10 Solve the linear system using substitution. 7. x + 2y = 11 3x 4y = x + y = 8 x + 2y = 5

11 Lesson 6.3 Solve Linear Systems by Adding or Subtracting Example 1: Solve the linear system by adding or subtracting. 2x + 4y = 2 Equation 1 4x 4y = 16 Equation 2 Example 2: Solve the linear system by adding or subtracting. 7x + 5y = 18 Equation 1 7x 3y = 34 Equation 2 Example 3: Solve the linear system by adding or subtracting.

12 6x 4y = 10 Equation 1 13y = 6x + 8 Equation 2

13 Name Date LESSON 6.3 Practice A For use with the lesson Solve Linear Systems by Adding or Subtracting Rewrite the linear system so that the like terms are arranged in columns. 1. 3x y =23 y + 8x = x = y + 1 3y + 8x = x 4y = 8 4y = 7x + 9 Solve the linear system by using elimination. 4. 6x y = 5 3x + y = 4 5. x + 4y = 9 x 2y = 3

14 6. 5x 3y = 14 x + 3y = x + y = 7 x + y = x + 3y = 18 4x 2y = x 3y = 36 5x = 3y + 30

15 Lesson 6.4: Solve Linear Systems by Multiplying First Least Common Multiple: Example 1: Solve the linear system. 3x 2y = 4 Equation 1 7x 4y = 6 Equation 2 Example 2: Solve the linear system. 5x +2y = 18 Equation 1 7y = 3x + 19 Equation 2

16

17 Name Date LESSON 6.4 Practice A For use with the lesson Solve Linear Systems by Multiplying First Describe the first step you would use to solve the linear system. 1. x + y = 4 3x 7y = x + 6y = 1 4x + 7y = x 6y = 1 x + y = 4 Solve the linear system by using elimination. 4. x + y = 3 2x + 4y = x + y = 8 3x + 3y = x y = 10 2x + 5y = x 4y = 42 x 6y = 24

18 Name CHAPTER 6 Practice Quiz 2 For use after the lessons Solve Linear Systems by Adding or Subtracting and Solve Linear Systems by Multiplying First Solve the linear system using elimination. 1. 7x + y = 6 6x y = 7 Answers x + 5y = 1 2x + y = x + 4y = 10 x + 2y = x + 2y = 2 6x + 3y = 5

19 Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5. 5x y = 7 4x y = 6 Answers x + y = 3 2x + 4y = x + y = 8 3x + 3y = x 4y = 42 x 6y = 24

20 Lesson 6.5: Solve Special Types of Linear Systems Inconsistent system: Consistent dependent system: Example 1: Show that the linear system has no solution. 5x + 4y = 16 Equation 1 5x 4y = 8 Equation 2 Example 2: Show that the linear system has infinitely many solutions. 2 y = x +5 Equation 1 3 2x + 3y = 15 Equation 2

21 Example 3: Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions. 7x 2y = 9 Equation 1 3x + y = 10 Equation 1 7x 2y = 1 Equation 2 6x 2y = 20 Equation 2 Number of Solutions? What are the lines doing? Characteristics of the lines

22 Name Date LESSON 6.5 Practice A For use with the lesson Solve Special Types of Linear Systems Identify the slope of the linear equation. 1. y 3x = x + 2y = x 3y = 15 Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions. 4. x + y = 4. y = x y 2x = 3 x + y = 2

23 6. 2x +2y = 4 y = x x y = 1 2x + y = x + 2y = 8 3x y = 3

24 Lesson 6.6 Solve Systems of Linear Inequalities System of linear inequalities: Solution of a system of linear inequalities: Graph of a system of linear inequalities: Example 1: Graph the system of inequalities. 1 y < x + 2 Inequality 1 2 y 2x + 5 Inequality 2 Example 2: Graph the system of inequalities. y 5 Inequality 1 x < 4 Inequality 2 y 2x + 2 Inequality 3

25 Example 3: Write a system of inequalities for the shaded region.

26 Name Date LESSON 6.6 Practice A For use with the lesson Solve Systems of Linear Inequalities Tell whether the ordered pair is a solution of the system of inequalities. 1. (2, 1) 2. ( 3, 2) Graph the system of inequalities. 3. x + y 4 4. x + y 4 x < 2 x < 2

27 5. x y y + x 4 y > 2 y < 2 7. x y 4 8. y + x 4 x > 2 y < 2

28 Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. Name CHAPTER 6 Practice Quiz 3 For use after the lessons Solve Special Types of Linear Systems and Solve Systems of Linear Inequalities Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions. 1. 2y 5x = x + 2y = 8 2y x = 4 3x y = 5 Answers x + y = 4 4x 2y = 8 Graph the system of inequalities. 4. x > 4 5. y x + 3 y < 2 y x 1 7. y x y > 2x y > 2 x > 3

29 Name Date CHAPTER 6 Practice Test For use after the chapter Systems of Equations and Inequalities Graph to solve the linear system. 1. y = (-1/2)x y = (2/3)x + 2 y = 2x + 1 x = 3 Answers Solve the linear system using substitution. 3. x = 4 4. y = 2x x + 2y = 4 x = - 2y 5. 3x + 2 = y 6. y = 2x 3 x = 11 2y x + 3y = 5

30 reserved. Name Date CHAPTER 6 Practice Test continued For use after the chapter Systems of Equations and Inequalities Solve the linear system using elimination. 7. x + y = x + 4y = -9 x y = 12 3x + y = 0 Answers x 4y = x + 2y = 5 4x + 2y = 6 4x 3y = 16 Determine whether the linear system has one solution, no solution, or infinitely many solutions. 11. y = 5x y = - 2x x y = 10 y = 5x + 1 y = 2x + 8 y = 6x 10 Graph the system of linear inequalities. 14. y > y > - 1 x < 3

### 0-5 Systems of Linear Equations and Inequalities

21. Solve each system of equations. Eliminate one variable in two pairs of the system. Add the first equation and second equations to eliminate x. Multiply the first equation by 3 and add it to the second

### 5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems

5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.1-1 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The

### Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

### 3.1 Solving Systems Using Tables and Graphs

Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

### Chapter 4 - Systems of Equations and Inequalities

Math 233 - Spring 2009 Chapter 4 - Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Solving simultaneous equations. Jackie Nicholas

Mathematics Learning Centre Solving simultaneous equations Jackie Nicholas c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Simultaneous linear equations We will introduce

### 2.1 Algebraic Expressions and Combining like Terms

2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like

### 12.2 Graphing Systems of Linear Inequalities

Name Class Date 1. Graphing Sstems of Linear Inequalities Essential Question: How do ou solve a sstem of linear inequalities? Resource Locker Eplore Determining Solutions of Sstems of Linear Inequalities

### Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

### Slope-Intercept Quiz. Name: Class: Date: 1. Graph the line with the slope 1 and y-intercept 3. a. c. b. d.

Name: Class: Date: ID: A Slope-Intercept Quiz 1. Graph the line with the slope 1 and y-intercept 3. a. c. b. d.. Write the equation that describes the line with slope = and y-intercept = 3 a. x + y = 3

### Number of Solutions to Simultaneous Equations

Worksheet 3.5 Simultaneous Equations Section 1 Number of Solutions to Simultaneous Equations In maths we are sometimes confronted with two equations in two variables and we want to find out which values

### Systems of Linear Equations - Introduction

Systems of Linear Equations - Introduction What are Systems of Linear Equations Use an Example of a system of linear equations If we have two linear equations, y = x + 2 and y = 3x 6, can these two equations

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### Mathematics Success Level H

T393 [OBJECTIVE] The student will solve two-step inequalities and graph the solutions on number lines. [MATERIALS] Student pages S132 S140 Transparencies T372 from Lesson 15, T405, T407, T409, T411, T413,

### Letter to the Student... 5 Letter to the Family... 6 Correlation of Mississippi Competencies and Objectives to Coach Lessons... 7 Pretest...

Table of Contents Letter to the Student........................................... 5 Letter to the Family............................................. 6 Correlation of Mississippi Competencies and Objectives

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### Topic 2 Solving Equations

Topic 2 Solving Equations Introduction: When you are given the value of a variable and an algebraic expression then you can evaluate the expression. For example, If you are told that x = 6 then the value

### 7.5 Systems of Linear Inequalities

7.5 Systems of Linear Inequalities In chapter 6 we discussed linear equations and then saw linear inequalities, so since here, in chapter 7, we are talking about systems of linear equations, it makes sense

### Solve Systems of Equations by the Addition/Elimination Method

Solve Systems of Equations by the Addition/Elimination Method When solving systems, we have found that graphing is very limited. We then considered the substitution method which has its limitations as

### Solving Systems by Elimination 35

10/21/13 Solving Systems by Elimination 35 EXAMPLE: 5x + 2y = 1 x 3y = 7 1.Multiply the Top equation by the coefficient of the x on the bottom equation and write that equation next to the first equation

### Solving Systems of Linear Equations Substitutions

Solving Systems of Linear Equations Substitutions Outcome (learning objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### 3-3 Solving Systems of Linear Inequalities

3-3 Solving Systems of Linear Inequalities Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Graph 2x y > 4. Determine if the given ordered pair is a solution of the system of equations. 2. (2, 2) yes

### Section 4.1 Inequalities & Applications. Inequalities. Equations. 3x + 7 = 13 y = 7 3x + 2y = 6. 3x + 7 < 13 y > 7 3x + 2y 6. Symbols: < > 4.

Section 4.1 Inequalities & Applications Equations 3x + 7 = 13 y = 7 3x + 2y = 6 Inequalities 3x + 7 < 13 y > 7 3x + 2y 6 Symbols: < > 4.1 1 Overview of Linear Inequalities 4.1 Study Inequalities with One

### Lesson 22: Solution Sets to Simultaneous Equations

Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

### Algebra II - Chapter 3 Review

Algebra II - Chapter 3 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. An independent system of two linear equations has an infinite number of solutions.

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Systems of Linear Equations: Elimination by Addition

OpenStax-CNX module: m21986 1 Systems of Linear Equations: Elimination by Addition Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License

### 5 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

### Algebra II New Summit School High School Diploma Program

Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

### Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

2-8 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Solve. 1. y + 7 < 11 2. 4m 12 3. 5 2x 17 y < 18 m 3 x 6 Use interval notation to indicate the graphed numbers. 4. (-2, 3] 5. (-, 1] Objectives Solve

### Algebra. Indiana Standards 1 ST 6 WEEKS

Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

### Systems of Equations Involving Circles and Lines

Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

### Graph Linear Inequalities in Two Variables

Graph Linear Inequalities in Two Variables Name Goal Graph linear inequalities in two variables. VOCABLARY Linear inequality in two variables Graph of an inequality in two variables Example 1 Check solutions

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 173.

Algebra Chapter : Systems of Equations and Inequalities - Systems of Equations Chapter : Systems of Equations and Inequalities System of Linear Equations: - two or more linear equations on the same coordinate

### Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

### 6-3 Solving Systems by Elimination

Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations

### Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,

Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 75-77 Es. 4-20. 23-28, 29-39 odd, 40-43, 49-52, 59-73 odd

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

### 5.2. Systems of linear equations and their solution sets

5.2. Systems of linear equations and their solution sets Solution sets of systems of equations as intersections of sets Any collection of two or more equations is called a system of equations. The solution

### Reasoning with Equations and Inequalities

Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving systems of linear equations using multiplication and addition Common Core Standards Algebra: Solve

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### Unit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system

Unit Stud Guide Sstems of Linear Equations and Inequalities 6- Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and - in for in

### Section Objective. Instruction. Math Standards Assignment (Each of these should be worked out by the teacher)

Chapter 1 Day 1 Day 2 Day 3 Day 4 Day 5 Chapter 1.0 - Skill Review and Unit Intro Practice and review simplifying numerical expressions, using formulas, and writing algebraic equations Chapter 1.1 - pg

### Activity 2. Tracing Paper Inequalities. Objective. Introduction. Problem. Exploration

Objective Graph systems of linear inequalities in two variables in the Cartesian coordinate plane Activity 2 Introduction A set of two or more linear equations is called a system of equations. A set of

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### Solving Systems of Linear Equations Substitutions

Solving Systems of Linear Equations Substitutions Outcome (lesson objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### REVIEW: Write each statement as an inequality and then graph the inequality.

LESSON 1-5 NOTES (Part A): SOLVING INEQUALITIES Words like "at most" and "at least" suggest a relationship in which two quantities may not be equal. These relationships can be represented by a mathematical

### Multiple Representations of Equations & What We Know. Note that the worked out our turn and your turn charts can also be used as a matching activity.

Multiple Representations of s & What We California State Standards: 7 AF., 7 AF 3.3, 7 AF 3.4, Alg. 6., Alg. 7., Alg. 8. CCSS: 8.EE.6, 8.F., 8.F., 8.F.4 The idea of this lesson is to have students make

### ( ) # 0. SOLVING INEQUALITIES and Example 1. Example 2

SOLVING INEQUALITIES 9.1.1 and 9.1.2 To solve an inequality in one variable, first change it to an equation and solve. Place the solution, called a boundary point, on a number line. This point separates

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.

### Section 8.1 System of Linear Equations: Substitution and Elimination

Identifying Linear Systems Linear Systems in two variables Solution, 0 1 1 0 Consistent/ Independent Consistent/ Dependent Inconsistent Exactly one solution Infinitely many solutions No solution Two lines

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### 7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

### Keystone Exams: Algebra I Assessment Anchors and Eligible Content. Pennsylvania

Keystone Exams: Algebra I Assessment Anchors and Pennsylvania Algebra 1 2010 STANDARDS MODULE 1 Operations and Linear Equations & Inequalities ASSESSMENT ANCHOR A1.1.1 Operations with Real Numbers and

### 3.1. Solving Two-Step Equations. Using Subtraction and Division to Solve. Goal: Solve two-step equations.

3.1 Solving Two-Step Equations Goal: Solve two-step equations. Notice in Example 1 that you isolate x by working backward. First you subtract from each side and then you divide. Example 1 Using Subtraction

### Algebra 1 Chapter 06 Review

Class: Date: Algebra 1 Chapter 06 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Solve the following system of equations by graphing. 4x + 3y =

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

### Arithmetic Series. or The sum of any finite, or limited, number of terms is called a partial sum. are shorthand ways of writing n 1

CONDENSED LESSON 9.1 Arithmetic Series In this lesson you will learn the terminology and notation associated with series discover a formula for the partial sum of an arithmetic series A series is the indicated

### Isolate the absolute value expression. Add 5 to both sides and then divide by 2.

11 of 21 8/14/2014 2:35 PM of a variable expression. You can use the definition of absolute value to solve absolute value equations algebraically. Since then the equation ax + b = c is equivalent to (ax

### Pre-AP Algebra 2 Lesson 2-5 Graphing linear inequalities & systems of inequalities

Lesson 2-5 Graphing linear inequalities & systems of inequalities Objectives: The students will be able to - graph linear functions in slope-intercept and standard form, as well as vertical and horizontal

### Study Resources For Algebra I. Unit 1D Systems of Equations and Inequalities

Study Resources For Algebra I Unit 1D Systems of Equations and Inequalities This unit explores systems of linear functions and the various methods used to determine the solution for the system. Information

### 3.5. Solving Inequalities. Introduction. Prerequisites. Learning Outcomes

Solving Inequalities 3.5 Introduction An inequality is an expression involving one of the symbols,, > or

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### SUMMER ASSIGNMENT FOR ALGEBRA 1 STUDENTS

SUMMER ASSIGNMENT FOR 2012-2013 ALGEBRA 1 STUDENTS This year, the summer math assignments are available online on the district homepage. To access the summer assignments, please go to the high school s

### 1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

### Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

### Name Class Date. Deriving the Equation of a Parabola

Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.

10 17 0 7 7 2 11 17 5 11 10 12 6 14 12 20 7 3 9 19 4 14 5 0 5 7 0 2 1 2 11 7 8 4 4 10 12 3 9 0 10 11 9 1 8 0 10 12 1 3 9 11 7 8 7 18 1 12 11 12 0 10 3 13 1 5 10 4 8 2 9 7 Sheet 1 10 17 0 7 7 14 2 9 Add

### Solving Systems of Linear Equations Elimination (Addition)

Solving Systems of Linear Equations Elimination (Addition) Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students

### Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Practice Math Placement Exam

Practice Math Placement Exam The following are problems like those on the Mansfield University Math Placement Exam. You must pass this test or take MA 0090 before taking any mathematics courses. 1. What

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### Equivalent Expressions. ESSENTIAL QUESTION How can you identify and write equivalent expressions? 6.7.C

? LESSON 11.3 Generating Equivalent Expressions ESSENTIAL QUESTION How can you identify and write equivalent expressions? Expressions, equations, and relationships 6.7.D Generate equivalent expressions

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### Systems of Linear Equations

DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

### Solving Systems of Linear Equations Elimination (Addition)

Solving Systems of Linear Equations Elimination (Addition) Outcome (learning objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Mathematics Common Core Sample Questions

New York State Testing Program Mathematics Common Core Sample Questions Grade The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### Variable. 1.1 Order of Operations. August 17, evaluating expressions ink.notebook. Standards. letter or symbol used to represent a number

1.1 evaluating expressions ink.notebook page 8 Unit 1 Basic Equations and Inequalities 1.1 Order of Operations page 9 Square Cube Variable Variable Expression Exponent page 10 page 11 1 Lesson Objectives

### High School Mathematics Algebra

High School Mathematics Algebra This course is designed to give students the foundation of understanding algebra at a moderate pace. Essential material will be covered to prepare the students for Geometry.

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Sect Solving and graphing inequalities

81 Sect 2.7 - Solving and graphing inequalities Concepts #1 & 2 Graphing Linear Inequalities Definition of a Linear Inequality in One Variable Let a and b be real numbers such that a 0. A Linear Inequality

### Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 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

### SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS

Name SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS STUDENT ACTIVITY Lesson 1 & Lesson 2 1. Use the equation x + 2 = 5 to answer questions (a) through (d). 1.a. Draw a representation of the