Linear Equations in One Variable

Size: px
Start display at page:

Download "Linear Equations in One Variable"

Transcription

1 Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012

2 Objectives In this section we will learn how to: Recognize and combine like terms. Solve linear equations of the form: ax + b = c. Solve absolute value equations of the form: ax + b = c.

3 Recognizing Like Terms A term in an equation is an expression combining variables and constants using the operations of multiplication and division. A term containing only a real number is called a constant term or just a constant. The numerical factor of a non-constant term is called a numerical coefficient or just the coefficient. If no explicit coefficient is written, it is assumed to be 1. If a negative sign precedes a non-constant term without an explicit numerical coefficient, the coefficient is assumed to be 1. If an equation contains more than one term, like terms have the same variables raised to the same powers.

4 Combining Like Terms To combine like terms, add or subtract the coefficients and keep the common variable expression. Unlike terms cannot be combined.

5 Linear Equations: ax + b = c Terminology: An algebraic expression is a combination of variables and numbers using the operations of addition, subtraction, multiplication, and division. An equation is a statement that two algebraic expressions are equal. If an equation contains a variable, any number which makes the equation true when substituted for the variable is called a solution. The set of all numbers making an equation true is called a solution set. Equations with the same solution sets are said to be equivalent. An equation of the form ax + b = c with a 0 is called a linear equation in x.

6 Solving Linear Equations (1 of 2) Equations are solved by using the following properties of equations. Addition/Subtraction Property: if the same expression is added to (or subtracted from) both sides of an equation, the two equations are equivalent. A = B A + C = B + C A C = B C Multiplication/Division Property: if both sides of an equation are multiplied (or divided) by the same non-zero expression, the two equations are equivalent. A = B AC = BC A C = B C if C 0.

7 Solving Linear Equations (2 of 2) To solve a linear equation: 1 Simplify each side of the equation by removing grouping symbols and combining like terms. 2 Use the Addition/Subtraction Property to add the opposites of constants or variable expressions so that the variable expressions are on one side of the equation and the constants are on the other. 3 Use the Multiplication/Division Property to multiply both sides of the equation by the reciprocal of the coefficient of the variable, so that the new coefficient is 1. 4 Check your answer by substituting it into the original equation.

8 Types of Equations Equations can be classified depending on the number of solution they possess. If an equation has a finite number of solutions, it is called a conditional equation. If an equation is solved by every real number in R then the equation is called an identity. If an equation has no solutions (the solution set is the empty set ) the equation is called a contradiction.

9 Absolute Value Definition For any real number x, x = { x if x 0, x if x < 0. Solving equations involving absolute value. If x = c > 0 then x = c or x = c. If ax + b = c > 0 then ax + b = c or ax + b = c. If a = b then a = b or a = b. If ax + b = cx + d then ax + b = cx + d or ax + b = (cx + d).

COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1

COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1 10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010 Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable

More information

Solving univariate equations

Solving univariate equations Click on the links below to jump directly to the relevant section Solving univariate equations Solving for one variable in a multivariate equation Solving systems of multivariate equations Solving univariate

More information

equals equals equals equals

equals equals equals equals Addition of Integers Rules Same Sign ---------------- Add --------------- Keep the Sign Different Signs -------- Subtract ------- Take the sign of the integer with the larger absolute value plus plus plus

More information

Topic 2 Solving Equations

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

More information

Sect Addition, Subtraction, Multiplication, and Division Properties of Equality

Sect Addition, Subtraction, Multiplication, and Division Properties of Equality Sect.1 - Addition, Subtraction, Multiplication, and Division Properties of Equality Concept #1 Definition of a Linear Equation in One Variable An equation is a statement that two quantities are equal.

More information

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

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

More information

1-2 Properties of Real Numbers

1-2 Properties of Real Numbers 1-2 Properties of Real Numbers Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Simplify. 1. 5+5 0 2. 1 3. 1.81 4. Find 10% of $61.70. $6.17 5. Find the reciprocal of 4. Objective Identify and use properties

More information

Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra Hawkes Learning Systems: College Algebra Section 1.2: The Arithmetic of Algebraic Expressions Objectives o Components and terminology of algebraic expressions. o The field properties and their use in algebra.

More information

LINEAR EQUATIONS. Example: x + 2 = 4 Linear equation: highest exponent of the variable is 1.

LINEAR EQUATIONS. Example: x + 2 = 4 Linear equation: highest exponent of the variable is 1. LINEAR EQUATIONS A linear equation can be defined as an equation in which the highest exponent of the equation variable is one. When graphed, the equation is shown as a single line. Example: x + = 4 Linear

More information

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 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

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

No Solution Equations Let s look at the following equation: 2 +3=2 +7

No Solution Equations Let s look at the following equation: 2 +3=2 +7 5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

Solving 1 and 2 Step Equations

Solving 1 and 2 Step Equations Section 2 1: Solving 1 and 2 Step Equations Epressions The last chapter in this book contained epressions. The net type of algebraic statement that we will eamine is an equation. At the start of this section

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations

Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapters 1-2 Learning Objectives: In chapter 1 students

More information

JUST THE MATHS UNIT NUMBER 1.7. ALGEBRA 7 (Simultaneous linear equations) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.7. ALGEBRA 7 (Simultaneous linear equations) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.7 ALGEBRA 7 (Simultaneous linear equations) by A.J.Hobson 1.7.1 Two simultaneous linear equations in two unknowns 1.7.2 Three simultaneous linear equations in three unknowns

More information

For Students Entering Algebra 1

For Students Entering Algebra 1 Summer Math Packet Student Name: 1 For Students Entering Algebra 1 This summer math booklet was developed to provide students in middle school an opportunity to review grade level math objectives and to

More information

3.1 Solving Systems Using Tables and Graphs

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

More information

EQUATIONS and INEQUALITIES

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

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

The Addition Property of Equality. The Addition Property of Equality. Adding the same number to both sides Solve x 3 7. x 3 7

The Addition Property of Equality. The Addition Property of Equality. Adding the same number to both sides Solve x 3 7. x 3 7 dug508_ch0a.qxd 8/9/0 :55 PM Page 66 66 ( ) Chapter Linear Equations in One Variable In this helpful section The Addition Property of Equality The Multiplication Property of Equality Variables on Both

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

More information

Quadratics - Quadratic Formula

Quadratics - Quadratic Formula 9.4 Quadratics - Quadratic Formula Objective: Solve quadratic equations by using the quadratic formula. The general from of a quadratic is ax + bx + c = 0. We will now solve this formula for x by completing

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation. 1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

More information

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

Properties of Real Numbers

Properties of Real Numbers 16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

More information

Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

More information

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

More information

Algebra Cheat Sheets

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

More information

Algebra I Notes Review Real Numbers and Closure Unit 00a

Algebra I Notes Review Real Numbers and Closure Unit 00a Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,

More information

troduction to Algebra

troduction to Algebra Chapter Three Solving Equations and Problem Solving Section 3.1 Simplifying Algebraic Expressions In algebra letters called variables represent numbers. The addends of an algebraic expression are called

More information

MATH REFRESHER I. Notation, Solving Basic Equations, and Fractions. Rockefeller College MPA Welcome Week 2016

MATH REFRESHER I. Notation, Solving Basic Equations, and Fractions. Rockefeller College MPA Welcome Week 2016 MATH REFRESHER I Notation, Solving Basic Equations, and Fractions Rockefeller College MPA Welcome Week 2016 Lucy C. Sorensen Assistant Professor of Public Administration and Policy 1 2 Agenda Why Are You

More information

Equations, Inequalities, Solving. and Problem AN APPLICATION

Equations, Inequalities, Solving. and Problem AN APPLICATION Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a $0 setup fee plus $ per person. The cost of Hotel

More information

Graphing Quadratic Functions: Parabolas

Graphing Quadratic Functions: Parabolas Graphing Quadratic Functions: Parabolas MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: graph a parabola (a quadratic function)

More information

6-3 Solving Systems by Elimination

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

More information

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations 1 Solving Equations An equation is a statement expressing the equality of two mathematical expressions. It may have numeric

More information

MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix

MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix MATH 05: Finite Mathematics 2-6: The Inverse of a Matrix Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of

More information

Equations of Lines Derivations

Equations of Lines Derivations Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Math Help and Additional Practice Websites

Math Help and Additional Practice Websites Name: Math Help and Additional Practice Websites http://www.coolmath.com www.aplusmath.com/ http://www.mathplayground.com/games.html http://www.ixl.com/math/grade-7 http://www.softschools.com/grades/6th_and_7th.jsp

More information

Sect Properties of Real Numbers and Simplifying Expressions

Sect Properties of Real Numbers and Simplifying Expressions Sect 1.6 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a.34 + 2.5 Ex. 1b 2.5 + (.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a).34 + 2.5 = 6.84

More information

Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

More information

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.

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

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

1.4 Compound Inequalities

1.4 Compound Inequalities Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Learning Objectives for Section 1.1 Linear Equations and Inequalities

Learning Objectives for Section 1.1 Linear Equations and Inequalities Learning Objectives for Section 1.1 Linear Equations and Inequalities After this lecture and the assigned homework, you should be able to solve linear equations. solve linear inequalities. use interval

More information

Reasoning with Equations and Inequalities

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

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

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

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

More information

Chapter 1.1 Rational and Irrational Numbers

Chapter 1.1 Rational and Irrational Numbers Chapter 1.1 Rational and Irrational Numbers A rational number is a number that can be written as a ratio or the quotient of two integers a and b written a/b where b 0. Integers, fractions and mixed numbers,

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N

I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N I N D U BRIDGING THE GAP TO A/S LEVEL MATHS C T I O N INTRODUCTION TO A LEVEL MATHS The Mathematics Department is committed to ensuring that you make good progress throughout your A level or AS course.

More information

Fractions and Linear Equations

Fractions and Linear Equations Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

More information

IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

A Systematic Approach to Factoring

A Systematic Approach to Factoring A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

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

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.

More information

Use your TI-84 graphing calculator to check as many problems as possible.

Use your TI-84 graphing calculator to check as many problems as possible. Name: Date: Period: Dear Future Algebra Honors student, We hope that you enjoy your summer vacation to the fullest. We look forward to working with you next year. As you enter your new math class, you

More information

Algebra 2: Q1 & Q2 Review

Algebra 2: Q1 & Q2 Review Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

More information

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

More information

3. Solve the equation containing only one variable for that variable.

3. Solve the equation containing only one variable for that variable. Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated

More information

1.1 Solving a Linear Equation ax + b = 0

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)

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

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,

More information

Florida Math for College Readiness

Florida Math for College Readiness Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

More information

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

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.

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

2.3. Finding polynomial functions. An Introduction:

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

More information

Solving Equations by the Multiplication Property

Solving Equations by the Multiplication Property 2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean

More information

MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

Systems of Linear Equations: Elimination by Addition

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

More information

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

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

More information

4.2 Algebraic Properties: Combining Expressions

4.2 Algebraic Properties: Combining Expressions 4.2 Algebraic Properties: Combining Expressions We begin this section with a summary of the algebraic properties of numbers. Property Name Property Example Commutative property (of addition) Commutative

More information

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. 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

More information

Unit 3 Polynomials Study Guide

Unit 3 Polynomials Study Guide Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial

More information

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

More information

Solving simultaneous equations. Jackie Nicholas

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

More information

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12 Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

More information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

More information

Foundations for Middle School 6 th Grade Syllabus

Foundations for Middle School 6 th Grade Syllabus Unit 1:, and Percents (Lessons 0 19) and Percents (Lessons 0 19) Foundations for Middle School 6 th Grade Syllabus Solving for every variable Foundations for Middle School 6 th Grade Syllabus 1 Unit 1:

More information

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

More information

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

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

More information

MATH Fundamental Mathematics II.

MATH Fundamental Mathematics II. MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

More information

Chapter 5 - Polynomials and Polynomial Functions

Chapter 5 - Polynomials and Polynomial Functions Math 233 - Spring 2009 Chapter 5 - Polynomials and Polynomial Functions 5.1 Addition and Subtraction of Polynomials Definition 1. A polynomial is a finite sum of terms in which all variables have whole

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information