5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality
|
|
|
- Kristopher Casey
- 5 years ago
- Views:
Transcription
1 5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality Now that we have studied the Addition Property of Equality and the Multiplication Property of Equality, we can solve equations that require the use of both properties. In this chapter, we have been solving linear equations in one variable, which are equations that contain only one variable, and that variable is raised only to the first power. The following are examples of linear equations in one variable that we will study in this section. x + = = = x 6x 8x + 2 = x (x + 2) = 7 STEPS TO SOLVING LINEAR EQUATIONS IN ONE VARIABLE. If the expression on either side of the equation has like terms, combine those like terms. 2. Move all variable terms to one side of the equation and all constant terms to the other side of the equation using the Addition Property of Equality.. Get the variable by itself on one side of the equation using the Multiplication Property of Equality.. Check the answer in the original equation. If that answer produces a true statement, then the answer is the solution. 0
2 Example : Solve the equation x + =. Then check the solution. Let s try to do this by thinking about what the equation says. times what number plus equals? First, ask yourself what number plus equals? We know that 8 plus equals. So x must equal 8. So, times what number equals 8? The number is 2. This was a little more difficult to work through. Now, we will learn how to solve this equation algebraically. To solve the equation means to determine the value of the variable that makes the equation a true statement. To do this, we want to get the variable on one side of the equation by itself; we call this isolating the variable. We will do this using the steps for solving equations. x x - - x 8 x 8 x 2. The expressions on the right and left of the equal sign are already simplified. Skip to step Subtract from both sides.. Divide both sides of the equation by. Check: Substitute 2 for x in the original equation. ( 2) 8 This is a true statement. Therefore, x = 2 is the solution of x. Looking at Example, we can see that each equation in the solving process looks a little different from the preceding one. What is interesting and useful is that each of the equations says the same thing about x: each one says that x is 2. The last equation, of course, is the easiest to read, which is why our goal is to end up with x isolated on one side of the equation. Practice : Solve the equation 7x + 2 = 2. Answer: x = 05
3 Example 2: Solve the equation = -. Then check the solution x 2. The expressions on the right and left of the equal sign are already simplified. Skip to step Add to both sides of the equation.. Divide both sides of the equation by 2. Check: Substitute - 2 for x in the original equation Substitute for x. 2 Write as a fraction multiplication problem. Divide out a factor of 2 in the numerator and denominator. This is a true statement. Therefore, x = - 2 is the solution of = -. Practice 2: Solve the equation 5x 9 = 2. Answer: x =
4 Example : Solve the equation 2 2 x. x 2 2 x x 0 x (0) The check is left to you. x 0 0 x 0 x. The expressions on the right and left of the equal sign are already simplified. Skip to step Subtract 2 from both sides of the equation.. Multiply both sides of the equation by, the reciprocal of the coefficient of x. Divide out common factors in the numerator and denominator. Practice : Solve the equation 2 x 2 0. Answer: x = 8 Example : Solve the equation The expressions on the right and left of the equal sign are already simplified. Skip to step Add.7 to both sides of the equation x Divide both sides of the equation by 0.2. Simplify The check is left to you. Practice : Solve the equation 0.5x Answer: x =
5 Example 5: Solve the equation x. Then check the solution x x.5 0.5x.5 0.5x x Check: Substitute 7 for x in the original equation x ( 7) The expressions on the right and left of the equal sign are already simplified. Skip to step Add.5 to both sides of the equation.. Divide both sides of the equation by 0.5. Simplify 2 2 This is a true statement. Therefore, x = 7 is the solution of x. Practice 5: Solve the equation Answer: x = Example 6: Solve the equation 5xx 5 9. Then check the solution. 5xx 5 9 Combine like terms on the left side of the equal sign. Combine like terms on the right side of the equal sign. Subtract from both sides of the equation x Divide both sides of the equation by 2, the coefficient of x. Check: Substitute - for x in the original equation. 5x x 5 9 5( -) ( -) This is a true statement. Therefore, x = - is the solution of 5xx
6 Practice 6: Solve the equation5x 0. Answer: x = Example 7: Solve the equation 2 6x. Then check the solution x x 22 To combine like terms on the left side of the equal sign the fractions must have common denominators. Multiply the first fraction s numerator and denominator by 2 in order to make a common denominator of in both fractions x 5 Add the numerators and keep the same 2 6x denominator x Subtract 2 from both sides of the equation x 5 2 6x Write 2 as the fraction 2 in order to subtract the fractions. Multiply the first fraction s numerator and denominator by in order to make a common denominator of in both fractions x Add the numerators and keep the same 6x denominator. ( 6 x) Multiply both sides of the equation by the 6 6 reciprocal of the coefficient of x. On the left side, divide out a in the numerator x and denominator. On the right side, use the 2 6 inverse property of multiplication to simplify. x 8 09
7 Check: Substitute 8 for x in the original equation. 2 6x This is a true statement. Therefore, x is the solution of 2 6x. 8 2 Practice 7: Solve the equation 5. 8 Answer: Example 8: Solve the equation5( x) x 5 8. Then check the solution. 5( x) x x - x x Distribute.. Simplify the left and right sides of the equation by combining like terms. 2. Subtract 5 from both sides of the equation.. Divide both sides of the equation by 2. 0
8 Check: Substitute - for x in the original equation. 5( x) x 5 8 5( ( ) ) ( ) 5 8 5( ) ( ) ( 2) This is a true statement. Therefore, x = - is the solution of 5( x) x 5 8. Practice 8: Solve the equation 2( x) x 7. Answer: x = - Watch All:
9 5.6 Solving Equations using the Addition and Multiplication Property Exercises Solve each equation.. 5x = x = = 2y. x = = y x 7. 2 x x a + = 5. 9 = 7y = x x. 6x 9x + 2 = x = x t x 7x 9. ( + ) x = (x + 5) 6 = 2 2
10 5.6 Addtion and Multiplication Property Exercises Answers. x = 8 2. x = = y. x = = y 6. x = x or 7 8. x x.6 0. a =. -6 = y 2. x = -. x =. x = 0 5. x = x 0 7. t 8. x 2 9. x = x =
Chapter 7 Rational Expressions, Equations, and Functions
Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions
Solving Two-Step Equations
Solving Two-Step Equations Warm Up Problem of the Day Lesson Presentation 3 Warm Up Solve. 1. x + 12 = 35 2. 8x = 120 y 9 3. = 7 4. 34 = y + 56 x = 23 x = 15 y = 63 y = 90 Learn to solve two-step equations.
3.5 Solving Equations Involving Integers II
208 CHAPTER 3. THE FUNDAMENTALS OF ALGEBRA 3.5 Solving Equations Involving Integers II We return to solving equations involving integers, only this time the equations will be a bit more advanced, requiring
MA Lesson 25 Section 2.6
MA 1500 Lesson 5 Section.6 I The Domain of a Function Remember that the domain is the set of x s in a function, or the set of first things. For many functions, such as f ( x, x could be replaced with any
A field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
Chapter 1.6. Perform Operations with Complex Numbers
Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each
Before this course is over we will see the need to split up a fraction in a couple of ways, one using multiplication and the other using addition.
CH MORE FRACTIONS Introduction I n this chapter we tie up some loose ends. First, we split a single fraction into two fractions, followed by performing our standard math operations on positive and negative
Partial 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
Summer MA Lesson 19 Section 2.6, Section 2.7 (part 1)
Summer MA 100 Lesson 1 Section.6, Section.7 (part 1) I The Domain of a Function Remember that the domain is the set of x s in a function, or the set of first things. For many functions, such as f ( x,
Fractions. Review R.7. Dr. Doug Ensley. January 7, Dr. Doug Ensley Review R.7
Review R.7 Dr. Doug Ensley January 7, 2015 Equivalence of fractions As long as c 0, a b = a c b c Equivalence of fractions As long as c 0, a b = a c b c Examples True or False? 10 18 = 2 5 2 9 = 5 9 10
Algebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
Solving Quadratic & Higher Degree Inequalities
Ch. 10 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic
There are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality
Chapter 1.1: Solving Linear and Literal Equations Linear Equations Linear equations are equations of the form ax + b = c, where a, b and c are constants, and a zero. A hint that an equation is linear is
Ch. 11 Solving Quadratic & Higher Degree Inequalities
Ch. 11 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic
Geometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
Algebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
ALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations
ALGEBRA 1 Interactive Notebook Chapter 2: Linear Equations 1 TO WRITE AN EQUATION: 1. Identify the unknown (the variable which you are looking to find) 2. Write the sentence as an equation 3. Look for
Before this course is over we will see the need to split up a fraction in a couple of ways, one using multiplication and the other using addition.
CH 0 MORE FRACTIONS Introduction I n this chapter we tie up some loose ends. First, we split a single fraction into two fractions, followed by performing our standard math operations on positive and negative
We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).
College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite
Algebra 31 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE. A rational expression is just a fraction involving polynomials, for example 3x2 2
MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE SECTION 1.2: PRECALCULUS REVIEW II Practice: 3, 7, 13, 17, 19, 23, 29, 33, 43, 45, 51, 57, 69, 81, 89 1. Rational Expressions and Other Algebraic Fractions A rational
Numerical and Algebraic Fractions
Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core
The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.
Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify 7 6 +3 8 4 3 4 The
264 CHAPTER 4. FRACTIONS cm in cm cm ft pounds
6 CHAPTER. FRACTIONS 9. 7cm 61. cm 6. 6ft 6. 0in 67. 10cm 69. pounds .. DIVIDING FRACTIONS 6. Dividing Fractions Suppose that you have four pizzas and each of the pizzas has been sliced into eight equal
Unit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
Factoring and Algebraic Fractions
Worksheet. Algebraic Fractions Section Factoring and Algebraic Fractions As pointed out in worksheet., we can use factoring to simplify algebraic expressions, and in particular we can use it to simplify
Section 4.6 Negative Exponents
Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals
Surds, and other roots
Surds, and other roots Roots and powers are closely related, but only some roots can be written as whole numbers. Surds are roots which cannot be written in this way. Nevertheless, it is possible to manipulate
1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.
1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors
Solving an equation involves undoing these things. We work backward through this verbal model.
To solve an equation, undo what was done to the variable. Intermediate algebra Class notes Solving Radical Equations and Problem Solving (section 10.6) Main idea: To solve most linear equations (and some
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 16 Solving Single Step Equations
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Please watch Section 16 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item67.cfm
SOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
Unit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
Inverse Operations. What is an equation?
Inverse Operations What is an equation? An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in 2+=5 9
DIRECTED NUMBERS ADDING AND SUBTRACTING DIRECTED NUMBERS
DIRECTED NUMBERS POSITIVE NUMBERS These are numbers such as: 3 which can be written as +3 46 which can be written as +46 14.67 which can be written as +14.67 a which can be written as +a RULE Any number
Class 8: Numbers Exercise 3B
Class : Numbers Exercise B 1. Compare the following pairs of rational numbers: 1 1 i First take the LCM of. LCM = 96 Therefore: 1 = 96 Hence we see that < 6 96 96 1 1 1 1 = 6 96 1 or we can say that
Chapter 1 Indices & Standard Form
Chapter 1 Indices & Standard Form Section 1.1 Simplifying Only like (same letters go together; same powers and same letter go together) terms can be grouped together. Example: a 2 + 3ab + 4a 2 5ab + 10
Sect Properties of Real Numbers and Simplifying Expressions
Sect 1.7 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a 9.34 + 2.5 Ex. 1b 2.5 + ( 9.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a) 9.34 + 2.5
Powers, Algebra 1 Teacher Notes
Henri Picciotto Powers, Algebra 1 Teacher Notes Philosophy The basic philosophy of these lessons is to teach for understanding. Thus: - The lessons start by describing a situation without invoking new
Some of the more common mathematical operations we use in statistics include: Operation Meaning Example
Introduction to Statistics for the Social Sciences c Colwell and Carter 206 APPENDIX H: BASIC MATH REVIEW If you are not using mathematics frequently it is quite normal to forget some of the basic principles.
Rational Numbers CHAPTER. 1.1 Introduction
RATIONAL NUMBERS Rational Numbers CHAPTER. Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + = () is solved when x =, because this value
College Algebra. Chapter 5 Review Created by: Lauren Atkinson. Math Coordinator, Mary Stangler Center for Academic Success
College Algebra Chapter 5 Review Created by: Lauren Atkinson Math Coordinator, Mary Stangler Center for Academic Success Note: This review is composed of questions from the chapter review at the end of
Supplemental Worksheet Problems To Accompany: The Algebra 2 Tutor Section 13 Fractional Exponents
Section Fractional Eponents Supplemental Worksheet Problems To Accompany: The Algebra 2 Tutor Section Fractional Eponents Please watch Section of this DVD before working these problems. The DVD is located
Factorizing Algebraic Expressions
1 of 60 Factorizing Algebraic Expressions 2 of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. Expanding or multiplying out a(b + c) ab + ac Factorizing Often: When
Section 2.3 Solving Linear Equations
Variable: Defined as A symbol (or letter) that is used to represent an unknown numbers Examples: a, b, c, x, y, z, s, t, m, n, Constant: Defined as A single number Examples: 1, 2, 3, 6, 1, π, e, π, 1.6,
Unit 9 Study Sheet Rational Expressions and Types of Equations
Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by
Sect Least Common Denominator
4 Sect.3 - Least Common Denominator Concept #1 Writing Equivalent Rational Expressions Two fractions are equivalent if they are equal. In other words, they are equivalent if they both reduce to the same
Ch. 12 Rational Functions
Ch. 12 Rational Functions 12.1 Finding the Domains of Rational F(n) & Reducing Rational Expressions Outline Review Rational Numbers { a / b a and b are integers, b 0} Multiplying a rational number by a
5-9. Complex Numbers. Key Concept. Square Root of a Negative Real Number. Key Concept. Complex Numbers VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
TEKS FOCUS 5-9 Complex Numbers VOCABULARY TEKS (7)(A) Add, subtract, and multiply complex TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(D),
Lesson 26: Solving Rational Equations
Lesson 2: Solving Rational Equations Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to
Solving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically
10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities
Lesson 7: Watch Your Step!
In previous lessons, we have looked at techniques for solving equations, a common theme throughout algebra. In this lesson, we examine some potential dangers where our intuition about algebra may need
Suppose we have the set of all real numbers, R, and two operations, +, and *. Then the following are assumed to be true.
Algebra Review In this appendix, a review of algebra skills will be provided. Students sometimes think that there are tricks needed to do algebra. Rather, algebra is a set of rules about what one may and
Section 2.4: Add and Subtract Rational Expressions
CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall
Student Instruction Sheet: Unit 1 Lesson 3. Polynomials
Student Instruction Sheet: Unit 1 Lesson 3 Suggested time: 150 min Polynomials What s important in this lesson: You will use algebra tiles to learn how to add/subtract polynomials. Problems are provided
we first add 7 and then either divide by x - 7 = 1 Adding 7 to both sides 3 x = x = x = 3 # 8 1 # x = 3 # 4 # 2 x = 6 1 =?
. Using the Principles Together Applying Both Principles a Combining Like Terms a Clearing Fractions and Decimals a Contradictions and Identities EXAMPLE Solve: An important strategy for solving new problems
Chapter 9 Notes SN AA U2C9
Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation
5.7 Translating English Sentences into Mathematical Equations and Solving
5.7 Translating English Sentences into Mathematical Equations and Solving Mathematical equations can be used to describe many situations in the real world. To do this, we must learn how to translate given
Roots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
Lesson 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
9.4 Radical Expressions
Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.
LS.2 Homogeneous Linear Systems with Constant Coefficients
LS2 Homogeneous Linear Systems with Constant Coefficients Using matrices to solve linear systems The naive way to solve a linear system of ODE s with constant coefficients is by eliminating variables,
Inverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics
1.4 Properties of Real Numbers and Algebraic Expressions
0 CHAPTER Real Numbers and Algebraic Expressions.4 Properties of Real Numbers and Algebraic Expressions S Use Operation and Order Symbols to Write Mathematical Sentences. 2 Identify Identity Numbers and
Appendix. Using Your Calculator. Squares, Square Roots, Reciprocals, and Logs. Addition, Subtraction, Multiplication, and Division
370770_app.qxd 1/9/03 7:2 PM Page A1 mac114 Mac 114:2nd shift:4_rst: Using Your Calculator In this section we will review how to use your calculator to perform common mathematical operations. This discussion
P.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, i. DEFINITION Complex Number. Operations with Complex Numbers
SECTION P.6 Complex Numbers 49 P.6 Complex Numbers What you ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations... and
Name: Chapter 7: Exponents and Polynomials
Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You
Algebra III and Trigonometry Summer Assignment
Algebra III and Trigonometry Summer Assignment Welcome to Algebra III and Trigonometry! This summer assignment is a review of the skills you learned in Algebra II. Please bring this assignment with you
Equations and Inequalities
Equations and Inequalities 2 Figure 1 CHAPTER OUTLINE 2.1 The Rectangular Coordinate Systems and Graphs 2.2 Linear Equations in One Variable 2.3 Models and Applications 2.4 Complex Numbers 2.5 Quadratic
27 = 3 Example: 1 = 1
Radicals: Definition: A number r is a square root of another number a if r = a. is a square root of 9 since = 9 is also a square root of 9, since ) = 9 Notice that each positive number a has two square
Lesson 3: Solving Equations A Balancing Act
Opening Exercise Let s look back at the puzzle in Lesson 1 with the t-shape and the 100-chart. Jennie came up with a sum of 380 and through the lesson we found that the expression to represent the sum
Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.
Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational
base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
( 4 p 3. ( 2 p 2. ( x 3 y 4. ( y. (2 p 2 ) 2 ( q 4 ) 2. ( x 2 ) POLYNOMIALS, PAGES CHECK IT OUT! PAGES
8. _ x 4 y 8 x 4-6 y 8-6 x 6 y 6 x - y y x 9. 5 m n 4 5 m - n 4-1 m n 5 m 0 n 3 30. ( 3 5) 3 3 3 31. _ ( 4 p 3 5 1 n 3 5 n 3 5 3 _ 7 15 4) p q ( 4 p 3-1 q -4 ) ( p q -4 ) ( p q 4 ) ( p ) ( q 4 ) _ ( p
Solving Equations with Addition and Subtraction
OBJECTIVE: You need to be able to solve equations by using addition and subtraction. In math, when you say two things are equal to each other, you mean they represent the same value. We use the = sign
Lesson 2. When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors.
Review of Exponential Notation: Lesson 2 - read to the power of, where is the base and is the exponent - if no exponent is denoted, it is understood to be a power of 1 - if no coefficient is denoted, it
Pre-Algebra 8 Notes Exponents and Scientific Notation
Pre-Algebra 8 Notes Eponents and Scientific Notation Rules of Eponents CCSS 8.EE.A.: Know and apply the properties of integer eponents to generate equivalent numerical epressions. Review with students
25. REVISITING EXPONENTS
25. REVISITING EXPONENTS exploring expressions like ( x) 2, ( x) 3, x 2, and x 3 rewriting ( x) n for even powers n This section explores expressions like ( x) 2, ( x) 3, x 2, and x 3. The ideas have been
CH 55 THE QUADRATIC FORMULA, PART I
1 CH 55 THE QUADRATIC FORMULA, PART I Introduction I n the Introduction to the previous chapter we considered the quadratic equation 10 + 16 0. We verified in detail that this equation had two solutions:
CH 14 MORE DIVISION, SIGNED NUMBERS, & EQUATIONS
1 CH 14 MORE DIVISION, SIGNED NUMBERS, & EQUATIONS Division and Those Pesky Zeros O ne of the most important facts in all of mathematics is that the denominator (bottom) of a fraction can NEVER be zero.
7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
2017 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
OPTIONAL: Watch the Flash version of the video for Section 6.1: Rational Expressions (19:09).
UNIT V STUDY GUIDE Rational Expressions and Equations Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to: 3. Perform mathematical operations on polynomials and
CHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
Adding and Subtracting Terms
Adding and Subtracting Terms 1.6 OBJECTIVES 1.6 1. Identify terms and like terms 2. Combine like terms 3. Add algebraic expressions 4. Subtract algebraic expressions To find the perimeter of (or the distance
NAME DATE PERIOD. Trigonometric Identities. Review Vocabulary Complete each identity. (Lesson 4-1) 1 csc θ = 1. 1 tan θ = cos θ sin θ = 1
5-1 Trigonometric Identities What You ll Learn Scan the text under the Now heading. List two things that you will learn in the lesson. 1. 2. Lesson 5-1 Active Vocabulary Review Vocabulary Complete each
Further Signed Numbers
Worksheet 1.7 Further Signed Numbers Section 1 Multiplication of signed numbers Multiplication is a shorthand way of adding together a large number of the same thing. For example, if I have 3 bags of oranges
Developed 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
Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3
Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring
1.2. Indices. Introduction. Prerequisites. Learning Outcomes
Indices 1.2 Introduction Indices, or powers, provide a convenient notation when we need to multiply a number by itself several times. In this Section we explain how indices are written, and state the rules
Basic Principles of Algebra
Basic Principles of Algebra Algebra is the part of mathematics dealing with discovering unknown numbers in an equation. It involves the use of different types of numbers: natural (1, 2, 100, 763 etc.),
5.2. November 30, 2012 Mrs. Poland. Verifying Trigonometric Identities
5.2 Verifying Trigonometric Identities Verifying Identities by Working With One Side Verifying Identities by Working With Both Sides November 30, 2012 Mrs. Poland Objective #4: Students will be able to
Equations and Solutions
Section 2.1 Solving Equations: The Addition Principle 1 Equations and Solutions ESSENTIALS An equation is a number sentence that says that the expressions on either side of the equals sign, =, represent
Toss 1. Fig.1. 2 Heads 2 Tails Heads/Tails (H, H) (T, T) (H, T) Fig.2
1 Basic Probabilities The probabilities that we ll be learning about build from the set theory that we learned last class, only this time, the sets are specifically sets of events. What are events? Roughly,
When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.
Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules
AQA Level 2 Further mathematics Further algebra. Section 4: Proof and sequences
AQA Level 2 Further mathematics Further algebra Section 4: Proof and sequences Notes and Examples These notes contain subsections on Algebraic proof Sequences The limit of a sequence Algebraic proof Proof
MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
Spring Nikos Apostolakis
Spring 07 Nikos Apostolakis Review of fractions Rational expressions are fractions with numerator and denominator polynomials. We need to remember how we work with fractions (a.k.a. rational numbers) before