10.6 Systems of Nonlinear Equations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "10.6 Systems of Nonlinear Equations"

Transcription

1 SECTION 0.6 Systems of Nonlinear Equations Systems of Nonlinear Equations PREPARING FOR THIS SECTION Before getting started, review the following: Lines (Section., pp. 7 38) Ellipses (Section 9.3, pp ) Circles (Section.5, pp. 9) Hyperbolas (Section 9., pp ) Parabolas (Section 9., pp ) Now work the Are You Prepared? problems on page 800. OBJECTIVES Solve a System of Nonlinear Equations Using Substitution Solve a System of Nonlinear Equations Using Elimination Solve a System of Nonlinear Equations Using Substitution In Section 0. we observed that the solution to a system of linear equations could be found geometrically by determining the point(s) of intersection (if any) of the equations in the system. Similarly, when solving systems of nonlinear equations, the solution(s) also represents the point(s) of intersection (if any) of the graphs of the equations. There is no general methodology for solving a system of nonlinear equations. At times substitution is best; other times, elimination is best; and sometimes neither of these methods works. Experience and a certain degree of imagination are your allies here. Before we begin, two comments are in order.. If the system contains two variables and if the equations in the system are easy to graph, then graph them. By graphing each equation in the system, you can get an idea of how many solutions a system has and approximately where they are located.. Extraneous solutions can creep in when solving nonlinear systems, so it is imperative that all apparent solutions be checked.

2 79 CHAPTER 0 Analytic Geometry EXAMPLE Solve the following system of equations: 3x - y = - () A line b x - y = 0 () A parabola Algebraic Solution Using Substitution First, we notice that the system contains two variables and that we know how to graph each equation by hand. See Figure 8. The system apparently has two solutions. Figure 8 x y = 0 (y = x ) y 0 3x y = (y = 3x + ) (, 8) We use a graphing utility to graph Y and Y = x = 3x +. From Figure 9 we see that the system apparently has two solutions. Using INTERSECT, the solutions to the system of equations are -0.5, 0.5 and, 8. (, ) 6 6 We will use substitution to solve the system. Equation () is easily solved for y. 3x - y = - Equation () y = 3x + We substitute this expression for y in equation (). The result is an equation containing just the variable x, which we can then solve. x - y = 0 x - 3x + = 0 Equation () Substitute 3x + for y x - 3x - = 0 x + x - = 0 Remove parentheses. Factor. x + = 0 or x - = 0 Apply the Zero-Product Property. x = - or x = Using these values for x in y = 3x +, we find y = 3a - b + = or y = 3 + = 8 The apparent solutions are x = - and x =, y = 8., y = CHECK: For x = -, y = : 3a - b - = = - d a - b - = a b - = 0 For x =, y = 8: 3-8 = 6-8 = - b - 8 = - 8 = 0 Each solution checks. Now we know that the graphs in Figure 8 intersect at a - and at, 8., b x () () () () Figure 9 Y x Y 0 3x 6 6 NOW WORK PROBLEM 5 USING SUBSTITUTION.

3 SECTION 0.6 Systems of Nonlinear Equations 795 Solve a System of Nonlinear Equations Using Elimination Our next example illustrates how the method of elimination works for nonlinear systems. EXAMPLE Solve: b x + y = 3 x - y = 7 () A circle () A parabola Algebraic Solution Using Elimination First, we graph each equation, as shown in Figure 0. Based on the graph, we expect four solutions. By subtracting equation () from equation (), the variable x can be eliminated. b x + y = 3 x - y = 7 y + y = 6 This quadratic equation in y can be solved by factoring. y + y - 6 = 0 y + 3y - = 0 y = -3 or y = Subtract We use these values for y in equation () to find x. If y =, then x = y + 7 = 9, so x = 3 or -3. If y = -3, then x = y + 7 =, so x = or -. We have four solutions: x = 3, y = ; x = -3, y = ; x =, y = -3; and x = -, y = -3. You should verify that, in fact, these four solutions also satisfy equation (), so all four are solutions of the system. The four points, 3,, -3,,, -3, and -, -3, are the points of intersection of the graphs. Look again at Figure 0. We use a graphing utility to graph x + y = 3 and x - y = 7. (Remember that to graph x + y = 3 requires two functions, Y and Y =-33 - x = 33 - x, and a square screen.) From Figure we see that the system apparently has four solutions. Using INTERSECT, the solutions to the system of equations are -3,, 3,, -, -3, and, -3. Figure Y 3 x 5 7 Y 3 x Y 3 x Figure 0 y x y = 7 (y = x 7) ( 3, ) (3, ) x + y = x (, 3) (, 3) 8 NOW WORK PROBLEM 3 USING ELIMINATION.

4 796 CHAPTER 0 Analytic Geometry EXAMPLE 3 Solve: x + x + y - 3y + = 0 c x + + y - y = 0 x () () Algebraic Solution Using Elimination Since it is not straightforward how to graph the equations in the system, we proceed directly to use the method of elimination. First, we multiply equation () by x to eliminate the fraction. The result is an equivalent system because x cannot be 0. [Look at equation () to see why.] b x + x + y - 3y + = 0 x + x + y - y = 0 Now subtract equation () from equation () to eliminate x. The result is -y + = 0 y = Solve for y. To find x, we back-substitute y = in equation (): x + x + y - 3y + = 0 x + x = 0 x = 0 or x = - x + x = 0 xx + = 0 () () Equation () Substitute for y in (). Simplify. Factor. Apply the Zero-Product Property. Because x cannot be 0, the value x = 0 is extraneous, and we discard it. The solution is x = -, y =. CHECK: We now check x = -, y = : = = 0 c = = 0 () () First, we multiply equation () by x to eliminate the fraction. The result is an equivalent system because x cannot be 0 [look at equation () to see why]: b x + x + y - 3y + = 0 x + x + y - y = 0 We need to solve each equation for y. First, we solve equation () for y: x + x + y - 3y + = 0 y - 3y = -x - x - y - 3y + 9 = -x - x ay - 3 b = -x - x + y - 3 = ; A -x - x + y = 3 ; A -x - x + Now we solve equation () for y: x + x + y - y = 0 y - y = -x - x y - y + = -x - x + ay - b = -x - x + y - = ; A -x - x + () () Equation () Rearrange so that terms involving y are on left side. Complete the square involving y. Factor; simplify Square Root Method Solve for y. Equation () Rearrange so that terms involving y are on left side. Complete the square involving y. Factor Square Root Method y = ; A -x - x + Solve for y. Now graph each equation using a graphing utility. See Figure. Using INTERSECT, the points of intersection are -, and 0,. Since x Z 0 [look back at the original equation ()], the graph of Y 3 has a hole at the point 0, and Y has a hole at 0, 0. The value x = 0 is extraneous, and we discard it. The only solution is x = - and y =.

5 SECTION 0.6 Systems of Nonlinear Equations 797 Figure Y 3 x x Y 3 x x Y 3 x x Y x x NOW WORK PROBLEMS 9 AND 9. EXAMPLE Solve: b x - y = y = x () A hyperbola () A parabola Algebraic Solution Either substitution or elimination can be used here. We use substitution and replace x by y in equation (). The result is y - y = y - y + = 0 This is a quadratic equation whose discriminant is - - # # = - # = The equation has no real solutions, so the system is inconsistent. The graphs of these two equations do not intersect. See Figure 3. We graph Y and x - y = x = in Figure. You will need to graph x - y = as two functions: Y = 3x - and Y 3 = -3x - From Figure we see that the graphs of these two equations do not intersect. The system is inconsistent. Figure Y x Figure 3 Y x (, ) y 5 y = x (, ) 6 6 ( 3, 5) (3, 5 ) 5 5 ( 3, 5 ) x (3, 5 ) Y 3 x 5 x y =

6 798 CHAPTER 0 Analytic Geometry EXAMPLE 5 Algebraic Solution Solve: b 3xy - y = - 9x + y = 0 We multiply equation () by and add the result to equation () to eliminate the y terms. () b 6xy - y = - 9x + y = 0 () 9x + 6xy = 6 Add. 3x + xy = Divide each side by 3. Since x Z 0 (do you see why?), we can solve for y in this equation to get y = - 3x, x Z 0 (3) x Now substitute for y in equation () of the system. 9x + y = 0 Equation () 9x + - 3x = 0 Substitute y = - 3x in (). x x 9x + - x + 9x x = 0 9x + - x + 9x = 0x Multiply both sides by x. 8x - x + = 0 Subtract 0x from both sides. 9x - x + = 0 Divide both sides by. This quadratic equation (in x ) can be factored: 9x - x - = 0 9x - = 0 or x - = 0 x = 9 x = () () To graph 3xy - y = -, we need to solve for y.in this instance, it is easier to view the equation as a quadratic equation in the variable y. 3xy - y = - y - 3xy - = 0 Place in standard form. y = --3x ; -3x - - Use the quadratic formula with a =, b = -3x, c = -. y = 3x ; 39x + 6 Simplify. Using a graphing utility, we graph Y = 3x + 39x + 6 and From equation (), we graph Y 3 = Y = x. Figure 5 Y 3 See Figure x Y = 3x - 39x x Y 3x 9x 6 and x = ; A 9 = ; x = ; 3 To find y, we use equation (3): 3 3 If If If x = 3 : y = - 3x x x = - 3 : y = - 3x x x = : y = - 3x x = - 3 = - If x = -: y = - 3x = - 3- = x - The system has four solutions. Check them for yourself. = = - 3 a 3 b a- - 3 = 3 b = = - = - Y 0 9x Y 3x 9x 6 Using INTERSECT, the solutions to the system of equations are -, 0.5, 0.7,.,, -0.5, and -0.7, -., each rounded to two decimal places. NOW WORK PROBLEM 7.

7 SECTION 0.6 Systems of Nonlinear Equations 799 EXAMPLE 6 Running a Long Distance Race In a 50-mile race, the winner crosses the finish line mile ahead of the second-place runner and miles ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how many miles does the second-place runner beat the third-place runner? 3 miles mile Solution Let v, v, and v 3 denote the speeds of the first-, second-, and third-place runners, respectively. Let t and t denote the times (in hours) required for the first-place runner and second-place runner to finish the race. Then we have the system of equations 50 = v t 9 = v d t 6 = v 3 t 50 = v t () First-place runner goes 50 miles in t. () Second-place runner goes 9 miles in t. (3) Third-place runner goes 6 miles in t. () Second-place runner goes 50 miles in t. We seek the distance d of the third-place runner from the finish at time t. At time t, the third-place runner has gone a distance of v 3 t miles, so the distance d remaining is 50 - v 3 t. Now d = 50 - v 3 t = 50 - v 3 t # t t = 50 - v 3 t # t t = 50-6 # = 50-6 # v v = 50-6 # 50 9 L 3.06 miles 50 v 50 v From (3), v 3 t = 6 From (), t = 50 e v From (), t = 50 v Form the quotient of () and ().

8 800 CHAPTER 0 Analytic Geometry HISTORICAL FEATURE In the beginning of this section, we said that imagination and experience are important in solving systems of nonlinear equations. Indeed, these kinds of problems lead into some of the deepest and most difficult parts of modern mathematics. Look again at the graphs in Examples and of this section (Figures 8 and 0). We see that Example has two solutions, and Example has four solutions. We might conjecture that the number of solutions is equal to the product of the degrees of the equations involved. Historical Problem A papyrus dating back to 950 BC contains the following problem: A given surface area of 00 units of area shall be represented as the sum of two squares whose sides are to each other as : 3. This conjecture was indeed made by Etienne Bezout ( ), but working out the details took about 50 years. It turns out that, to arrive at the correct number of intersections, we must count not only the complex number intersections, but also those intersections that, in a certain sense, lie at infinity. For example, a parabola and a line lying on the axis of the parabola intersect at the vertex and at infinity. This topic is part of the study of algebraic geometry. Solve for the sides by solving the system of equations x + y = 00 c x = 3 y

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

Math Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.

Math Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices. Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that

More information

Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

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

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square 9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

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

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

7.2 Quadratic Equations

7.2 Quadratic Equations 476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

Solving Logarithmic Equations

Solving Logarithmic Equations Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

More information

Systems of Equations Involving Circles and Lines

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

More information

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

More information

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Year 9 set 1 Mathematics notes, to accompany the 9H book. Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

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

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

More information

Algebra 1-2. A. Identify and translate variables and expressions.

Algebra 1-2. A. Identify and translate variables and expressions. St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

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

Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole

More information

Analytic Geometry Section 2-6: Circles

Analytic Geometry Section 2-6: Circles Analytic Geometry Section 2-6: Circles Objective: To find equations of circles and to find the coordinates of any points where circles and lines meet. Page 81 Definition of a Circle A circle is the set

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Equations Involving Fractions

Equations Involving Fractions . Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

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

Unit 10: Quadratic Relations

Unit 10: Quadratic Relations Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

More information

Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination

Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used

More information

Assessment Schedule 2013

Assessment Schedule 2013 NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

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

What is a parabola? It is geometrically defined by a set of points or locus of points that are

What is a parabola? It is geometrically defined by a set of points or locus of points that are Section 6-1 A Parable about Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directrix).

More information

Quadratic Modeling Business 10 Profits

Quadratic Modeling Business 10 Profits Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

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

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

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

Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate) New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

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

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

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479 Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

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

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

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

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

MATH 143 Pre-calculus Algebra and Analytic Geometry

MATH 143 Pre-calculus Algebra and Analytic Geometry MATH 143 Pre-calculus Algebra and Analytic Geometry Course Guide Self-paced study. Anytime. Anywhere! Math 143 Pre-calculus Algebra and Analytic Geometry University of Idaho 3 Semester-Hour Credits Prepared

More information

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides: In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

More information

CAHSEE on Target UC Davis, School and University Partnerships

CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

Math Common Core Sampler Test

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

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

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM

Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM NAME: Calculus with Analytic Geometry I Exam 5-Take Home Part Due: Monday, October 3, 2011; 12PM INSTRUCTIONS. As usual, show work where appropriate. As usual, use equal signs properly, write in full sentences,

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A.

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A. Order of Operations Section 1: Introduction You know from previous courses that if two quantities are added, it does not make a difference which quantity is added to which. For example, 5 + 6 = 6 + 5.

More information

Simplifying Algebraic Fractions

Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

More information

10.1 Systems of Linear Equations: Substitution and Elimination

10.1 Systems of Linear Equations: Substitution and Elimination 726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

More information

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Algebra I. In this technological age, mathematics is more important than ever. When students

Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

More information

Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring 4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

More information

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

More information

Solving Quadratic & Higher Degree Inequalities

Solving Quadratic & Higher Degree Inequalities Ch. 8 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

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

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2 Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

More information

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. 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

More information

ModuMath Algebra Lessons

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

More information

12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following:

12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Section 1.6 Logarithmic and Exponential Equations 811 1.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solve Quadratic Equations (Section

More information

Equations and Inequalities

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.

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

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

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

More information

5 Systems of Equations

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

More information

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

More information

Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

More information

Linear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

Linear Equations ! 25 30 35$ &  350 150% &  11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

More information

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2 Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

More information

Algebra I Credit Recovery

Algebra I Credit Recovery Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

More information

Utah Core Curriculum for Mathematics

Utah Core Curriculum for Mathematics Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions

More information

Sequences. A sequence is a list of numbers, or a pattern, which obeys a rule.

Sequences. A sequence is a list of numbers, or a pattern, which obeys a rule. Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the

More information

8.9 Intersection of Lines and Conics

8.9 Intersection of Lines and Conics 8.9 Intersection of Lines and Conics The centre circle of a hockey rink has a radius of 4.5 m. A diameter of the centre circle lies on the centre red line. centre (red) line centre circle INVESTIGATE &

More information

2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

More information

Algebra 2. Curriculum Map

Algebra 2. Curriculum Map Algebra 2 Curriculum Map Table of Contents Unit 1: Basic Concepts of Algebra Unit 7: Solving Quadratic Unit 2: Inequalities Unit 8: Variation and Proportion Unit 3: Linear and Functions Unit 9: Analytic

More information

Mathematical Procedures

Mathematical Procedures CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

More information

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved. 2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

More information