Example 1: Model A Model B Total Available. Gizmos. Dodads. System:

Size: px
Start display at page:

Download "Example 1: Model A Model B Total Available. Gizmos. Dodads. System:"

Transcription

1 Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world problems. Eample : Gimos Model A Model B Total Available Dodads Sstem: The goal of solving a sstem is to find the values of and that satisf both equations simultaneousl. Three methods of solving a sstem we will investigate include graphing, solving algebraicall, and b using matrices. I. Graphical Solution When solving sstems of two linear equations, there are three possible outcomes:... In an attempt to solve the sstem in the calculator, we must first solve each equation for and then enter the equations in the calculator into Y and Y to find the intersection, if it eists. II. Algebraic Solution Elimination Method Eample: Substitution Method Eample: 9

2 III. Matrices An individual element in a matri is identified b its location in the matri. Coefficient matri Column vector Solution vector An identit matri is a matri that has the following characteristics: Back to Eample : or AX Y Inverse Matri Method: Just as an inverse function undoes what a function does, the inverse matri undoes what the matri does. AX Y AA I A AX A Y X A Y We can get a solution to a sstem of equations b multipling the inverse of the coefficient matri b the (solution) column vector in the proper order (and order is critical!) on the graphing calculator. To set up the matrices on the calculator: Access the MATRIX command (over the ke) b using the kestrokes nd EDIT (enter the sie of the coefficient matri A b # rows #columns) ENTER (enter individual entries from the coefficient matri) ( nd quit). Then, MATRIX EDIT (arrow down to matri B) (enter sie of column vector) (enter individual entries from the column vector) ( nd quit). Now, to do the calculation, enter: MATRIX (select matri [A] if not automaticall selected) ENTER (times) MATRIX (select matri [B]) ENTER ENTER and read the result of the variable column vector. An augmented matri is the coefficient matri combined with the solution column vector. In order to solve a sstem in reduced row echelon form on the calculator, we must input the augmented matri. 5

3 Reduced Row Echelon Form Method: A matri is said to be in row echelon form provided all of the following conditions hold:... A matri is said to be in reduced row echelon form if both the following conditions hold:.. Calculator kestrokes for reduced row echelon form: MATRIX EDIT (enter augmented matri sie) ENTER (enter augmented matri individual entries) nd QUIT MATRIX MATH (scroll down to rref) ENTER MATRIX (select the augmented matri and close parentheses) ENTER and read the solution. Eample: Vitamin A Vitamin D Vitamin E Brand X Brand Y Brand Z Total Potential Solutions: There are three possibilities for solutions: A Consistent Sstem with One Solution: How can I tell? Geometric Interpretation: A Consistent Solution with Infinite Solutions: How can I tell? Geometric Interpretation: An Inconsistent Sstem: How can I tell? Geometric Interpretation: 5

4 Sstems of Equations and Matrices Activit Objectives: Solve sstem using substitution, elimination and graphing Solve sstem using matrices Solve sstem using matrices Set up sstem of equations and solve for applications 5

5 Solving Sstems. Each person in our group should use one of the following methods. Make sure ou all get the same result. Solve the following sstem of equations: a. Using elimination b. Using substitution c. Graphicall d. Using matrices 5. Solve the following sstem of equations: a. Using elimination b. Using substitution c. Graphicall d. Using matrices 5 5

6 Solving a Sstem Using an Inverse Matri: Eample: Consider: Create matri: [A] to be and [B] to be. [A] - [B]= 5. Find the inverse matri using our calculator and them perform the multiplication to get the solution.. Now tr the method on the following sstem: 8 5

7 Applications Involving Sstems. A compan develops two different tpes of snack mi. Tpe A requires ounces of peanuts and 9 ounces of che mi while tpe B requires 5 ounces of peanuts and ounces of che mi. There is a total of 5 ounces of peanuts available and 85 ounces of che mi. How much of each tpe can the compan produce? Set up a sstem for the problem and solve it b an method ou like.. A person flies from Phoeni to Tucson and back (about miles each wa). He finds that it takes him hours to fl to Tucson against a headwind and onl hour to fl back with the wind. What was the airspeed of the plane (speed if there was no wind) and the speed of the wind? 55

8 . Two numbers when added together are 96 and when subtracted are 9. Find the two numbers. Set up a sstem for the situation and solve it b an method ou like.. A man has 9 coins in his pocket, all of which are dimes and quarters. If the total value of his change is $.55, how man dimes and how man quarters does he have? 56

9 57 Solving Sstems. Solve the following sstems using our graphing calculator. Determine if the sstem is consistent (one solution or infinitel man solutions) or inconsistent (no solution). If it has infinitel man solutions, write the dependent variable(s) in terms of the independent variable(s). a. 6 b c.

10 58 d. 7 5 e f. g

11 Finding a Polnomial Given Points Eample: Set up a sstem of equations to find a parabola that passes through (,96), (, 9), and (5, 96). For the first point we know that a b c becomes 96 a b c so a b c 96 For the second point we know that a b c becomes 9 a b c so a b c 9 For the second point we know that a b c becomes 96 a 5 b5 c so 5a 5b c 96 So the sstem becomes: a b c 96 a b c 9 5a 5b c So our quadratic would be a b c. Find a parabola that passes through the points: (-,), (,5), and (-6,5). Find a cubic function, a b c d, that passes through the points: (-,), (,5), (,), and (-5,). Note: In this case ou will have equations with variables so set up our sstem and use our graphing calculator to solve. 59

12 Applications Involving Sstems. In a particular factor, skilled workers are paid $5 per hour, unskilled workers are paid $9 per hour and shipping clerks are paid $ per hour. Recentl the compan has received an increase in orders and will need to hire a total of 7 workers. The compan has budgeted a total of $88 per hour for these new hires. Due to union requirements, the must hire twice as man skilled emploees as unskilled. How man of each tpe of worker should the compan hire?. I have 7 coins (some are pennies, some are dimes, and some are quarters). I have twice as man pennies as dimes. I have $.97 in coins. How man of each tpe of coin do I have? 6

13 . You have $ to invest in stocks. Stock A is predicted to ield % per ear. Stock B is the safest and is predicted to ield 5% per ear. Stock C is risk but is epected to ield 7% per ear. You decide to spend twice as much on stock B than on stock C. You hope to make % each ear on our stock. Use a matri (and our calculator) to find the amount of each tpe of stock that ou should bu.. An inheritance of $9,6 is to be split among children. To pa back mone owed to the two oldest children, it is written that the oldest child gets $, more than the oungest and the middle child get $, more than the oungest. How much should each child get? 6

14 5. There are candidates to choose for president and 8, people are epected to vote. Candidate A is epected to receive twice as man votes as candidate C. Candidate C is epected to receive times the votes as candidate B. Predict how man votes each candidate will receive. 6. Mar invested $, in three separate investments. At the end of the ear the earned %, 5% and 6%, respectivel. She invested twice as much in the account earning 5% as she did in the one paing %. She made a total of $5 in interest during the ear. How much did she allocate to each investment? 6

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented

More information

Solving Systems of Equations

Solving Systems of Equations Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that

More information

Solving Systems of Linear Equations With Row Reductions to Echelon Form On Augmented Matrices. Paul A. Trogdon Cary High School Cary, North Carolina

Solving Systems of Linear Equations With Row Reductions to Echelon Form On Augmented Matrices. Paul A. Trogdon Cary High School Cary, North Carolina Solving Sstems of Linear Equations With Ro Reductions to Echelon Form On Augmented Matrices Paul A. Trogdon Car High School Car, North Carolina There is no more efficient a to solve a sstem of linear equations

More information

SYSTEMS OF LINEAR EQUATIONS

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

More information

{ } Sec 3.1 Systems of Linear Equations in Two Variables

{ } Sec 3.1 Systems of Linear Equations in Two Variables Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination

More information

4.3-4.4 Systems of Equations

4.3-4.4 Systems of Equations 4.3-4.4 Systems of Equations A linear equation in 2 variables is an equation of the form ax + by = c. A linear equation in 3 variables is an equation of the form ax + by + cz = d. To solve a system of

More information

Systems of Equations and Matrices

Systems of Equations and Matrices Sstems of Equations and Matrices A sstem of equations is a collection of two or more variables In this chapter, ou should learn the following How to use the methods of substitution and elimination to solve

More information

Systems of Linear Equations: Solving by Substitution

Systems of Linear Equations: Solving by Substitution 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

More information

Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true.

Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true. Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

2 Solving Systems of. Equations and Inequalities

2 Solving Systems of. Equations and Inequalities Solving Sstems of Equations and Inequalities. Solving Linear Sstems Using Substitution. Solving Linear Sstems Using Elimination.3 Solving Linear Sstems Using Technolog.4 Solving Sstems of Linear Inequalities

More information

Math 152, Intermediate Algebra Practice Problems #1

Math 152, Intermediate Algebra Practice Problems #1 Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work

More information

Systems of Linear Equations

Systems of Linear Equations Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing. Solving Sstems of Equations b Using the Substitution Method. Solving Sstems of Equations b Using the Addition Method. Applications

More information

Solving Special Systems of Linear Equations

Solving Special Systems of Linear Equations 5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

More information

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems

More information

Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10

Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10 Chapter 3 & 8.1-8.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

More information

Solving Systems of Equations. 11 th Grade 7-Day Unit Plan

Solving Systems of Equations. 11 th Grade 7-Day Unit Plan Solving Sstems of Equations 11 th Grade 7-Da Unit Plan Tools Used: TI-83 graphing calculator (teacher) Casio graphing calculator (teacher) TV connection for Casio (teacher) Set of Casio calculators (students)

More information

Row Echelon Form and Reduced Row Echelon Form

Row Echelon Form and Reduced Row Echelon Form These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Algebra II Notes Piecewise Functions Unit 1.5. Piecewise linear functions. Math Background

Algebra II Notes Piecewise Functions Unit 1.5. Piecewise linear functions. Math Background Piecewise linear functions Math Background Previousl, ou Related a table of values to its graph. Graphed linear functions given a table or an equation. In this unit ou will Determine when a situation requiring

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

1.5 SOLUTION SETS OF LINEAR SYSTEMS

1.5 SOLUTION SETS OF LINEAR SYSTEMS 1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as

More information

How To Understand And Solve A Linear Programming Problem

How To Understand And Solve A Linear Programming Problem At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

More information

4-5 Matrix Inverses and Solving Systems. Warm Up Lesson Presentation Lesson Quiz

4-5 Matrix Inverses and Solving Systems. Warm Up Lesson Presentation Lesson Quiz Warm Up Lesson Presentation Lesson Quiz Holt Holt Algebra 2 2 Warm Up Find the determinant and the inverse. Warm Up Find the determinant. How To Find the Inverse Matrix Warm Up Find the inverse. 1. Find

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

How To Understand And Solve Algebraic Equations

How To Understand And Solve Algebraic Equations 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

THE POWER RULES. Raising an Exponential Expression to a Power

THE POWER RULES. Raising an Exponential Expression to a Power 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

9.1 9.1. Graphical Solutions to Equations. A Graphical Solution to a Linear Equation OBJECTIVES

9.1 9.1. Graphical Solutions to Equations. A Graphical Solution to a Linear Equation OBJECTIVES SECTION 9.1 Graphical Solutions to Equations in One Variable 9.1 OBJECTIVES 1. Rewrite a linear equation in one variable as () () 2. Find the point o intersection o () and () 3. Interpret the point o intersection

More information

Plotting Lines in Mathematica

Plotting Lines in Mathematica Lines.nb 1 Plotting Lines in Mathematica Copright 199, 1997, 1 b James F. Hurle, Universit of Connecticut, Department of Mathematics, 196 Auditorium Road Unit 39, Storrs CT 669-39. All rights reserved.

More information

Solving Systems. of Linear Equations

Solving Systems. of Linear Equations Solving Sstems 8 MODULE of Linear Equations? ESSENTIAL QUESTION How can ou use sstems of equations to solve real-world problems? LESSON 8.1 Solving Sstems of Linear Equations b Graphing 8.EE.8, 8.EE.8a,

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

Question 2: How do you solve a matrix equation using the matrix inverse?

Question 2: How do you solve a matrix equation using the matrix inverse? Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A

More information

Systems of Linear Equations in Three Variables

Systems of Linear Equations in Three Variables 5.3 Systems of Linear Equations in Three Variables 5.3 OBJECTIVES 1. Find ordered triples associated with three equations 2. Solve a system by the addition method 3. Interpret a solution graphically 4.

More information

Linear Equations in Two Variables

Linear Equations in Two Variables Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

More information

Unit 1 Equations, Inequalities, Functions

Unit 1 Equations, Inequalities, Functions Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious

More information

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

ALGEBRA 1 SKILL BUILDERS

ALGEBRA 1 SKILL BUILDERS ALGEBRA 1 SKILL BUILDERS (Etra Practice) Introduction to Students and Their Teachers Learning is an individual endeavor. Some ideas come easil; others take time--sometimes lots of time- -to grasp. In addition,

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

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

SECTION 7-4 Algebraic Vectors

SECTION 7-4 Algebraic Vectors 7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson 1.1 Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of

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

Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

More information

LAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING

LAB 11: MATRICES, SYSTEMS OF EQUATIONS and POLYNOMIAL MODELING LAB 11: MATRICS, SYSTMS OF QUATIONS and POLYNOMIAL MODLING Objectives: 1. Solve systems of linear equations using augmented matrices. 2. Solve systems of linear equations using matrix equations and inverse

More information

Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. When we group two such equations together, we know from geometry what can

More information

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above: Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

More information

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM . Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

More information

Pre-AP Algebra 2 Lesson 2-6 Linear Programming Problems

Pre-AP Algebra 2 Lesson 2-6 Linear Programming Problems Lesson 2-6 Linear Programming Problems Objectives: The students will be able to: use sstems of linear inequalities to solve real world problems. set up constraints & objective functions for linear programming

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) = Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

More information

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

More information

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

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

Linear Inequality in Two Variables

Linear Inequality in Two Variables 90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.

More information

The Slope-Intercept Form

The Slope-Intercept Form 7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph

More information

Solving Absolute Value Equations and Inequalities Graphically

Solving Absolute Value Equations and Inequalities Graphically 4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

More information

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system 1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

More information

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship 6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works

More information

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

More information

Shake, Rattle and Roll

Shake, Rattle and Roll 00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar

More information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

More information

Classifying Solutions to Systems of Equations

Classifying Solutions to Systems of Equations CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham

More information

Addition and Subtraction of Vectors

Addition and Subtraction of Vectors ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

More information

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

4.9 Graph and Solve Quadratic

4.9 Graph and Solve Quadratic 4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

Complex Numbers. w = f(z) z. Examples

Complex Numbers. w = f(z) z. Examples omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

Complex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No real-number solutions. From the definition, it follows that i 2 1.

Complex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No real-number solutions. From the definition, it follows that i 2 1. 7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl

More information

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

More information

3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533

3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533 Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might

More information

5.3 Graphing Cubic Functions

5.3 Graphing Cubic Functions Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

More information

Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027

Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 Answer Ke Masters Copright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition

More information

Midterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013

Midterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013 Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in

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

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics A Semester Course in Finite Mathematics for Business and Economics Marcel B. Finan c All Rights Reserved August 10,

More information