12.1 Systems of Linear equations: Substitution and Elimination

Size: px
Start display at page:

Download "12.1 Systems of Linear equations: Substitution and Elimination"

Transcription

1 . Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables is a pair of numbers that satisfies all the equations in the sstem. Eample: is a sstem whose solution is the pair ) There are two tpes of sstems: consistent have a solution) and inconsistent do not have a solution) Consistent sstem can be dependent have infinitel man solutions) and independent have onl one solution) Sstem of two linear equations with two variables can be represented b two straight lines. In the sstem above one line is represented b the equation + = and the other line b the equation + = If the lines intersect the point of intersection is the solution of the sstem. The sstem is consistent independent. If the lines are identical then there are infinitel man solutions. Each point on the line is a solution. The sstem is consistent dependent. If the lines are parallel there is no solution. The sstem is inconsistent. Sstem of equations can be solved graphicall ou need a ver accurate graph for that) b substitution b elimination addition) using matrices we will not stud this method) and using Cramer s Rule we ll learn it in sec.). Graphical method: Carefull graph both equations in the same coordinate plane. Use graph paper and a ruler. The intersection of the two graphs if an is the solution. Eample: Solve b graphing Line + = = - + slope m = - intercept : 0) Line + = = - + slope m = - -intercept : 0)

2 Substitution method: Solve one of the equations for one of the variables sa substitute to the other equation and solve for. Then go back to the first equation to find value of. Eample: Solve ) substitute ) 0 solution : 0) Elimination addition) method: Multipl one or both equations b suitable numbers so that after the multiplication is done coefficients of one of the variables sa are opposite numbers. Add both equations side b side to eliminate that variable. Solve the resulting equation for. Substitute in one of the equations and solve for. Eample: we eliminate ) Multipl first equation b so the coefficients of are and - Add side b side Solve for Substitute = into the first equation Solve for Solution: 0) Remark: - if after substitution or elimination ou obtain an equation that is alwas true like 0 = 0) then the sstem has infinitel man solutions. The line represented b either equation contains all solutions - If after substitution or elimination ou obtain an equation that is alwas false like 0 = ) then the sstem has no solutions.

3 Sstems of three linear equations with three variables Eample: Solution of such a sstem is a triple of numbers abc) that when substituted for respectivel in each equation makes each equation a true statement. A sstem of three linear equation can have one solution infinitel man solutions and no solutions Sstem of three equations with three variables can be solved b substitution and elimination. Substitution method Eample: Solve ) Solve one of the equations for one of the variables choose an equation that is easiest to solve for one of the variables). First equation can be easil solved for : = - + ) Substitute in the remaining two equations with the epression obtained in step ) ) ) ) Simplif the two equations. Notice that the last two equations contain onl and. ) Solve the resulting sstem of two equations with two variables and ) The solution is = = - ) use the first equation to find value of : = + = - = ) The solution is -)

4 Elimination method Eample: Solve the sstem The idea is to eliminate one of the variables b multipling the equations b suitable constants adding two equations side b side and reducing the sstem to a sstem with two equations and two variables. We ll eliminate. ) Multipl the first equation b -) and add to the second equation = - + = - + = -7 this is now the equation ) ) Multipl the first equation b -) and add to the third equation = 0 + = -7 = this is now the equation ) ) Solve the sstem 7 7 Using elimination we get = = - ) Use the first equation to find - - = - = ) solution is -) Eample: Infinitel man solutions) Solve Using the elimination method eliminate using the first equation) we get the sstem Solving b elimination the sstem leads to the equation 0 = 0. This means that there are infinitel man solutions. To find the solution set notice that the sstem reduces to or. We get = + and = - = +) - - = -. Therefore if is an real number then = - = + = is a solution. The solution set is {) = - = + is an real number}

5 . Sstem of linear equations: determinants and Cramer s Rule Consider a sstem of equations a b s c d t Let s use the Elimination method to solve this general sstem. Multipling the first equation b d and the second equation b -b) gives ad bd sd b) c b) d b) t Adding the equations side b side eliminates the variable and gives the equation for ad bc) sd bt sd bt When ad bc 0 we can solve that equation for :. If ad bc 0 then this last equation ad bc has either no solution when sd bt 0 ) or infinitel man solutions can be an real number when sd bt 0) at sc If ad bc 0 then plugging in to the first equation) we get and hence the solution of ad bc sd bt at sc the sstem is ad bc ad bc Note that the and values of the solution if it eists) have a ver similar structure. efinition: A b determinant denoted a b ad bc c d Eample: Evaluate a b is defined as follows: c d ) ) = s b a s Note that sd bt and at cs t d c t We can use this new notation to discuss the solutions of a sstem of two equations with two variables. Theorem Cramer s Rule) Consider a sstem of two equations with two variables Let

6 If 0 then the sstem has eactl one solution If 0 and 0 and 0 then the sstem has infinitel man solutions If 0 and either 0 or 0 then the sstem has no solutions Eample: Use Cramer s Rule to solve the sstem ) Compute the determinant of the sstem ) Since 0 the sstem has a solution. ) Compute and ) ) Write the solution: ) Cramer s Rule is also applicable in modified form for larger sstems of linear equations. We will discuss onl sstems of three linear equations with three variables. We start with defining a determinant. efinition A determinant denoted a d g b e h c f i is defined as a d g b e h c f i e a h f i d b g f i d c g e h aei cdh bgf afh bdi cge Note that the determinants are obtained b crossing out the row and the column in which the numbers ab c are.

7 There are other was to evaluate a determinant but we will use this method onl. Eample : 7 Evaluate ) 9 ) 7 ) ) ) 7 ) ) ) 7 ) 0 Theorem Cramer s Rule) Consider the sstem Let If 0 then the sstem has eactl one solution If 0 and then the sstem has infinitel man solutions If 0 and either 0 0 or 0 then the sstem has no solutions Eample: Use Cramer s Rule to solve the sstem ) Compute the determinant of the sstem ) ) ) 9 0 ) Since 0 sstem has eactl one solution ) ) ) ) )) ) Compute

8 ) ) ) ) 9 ) ) Write the solution ) 9 9 Eample: Solve the sstem ) Compute the determinant of the sstem. 0 ) Since = 0 Cramer s Rule cannot be applied. Computing shows that = = =0 therefore sstem has infinitel man solutions. We need to use either Elimination or Substitution method to find the solution set. This has been done earlier when discussing the elimination method.. Sstem of nonlinear equations in two variables An equation is nonlinear if it can t be rewritten in the form a + b + c = 0. Eample: = 0 A sstem of nonlinear equations is a sstem that consists of one or more nonlinear equations. A sstem of nonlinear equations can be solved using the Elimination or Substitution method. Eample: Solve Graphical method: Graph both equations and find the intersection points.

9 solutions: 0) 7) Substitution method: Solve one of the equations for one of the variables substitute into the other equation and solve it. ) First equation is alread solved for ) Substitute = + into the second equation + = + ) Solve for = 0 -) = 0 = 0 or = Back substitute these values to compute = 0: = 0 + = solution 0) = : = + = 7 solution 7) Elimination method: can be applied onl when one of the variables can be eliminated. For some sstems it might not be possible. Eample: Solve the sstem 0 7 Multipl the second equation b -) and add the equations side b side ) ) ) 0 variable was elimanted. The equation simplifies to or Substituting these values of into the second equations we get

10 If 7 and and if 7 and Therefore the solutions are ) )

14.1 Systems of Linear Equations in Two Variables

14.1 Systems of Linear Equations in Two Variables 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination

More information

Unit 12 Study Notes 1 Systems of Equations

Unit 12 Study Notes 1 Systems of Equations You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2

1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2 78 Chapter Sstems of Linear Equations Section. Concepts. Solutions to Sstems of Linear Equations. Dependent and Inconsistent Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Solving

More information

7.5 Solve Special Types of

7.5 Solve Special Types of 75 Solve Special Tpes of Linear Sstems Goal p Identif the number of of a linear sstem Your Notes VOCABULARY Inconsistent sstem Consistent dependent sstem Eample A linear sstem with no Show that the linear

More information

Chapter 11. Systems of Equations Solving Systems of Linear Equations by Graphing

Chapter 11. Systems of Equations Solving Systems of Linear Equations by Graphing Chapter 11 Sstems of Equations 11.1 Solving Sstems of Linear Equations b Graphing Learning Objectives: A. Decide whether an ordered pair is a solution of a sstem of linear equations. B. Solve a sstem of

More information

Chapter 6: Systems of Equations and Inequalities

Chapter 6: Systems of Equations and Inequalities Chapter 6: Sstems of Equations and Inequalities 6-1: Solving Sstems b Graphing Objectives: Identif solutions of sstems of linear equation in two variables. Solve sstems of linear equation in two variables

More information

Graph the linear system and estimate the solution. Then check the solution algebraically.

Graph the linear system and estimate the solution. Then check the solution algebraically. (Chapters and ) A. Linear Sstems (pp. 6 0). Solve a Sstem b Graphing Vocabular Solution For a sstem of linear equations in two variables, an ordered pair (x, ) that satisfies each equation. Consistent

More information

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations MA 800, Summer 06 Lesson Notes Solving a Sstem of Equations b substitution (or elimination) Matrices Consider the graphs of the two equations below. A Sstem of Equations From our mathematics eperience,

More information

Section 3.1 Solving Linear Systems by Graphing

Section 3.1 Solving Linear Systems by Graphing Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures AB = BA = I,

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures AB = BA = I, FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 7 MATRICES II Inverse of a matri Sstems of linear equations Solution of sets of linear equations elimination methods 4

More information

( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing

( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00

More information

RELATIONS AND FUNCTIONS through

RELATIONS AND FUNCTIONS through RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or

More information

UNIT 5. SIMULTANEOUS EQUATIONS

UNIT 5. SIMULTANEOUS EQUATIONS 3º ESO. Definitions UNIT 5. SIMULTANEOUS EQUATIONS A linear equation with two unknowns is an equation with two unknowns having both of them degree one. Eamples. 3 + 5 and + 6 9. The standard form for these

More information

Chapter 5: Systems of Equations

Chapter 5: Systems of Equations Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

More information

Chapter 4 Analytic Trigonometry

Chapter 4 Analytic Trigonometry Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process

More information

Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.

Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing. NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations

More information

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,

More information

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

Lines and Planes 1. x(t) = at + b y(t) = ct + d

Lines and Planes 1. x(t) = at + b y(t) = ct + d 1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. Will we call this the ABC form. Recall that the slope-intercept

More information

MATH 115: Review for Chapter 6

MATH 115: Review for Chapter 6 MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING

More information

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.

Algebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations. Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems

More information

Mth Quadratic functions and quadratic equations

Mth Quadratic functions and quadratic equations Mth 0 - Quadratic functions and quadratic equations Name Find the product. 1) 8a3(2a3 + 2 + 12a) 2) ( + 4)( + 6) 3) (3p - 1)(9p2 + 3p + 1) 4) (32 + 4-4)(2-3 + 3) ) (4a - 7)2 Factor completel. 6) 92-4 7)

More information

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

More information

Limits. Calculus Module C06. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Limits. Calculus Module C06. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. e Calculus Module C Limits Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED March, Introduction to Limits Statement of Prerequisite Skills Complete

More information

Name Class Date. Solving Special Systems by Graphing. Does this linear system have a solution? Use the graph to explain.

Name Class Date. Solving Special Systems by Graphing. Does this linear system have a solution? Use the graph to explain. Name Class Date 5 Solving Special Sstems Going Deeper Essential question: How do ou solve sstems with no or infinitel man solutions? 1 A-REI.3.6 EXAMPLE Solving Special Sstems b Graphing Use the graph

More information

Linear algebra in turn is built on two basic elements, MATRICES and VECTORS.

Linear algebra in turn is built on two basic elements, MATRICES and VECTORS. M-Lecture():.-. Linear algebra provides concepts that are crucial to man areas of information technolog and computing, including: Graphics Image processing Crptograph Machine learning Computer vision Optimiation

More information

Math098 Practice Final Test

Math098 Practice Final Test Math098 Practice Final Test Find an equation of the line that contains the points listed in the table. 1) 0-6 1-2 -4 3-3 4-2 Find an equation of the line. 2) 10-10 - 10 - -10 Solve. 3) 2 = 3 + 4 Find the

More information

(2.5) 1. Solve the following compound inequality and graph the solution set.

(2.5) 1. Solve the following compound inequality and graph the solution set. Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.

More information

APPENDIX D Rotation and the General Second-Degree Equation

APPENDIX D Rotation and the General Second-Degree Equation APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the

More information

13.1 2X2 Systems of Equations

13.1 2X2 Systems of Equations . X Sstems of Equations In this section we want to spend some time reviewing sstems of equations. Recall there are two basic techniques we use for solving a sstem of equations: Elimination and Substitution.

More information

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

More information

McKinney High School AP Calculus Summer Packet

McKinney High School AP Calculus Summer Packet McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work

More information

Module 3, Section 4 Analytic Geometry II

Module 3, Section 4 Analytic Geometry II Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

More information

1.7 Inverse Functions

1.7 Inverse Functions 71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

More information

8.1 Exponents and Roots

8.1 Exponents and Roots Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

More information

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?

Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5? 3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear

More information

For questions 5-8, solve each inequality and graph the solution set. You must show work for full credit. (2 pts each)

For questions 5-8, solve each inequality and graph the solution set. You must show work for full credit. (2 pts each) Alg Midterm Review Practice Level 1 C 1. Find the opposite and the reciprocal of 0. a. 0, 1 b. 0, 1 0 0 c. 0, 1 0 d. 0, 1 0 For questions -, insert , or = to make the sentence true. (1pt each) A. 5

More information

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INEQUALITY = a mathematical statement that contains one of these four inequalit signs: ,.

More information

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

Systems of Linear Equations Monetary Systems Overload

Systems of Linear Equations Monetary Systems Overload Sstems of Linear Equations SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Interactive Word Wall Have ou ever noticed that when an item is popular and man people want to bu it, the price

More information

Name: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig*

Name: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig* Name: Richard Montgomer High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website

More information

Review of Prerequisite Skills, p. 350 C( 2, 0, 1) B( 3, 2, 0) y A(0, 1, 0) D(0, 2, 3) j! k! 2k! Section 7.1, pp

Review of Prerequisite Skills, p. 350 C( 2, 0, 1) B( 3, 2, 0) y A(0, 1, 0) D(0, 2, 3) j! k! 2k! Section 7.1, pp . 5. a. a a b a a b. Case If and are collinear, then b is also collinear with both and. But is perpendicular to and c c c b 9 b c, so a a b b is perpendicular to. Case If b and c b c are not collinear,

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

Solve each system by substitution or elimination. Check your solutions. b.

Solve each system by substitution or elimination. Check your solutions. b. Algebra: 10.3.1: Intersect or Intercept? Name Solutions Block Date Bell Work: a. = 4 2 3 = 3 2 3(4 ) = 3 2 12 + 3 = 3 5 12 = 3 5 = 15 Solve each sstem b substitution or elimination. Check our solutions.

More information

Section 1.2: A Catalog of Functions

Section 1.2: A Catalog of Functions Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as

More information

EOC Review. Algebra I

EOC Review. Algebra I EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.

More information

Chapter 1 Graph of Functions

Chapter 1 Graph of Functions Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane

More information

Linear Programming. Maximize the function. P = Ax + By + C. subject to the constraints. a 1 x + b 1 y < c 1 a 2 x + b 2 y < c 2

Linear Programming. Maximize the function. P = Ax + By + C. subject to the constraints. a 1 x + b 1 y < c 1 a 2 x + b 2 y < c 2 Linear Programming Man real world problems require the optimization of some function subject to a collection of constraints. Note: Think of optimizing as maimizing or minimizing for MATH1010. For eample,

More information

Linear Equation Theory - 2

Linear Equation Theory - 2 Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite

More information

University of Regina Department of Mathematics and Statistics

University of Regina Department of Mathematics and Statistics u z v 1. Consider the map Universit of Regina Department of Mathematics and Statistics MATH431/831 Differential Geometr Winter 2014 Homework Assignment No. 3 - Solutions ϕ(u, v) = (cosusinv, sin u sin

More information

An Introduction to Systems of Equations

An Introduction to Systems of Equations LESSON 17 An Introduction to Sstems of Equations LEARNING OBJECTIVES Toda I am: completing the Desmos activit Sstems of Two Linear Equations. So that I can: write and solve a sstem of two linear equations

More information

2. Bunny Slope:, Medium Trail: 4. 58; The starting height of the trail is 58 meters ; The length of the trail is 1450 meters.

2. Bunny Slope:, Medium Trail: 4. 58; The starting height of the trail is 58 meters ; The length of the trail is 1450 meters. ...7 Start Thinking! For use before Lesson.7 Sample answer: If ou need to write the equation of a line given the slope and a point on the line, ou can find the -intercept (either graphicall or algebraicall).

More information

QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS

QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS arxiv:1803.0591v1 [math.gm] QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS Aleander Panfilov stable spiral det A 6 3 5 4 non stable spiral D=0 stable node center non stable node saddle 1 tr A QUALITATIVE

More information

Eigenvectors and Eigenvalues 1

Eigenvectors and Eigenvalues 1 Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and

More information

7.1 Solving Linear Systems by Graphing

7.1 Solving Linear Systems by Graphing 7.1 Solving Linear Sstems b Graphing Objectives: Learn how to solve a sstem of linear equations b graphing Learn how to model a real-life situation using a sstem of linear equations With an equation, an

More information

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector. LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man

More information

Unit 2 Notes Packet on Quadratic Functions and Factoring

Unit 2 Notes Packet on Quadratic Functions and Factoring Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

More information

Functions and Graphs TERMINOLOGY

Functions and Graphs TERMINOLOGY 5 Functions and Graphs TERMINOLOGY Arc of a curve: Part or a section of a curve between two points Asmptote: A line towards which a curve approaches but never touches Cartesian coordinates: Named after

More information

Solving Linear Systems

Solving Linear Systems 1.4 Solving Linear Sstems Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent

More information

MATHEMATICS LEVEL 2 TEST FORM B Continued

MATHEMATICS LEVEL 2 TEST FORM B Continued Mathematics Level Test Form B For each of the following problems, decide which is the BEST of the choices given. If the eact numerical value is not one of the choices, select the choice that best approimates

More information

13.2 Solving Larger Systems by Gaussian Elimination

13.2 Solving Larger Systems by Gaussian Elimination . Solving Larger Sstems b Gaussian Elimination Now we want to learn how to solve sstems of equations that have more that two variables. We start with the following definition. Definition: Row echelon form

More information

Graphical Solutions of Linear Systems

Graphical Solutions of Linear Systems Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations

More information

MATHEMATICS LEVEL 2. MATHEMATICS LEVEL 2 Continued GO ON TO THE NEXT PAGE USE THIS SPACE FOR SCRATCHWORK. 1. If xy 0 and 3x = 0.

MATHEMATICS LEVEL 2. MATHEMATICS LEVEL 2 Continued GO ON TO THE NEXT PAGE USE THIS SPACE FOR SCRATCHWORK. 1. If xy 0 and 3x = 0. MATHEMATICS LEVEL For each of the following problems, decide which is the BEST of the choices given. If the eact numerical value is not one of the choices, select the choice that best approimates this

More information

Mt. Douglas Secondary

Mt. Douglas Secondary Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms.

More information

Can a system of linear equations have no solution? Can a system of linear equations have many solutions?

Can a system of linear equations have no solution? Can a system of linear equations have many solutions? 5. Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with

More information

Approximation and Error

Approximation and Error Approimation and Error The queue is quite long. We can ride on the roller coaster after 9 rounds. Each round is estimated to last for minutes, correct to the nearest minute. Are we sure that we can ride

More information

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

Systems of Equations and Inequalities

Systems of Equations and Inequalities Sstems of Equations and Inequalities 7 7. Linear and Nonlinear Sstems of Equations 7. Two-Variable Linear Sstems 7.3 Multivariable Linear Sstems 7. Partial Fractions 7.5 Sstems of Inequalities 7.6 Linear

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

More information

Linear Equations and Arithmetic Sequences

Linear Equations and Arithmetic Sequences CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

3-1. Solving Systems Using Tables and Graphs. Concept Summary. Graphical Solutions of Linear Systems VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

3-1. Solving Systems Using Tables and Graphs. Concept Summary. Graphical Solutions of Linear Systems VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING 3- Solving Sstems Using Tables and Graphs TEKS FOCUS VOCABULARY Foundational to TEKS (3)(A) Formulate sstems of equations, including sstems consisting of three linear equations in three variables and sstems

More information

Math Intermediate Algebra

Math Intermediate Algebra Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and

More information

3.7 InveRSe FUnCTIOnS

3.7 InveRSe FUnCTIOnS CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

More information

Section 5.1: Functions

Section 5.1: Functions Objective: Identif functions and use correct notation to evaluate functions at numerical and variable values. A relationship is a matching of elements between two sets with the first set called the domain

More information

2 x = You try: 1a) Simplify: 6x 5+ 8x 5. 1 a) Simplify: 5x 4+ 3x+ You try: 2a) Solve: x 6= 2 a) Solve: 4+ x = 10. b) Solve: 3 12.

2 x = You try: 1a) Simplify: 6x 5+ 8x 5. 1 a) Simplify: 5x 4+ 3x+ You try: 2a) Solve: x 6= 2 a) Solve: 4+ x = 10. b) Solve: 3 12. 1 a) Simplif: 5 + + 7 1 1a) Simplif: 6 5+ 5. b) Simplif: 6 ( + 5) ( 6+ 5) 1b) Simplif: 5 ( ) 65 ( ) A.SSE. a) Solve: + 10 a) Solve: 6 10 b) Solve: 1 b) Solve: 5 A.REI. Page 1 of 19 MCC@WCCUSD (AUSD) 10/1/1

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

Math 0308 Final Exam Review(answers) Solve the given equations. 1. 3x 14 8x 1

Math 0308 Final Exam Review(answers) Solve the given equations. 1. 3x 14 8x 1 Math 8 Final Eam Review(answers) Solve the given equations.. 8.. 9.. 9 9 8 8.. 8 8 all real numbers 8. 9. all real numbers no solution 8 8 9 9 9 Solve the following inequalities. Graph our solution on

More information

Essential Question How can you determine the number of solutions of a linear system?

Essential Question How can you determine the number of solutions of a linear system? .1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.A A.3.B Solving Linear Sstems Using Substitution Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent

More information

To plot the ordered pair (3, 2), start at the origin, then move 3 units to the right and 2 units up.

To plot the ordered pair (3, 2), start at the origin, then move 3 units to the right and 2 units up. 16 MODULE 6. GEOMETRY AND UNIT CONVERSION 6b Graphs Plotting Points Plot the following ordered pairs of whole numbers: (2, 2), (, ), and (7, 4). EXAMPLE 1. Plot the following ordered pairs of whole numbers:

More information

A Preview of College Algebra CHAPTER

A Preview of College Algebra CHAPTER hal9_ch_-9.qd //9 : PM Page A Preview of College Algebra CHAPTER Chapter Outline. Solving Sstems of Linear Equations b Using Augmented Matrices. Sstems of Linear Equations in Three Variables. Horizontal

More information

A Primer on Solving Systems of Linear Equations

A Primer on Solving Systems of Linear Equations A Primer on Solving Systems of Linear Equations In Signals and Systems, as well as other subjects in Unified, it will often be necessary to solve systems of linear equations, such as x + 2y + z = 2x +

More information

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1. 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

More information

x. 4. 2x 10 4x. 10 x

x. 4. 2x 10 4x. 10 x CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

More information

MATH STUDENT BOOK. 9th Grade Unit 8

MATH STUDENT BOOK. 9th Grade Unit 8 MATH STUDENT BOOK 9th Grade Unit 8 Unit 8 Graphing Math 908 Graphing INTRODUCTION 3. USING TWO VARIABLES 5 EQUATIONS 5 THE REAL NUMBER PLANE TRANSLATIONS 5 SELF TEST. APPLYING GRAPHING TECHNIQUES 5 LINES

More information

Worksheet #1. A little review.

Worksheet #1. A little review. Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves

More information

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1 Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

More information

CONSUMER CHOICES Madison is thinking about leasing a car for. Example 1 Solve the system of equations by graphing.

CONSUMER CHOICES Madison is thinking about leasing a car for. Example 1 Solve the system of equations by graphing. 2-1 BJECTIVES Solve sstems of equations graphicall. Solve sstems of equations algebraicall. Solving Sstems of Equations in Two Variables CNSUMER CHICES Madison is thinking about leasing a car for two ears.

More information

The letter m is used to denote the slope and we say that m = rise run = change in y change in x = 5 7. change in y change in x = 4 6 =

The letter m is used to denote the slope and we say that m = rise run = change in y change in x = 5 7. change in y change in x = 4 6 = Section 4 3: Slope Introduction We use the term Slope to describe how steep a line is as ou move between an two points on the line. The slope or steepness is a ratio of the vertical change in (rise) compared

More information

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information