Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
|
|
- Whitney Richards
- 5 years ago
- Views:
Transcription
1 Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of Linear Equations by the Addition (Elimination) Method Application Problem Involving Two Equations and Two Variables. A system consists of more than one equation. There is more than one way to solve a system of equations. They are all valid methods, but sometimes one or another will be the better choice for solving a system of equations. The solutions of a system of equations are the values that make all the equations in the system true. When we check our answers, we must be sure to check them in all the equations, since a correct answer for one equation may not be correct for the others. The Graphing Method This will involve graphing the equations on a coordinate plane. The solution(s) to the system will be all points that are common to both graphs (where the graphs intersect). A system of two linear equations in two variables is two equations considered together. To solve a system is to find all the ordered pairs that satisfy both equations. When solving a system three situations can occur: 1. The lines intersect. There is one solution, the point where they intersect. 2. The lines are parallel. There is no solution, the lines do not intersect. 3. The lines are the same. There are infinitely many solutions, the lines coincide. Systems of equations can be solved by various methods which will be the focus of the unit. Example 1. a) Use the graphing approach to solve the system. Tell whether the system has one, infinite or no solutions. 2x - y = 5 3x + 2y = 4 Graph the two lines using any method. 2x y = 5 3x + 2y = 4 y x y x y x The solution to the system is the point of intersection: (, ).
2 Example 2: 2x + 3y = 5 x 5y = -17 The first step will be to graph these two equations. 2x + 3y = 5 It looks like the answer is (-2, 3) x 5y = -17 After we graph the lines on the same coordinate plane, we notice that they intersect in a point. It looks like (-2, 3) is the solution to this system, but we need to check this potential solution. In equation #1: 2(-2) + 3(3) = = 5 Checks In equation #2: (-2) 5(3) = = -17 Checks (-2, 3) Answer Note: The graphing method is not a very accurate one. If our graphs are not perfect (or the answers not integers), we will probably not be able to get the right answer. Graphing is useful, however, for visualizing the problem, estimating answers, and sometimes for informing us of how many answers we should expect to find. Note also: If the two linear graphs do not intersect, we call the system inconsistent (this means the lines are parallel). If the two linear graphs turn out to be the same line (there are infinitely many solutions: all the points on the line), then the system is consistent and dependent. If the two linear graphs intersect in a point, the system is consistent and independent.
3 The Substitution Method To solve a system of equations by the substitution method: 1. Solve one of the equations for one of the variables.(choose a variable with a coefficient of 1 or -1 if possible.) 2. Substitute this expression into the other equation to produce an equation with only one variable. 3. Solve the equation in Step 2 for the remaining variable. 4. Substitute this solution into the expression obtained in Step Solve for the second variable. 6. Write your solution set as an ordered pair, and check in each equation. Example 2x - 3y = 5 Solve the system by the substitution method: 3x + y = 2 y = 2-3x 1) Solve the 2nd equation for y. 2x - 3( ) = 5 2) Substitute 2-3x into the other equation in place of y. 2x - + = 5 3) Solve for x. 11x = x = y = 2-3( ) 4) Substitute this value into step 1. y = 5) Solve for y. {(, )} 6) Write the solution set. Check: 2( ) - 3( ) = 5? 7) Check the solution in both equations. 3( ) + ( ) = 2?
4 Elimination-by-Addition Method In the elimination-by-addition method we use the following operations to produce an equivalent system which can be easily solved. 1. Any two equations can be interchanged. 2. Both sides of an equation can be multiplied by any nonzero real number. 3. Any equation can be replaced by the sum of that equation and a nonzero multiple of another equation. Steps for elimination-by-addition method: 1. Write each equation in standard form if needed. 2. If necessary, multiply one or both equations by some constant which will make the x or y coefficients opposites. 3. Add the equations from step 2 together eliminating one of the variables. 4. Solve for the remaining variable. 5. Substitute this solution into either of the original equations. 6. Solve for the second variable. 7. Write the solution set and check. Example Solve the system by the elimination-by-addition method: x + 3y =7 2x - 3y = 5 The equations are in standard form. 3x = x = Solve for x. + 3y = 7 Substitute x = 4 into either equation. 3y = Solve for y. y = When we add the 2 equations the y's will cancel. {(, )} Write the solution set and check. check: 4 + 3(1) = 7? 2(4) - 3(1) = 5?
5 No solution and infinite solutions No Solution: Sometimes you will not have a solution (an ordered pair) to a system of equations this occurs when the lines represented by the equations never intersect. What this looks like when you solve the system is a result that is. For example your solution might be 12 = 14 which is false! In other words, the solution set is Ǿ and the system has no solution. Infinite Solutions: Sometimes your solution shows that one value is equal to the same value. For example, your solution might be 12 = 12. In this case, the solution set to the system is infinite. In other words, the equation is always true and any real number will solve the system. This occurs in linear equations when both equations represent the same line. In this case, the solution set is all real numbers. Which Method to Use? 1. If one equation is already solved for one of the variables, substitution would probably be the easiest method. 2. If solving for either variable in either equation would produce fractions to substitute, use the addition method. 3. Always clear fractions in both equations before deciding which method to use. Practice Problems - Solve by either substitution or elimination method: 5x - y = x + 2y = 0 4x - 3y = x + 4y = -15
6 y = 2x x - 2y = 12 2x + y = x + 2y = 18 3x + y = 0 5. x - 2y = -7
7 Systems that do not have solutions Sometimes a linear system will not have any solutions. Here s a system that has no solutions: x + y = 4 2x + 2y = 10 What happens in our system solving methods, if the system actually has no solution? Substitution Method: Here s what happens for the example problem. First equation solved for y: y = 4 x Substitute into second equation: 2x + 2(4 x) = 10 Distribute and simplify: 2x + 8 2x = 10 8 = 10 We end up with an always-false equation, which tells us that there s no solution. Elimination Method: Here s what happens for the example problem. First equation multiplied by 2: 2x + 2y = 8 Second equation: 2x + 2y = 10 Subtract: 0x + 0y = 2 0 = 2 We end up with an always-false equation, which tells us that there s no solution. In general, using either of the methods, the indicator that a system has no solutions is that you reach an equation that is always false, such as a number equaling a different number. Since the equation is false no matter what values x and y have, that means the system has no solution. You try it 15. a. Solve by substitution: b. Solve by elimination: 2x + 4y = 20 2x + 4y = 20 3x + 6y = 50 3x + 6y = 50
8 Systems that have infinitely many solutions Sometimes a linear system will have infinitely many solutions. Here s such a system: x + y = 4 2x + 2y = 8 What happens in our system solving methods, if the system has infinitely many solutions? Substitution Method: Here s what happens for the example problem. First equation solved for y: y = 4 x Substitute into second equation: 2x + 2(4 x) = 8 Distribute and simplify: 2x + 8 2x = 8 8 = 8 We get an always-true equation, which tells that there are infinitely many solutions. Elimination Method: Here s what happens for the example problem. First equation multiplied by 2: 2x + 2y = 8 Second equation: 2x + 2y = 8 Subtract: 0x + 0y = 0 0 = 0 We get an always-true equation, which tells that there are infinitely many solutions. In general, using either of the methods, the indicator that a system has infinitely many solutions is that you reach an equation that is always true, such as a number equaling the same number. When that happens, any of the infinitely many (x, y) pairs that makes one of the equations true also makes the other equation true.
A. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationSections 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 informationA linear equation in two variables is generally written as follows equation in three variables can be written as
System of Equations A system of equations is a set of equations considered simultaneously. In this course, we will discuss systems of equation in two or three variables either linear or quadratic or a
More informationConsistent and Dependent
Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;
More informationDefinition: A "system" of equations is a set or collection of equations that you deal with all together at once.
System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems
More informationCHAPTER 1 Systems of Linear Equations
CHAPTER Systems of Linear Equations Section. Introduction to Systems of Linear Equations. Because the equation is in the form a x a y b, it is linear in the variables x and y. 0. Because the equation cannot
More information5 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 informationSystems of Linear Equations
Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
More informationSystems of Linear Equations
4 Systems of Linear Equations Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.1, Slide 1 1-1 4.1 Systems of Linear Equations in Two Variables R.1 Fractions Objectives 1. Decide whether an
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationYOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
More information5x 2 = 10. x 1 + 7(2) = 4. x 1 3x 2 = 4. 3x 1 + 9x 2 = 8
1 To solve the system x 1 + x 2 = 4 2x 1 9x 2 = 2 we find an (easier to solve) equivalent system as follows: Replace equation 2 with (2 times equation 1 + equation 2): x 1 + x 2 = 4 Solve equation 2 for
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationGraphing Linear Systems
Graphing Linear Systems Goal Estimate the solution of a system of linear equations by graphing. VOCABULARY System of linear equations A system of linear equations is two or more linear equations in the
More informationSolving Linear Systems Using Gaussian Elimination
Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n
More informationAnswers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets.
Name Geometry SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Geometry. We will use these concepts on a regular basis throughout
More information6.2. TWO-VARIABLE LINEAR SYSTEMS
6.2. TWO-VARIABLE LINEAR SYSTEMS What You Should Learn Use the method of elimination to solve systems of linear equations in two variables. Interpret graphically the numbers of solutions of systems of
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where
More informationx y = 2 x + 2y = 14 x = 2, y = 0 x = 3, y = 1 x = 4, y = 2 x = 5, y = 3 x = 6, y = 4 x = 7, y = 5 x = 0, y = 7 x = 2, y = 6 x = 4, y = 5
List six positive integer solutions for each of these equations and comment on your results. Two have been done for you. x y = x + y = 4 x =, y = 0 x = 3, y = x = 4, y = x = 5, y = 3 x = 6, y = 4 x = 7,
More informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More information6-3 Solving Systems by Elimination
Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two
More informationUNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationFinite Math - Fall Section Present Value of an Annuity; Amortization
Finite Math - Fall 016 Lecture Notes - 9/1/016 Section 3. - Present Value of an Annuity; Amortization Amortization Schedules. Suppose you are amortizing a debt by making equal payments, but then decided
More informationThis is Solving Linear Systems, chapter 3 from the book Advanced Algebra (index.html) (v. 1.0).
This is Solving Linear Systems, chapter 3 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
More informationIntroduction to systems of equations
Introduction to systems of equations A system of equations is a collection of two or more equations that contains the same variables. This is a system of two equations with two variables: In solving a
More informationModule 2 Study Guide. The second module covers the following sections of the textbook: , 4.1, 4.2, 4.5, and
Module 2 Study Guide The second module covers the following sections of the textbook: 3.3-3.7, 4.1, 4.2, 4.5, and 5.1-5.3 Sections 3.3-3.6 This is a continuation of the study of linear functions that we
More informationSection 8.1 Vector and Parametric Equations of a Line in
Section 8.1 Vector and Parametric Equations of a Line in R 2 In this section, we begin with a discussion about how to find the vector and parametric equations of a line in R 2. To find the vector and parametric
More informationLesson 3: Using Linear Combinations to Solve a System of Equations
Lesson 3: Using Linear Combinations to Solve a System of Equations Steps for Using Linear Combinations to Solve a System of Equations 1. 2. 3. 4. 5. Example 1 Solve the following system using the linear
More informationChapter 4. Systems of Linear Equations; Matrices. Opening Example. Section 1 Review: Systems of Linear Equations in Two Variables
Chapter 4 Systems of Linear Equations; Matrices Section 1 Review: Systems of Linear Equations in Two Variables Opening Example A restaurant serves two types of fish dinners- small for $5.99 and large for
More informationQ520: Answers to the Homework on Hopfield Networks. 1. For each of the following, answer true or false with an explanation:
Q50: Answers to the Homework on Hopfield Networks 1. For each of the following, answer true or false with an explanation: a. Fix a Hopfield net. If o and o are neighboring observation patterns then Φ(
More information6-4 Solving Special Systems
Warm Up Solve each equation. 1. 2x + 3 = 2x + 4 2. 2(x + 1) = 2x + 2 3. Solve 2y 6x = 10 for y Solve by using any method. 4. y = 3x + 2 2x + y = 7 5. x y = 8 x + y = 4 Know: Solve special systems of linear
More informationName Period Date Ch. 5 Systems of Linear Equations Review Guide
Reteaching 5-1 Solving Systems by Graphing ** A system of equations is a set of two or more equations that have the same variables. ** The solution of a system is an ordered pair that satisfies all equations
More informationChapter 1-2 Add and Subtract Integers
Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More informationSolving Linear and Rational Inequalities Algebraically. Definition 22.1 Two inequalities are equivalent if they have the same solution set.
Inequalities Concepts: Equivalent Inequalities Solving Linear and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.4).1 Equivalent Inequalities Definition.1
More informationPair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables Linear equation in two variables x and y is of the form ax + by + c= 0, where a, b, and c are real numbers, such that both a and b are not zero. Example: 6x +
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More information2x + 5 = x = x = 4
98 CHAPTER 3 Algebra Textbook Reference Section 5.1 3.3 LINEAR EQUATIONS AND INEQUALITIES Student CD Section.5 CLAST OBJECTIVES Solve linear equations and inequalities Solve a system of two linear equations
More informationMatrices and RRE Form
Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More information10.2 Systems of Linear Equations
10.2 Systems of Linear Equations in Several Variables Copyright Cengage Learning. All rights reserved. Objectives Solving a Linear System The Number of Solutions of a Linear System Modeling Using Linear
More informationChapter 1 Linear Equations. 1.1 Systems of Linear Equations
Chapter Linear Equations. Systems of Linear Equations A linear equation in the n variables x, x 2,..., x n is one that can be expressed in the form a x + a 2 x 2 + + a n x n = b where a, a 2,..., a n and
More informationMAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices
MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is
More informationSystems of Linear Equations
Systems of Linear Equations Linear Equation Definition Any equation that is equivalent to the following format a a ann b (.) where,,, n are unknown variables and a, a,, an, b are known numbers (the so
More informationExamples of linear systems and explanation of the term linear. is also a solution to this equation.
. Linear systems Examples of linear systems and explanation of the term linear. () ax b () a x + a x +... + a x b n n Illustration by another example: The equation x x + 5x 7 has one solution as x 4, x
More informationChapter 4. Systems of Linear Equations; Matrices
Chapter 4 Systems of Linear Equations; Matrices Section 1 Review: Sys of Linear Eg in Two Var Section 2 Sys of Linear Eq and Aug Matr Section 3 Gauss-Jordan Elimination Section 4 Matrices: Basic Operations
More information6-4 Solving Special Systems
6-4 Solving Special Systems Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 6-4 Solve the equation. 1. 2(x + 1) = 2x + 2 3 pts Solve by using any method. 2. y = 3x + 2 2x + y = 7 5 pts possible
More informationMath 2331 Linear Algebra
1.1 Linear System Math 2331 Linear Algebra 1.1 Systems of Linear Equations Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu, University
More information1.5 F15 O Brien. 1.5: Linear Equations and Inequalities
1.5: Linear Equations and Inequalities I. Basic Terminology A. An equation is a statement that two expressions are equal. B. To solve an equation means to find all of the values of the variable that make
More informationMath 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
More informationCollege Algebra Through Problem Solving (2018 Edition)
City University of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Community College Winter 1-25-2018 College Algebra Through Problem Solving (2018 Edition) Danielle Cifone
More informationSection 8.1 & 8.2 Systems of Equations
Math 150 c Lynch 1 of 5 Section 8.1 & 8.2 Systems of Equations Geometry of Solutions The standard form for a system of two linear equations in two unknowns is ax + by = c dx + fy = g where the constants
More informationPart 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2
Solving Systems of Equations Algebraically Teacher Notes Comment: As students solve equations throughout this task, have them continue to explain each step using properties of operations or properties
More informationA. Incorrect! This inequality is a disjunction and has a solution set shaded outside the boundary points.
Problem Solving Drill 11: Absolute Value Inequalities Question No. 1 of 10 Question 1. Which inequality has the solution set shown in the graph? Question #01 (A) x + 6 > 1 (B) x + 6 < 1 (C) x + 6 1 (D)
More information9.1 - Systems of Linear Equations: Two Variables
9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n
More informationSNAP Centre Workshop. Solving Systems of Equations
SNAP Centre Workshop Solving Systems of Equations 35 Introduction When presented with an equation containing one variable, finding a solution is usually done using basic algebraic manipulation. Example
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More informationAlgebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher
Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
More informationMath 154 :: Elementary Algebra
Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with
More informationName: Block: Unit 2 Inequalities
Name: Block: Unit 2 Inequalities 2.1 Graphing and Writing Inequalities 2.2 Solving by Adding and Subtracting 2.3 Solving by Multiplying and Dividing 2.4 Solving Two Step and Multi Step Inequalities 2.5
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationMATCHING. Match the correct vocabulary word with its definition
Name Algebra I Block UNIT 2 STUDY GUIDE Ms. Metzger MATCHING. Match the correct vocabulary word with its definition 1. Whole Numbers 2. Integers A. A value for a variable that makes an equation true B.
More informationPut the following equations to slope-intercept form then use 2 points to graph
Tuesday September 23, 2014 Warm-up: Put the following equations to slope-intercept form then use 2 points to graph 1. 4x - 3y = 8 8 x 6y = 16 2. 2x + y = 4 2x + y = 1 Tuesday September 23, 2014 Warm-up:
More informationMatrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.
Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced
More informationChapter 2 (Operations with Integers) Bringing It All Together #1
Chapter 2 (Operations with Integers) Bringing It All Together #1 Vocabulary Check Define the following vocabulary words: 1) Absolute Value: 2) Quadrant: State whether the statement is true or false. If
More informationObjective. The student will be able to: solve systems of equations using elimination with multiplication. SOL: A.9
Objective The student will be able to: solve systems of equations using elimination with multiplication. SOL: A.9 Designed by Skip Tyler, Varina High School Solving Systems of Equations So far, we have
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationAlgebra 2 Honors Unit 1 Review of Algebra 1
Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationLesson 12: Systems of Linear Equations
Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More informationUnit 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 informationGraphing Systems of Linear Equations
Graphing Systems of Linear Equations Groups of equations, called systems, serve as a model for a wide variety of applications in science and business. In these notes, we will be concerned only with groups
More informationSolving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically
10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities
More informationMath 2 Variable Manipulation Part 6 System of Equations
Name: Date: 1 Math 2 Variable Manipulation Part 6 System of Equations SYSTEM OF EQUATIONS INTRODUCTION A "system" of equations is a set or collection of equations that you deal with all together at once.
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are
More informationThis is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0).
This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
More informationSystem of Linear Equation: with more than Two Equations and more than Two Unknowns
System of Linear Equation: with more than Two Equations and more than Two Unknowns Michigan Department of Education Standards for High School: Standard 1: Solve linear equations and inequalities including
More informationAlgebra 1. Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots.
Standard 1: Operations With Real Numbers Students simplify and compare expressions. They use rational exponents and simplify square roots. A1.1.1 Compare real number expressions. A1.1.2 Simplify square
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationdegree -6x 3 + 5x 3 Coefficients:
Date P3 Polynomials and Factoring leading coefficient degree -6 3 + 5 3 constant term coefficients Degree: the largest sum of eponents in a term Polynomial: a n n + a n-1 n-1 + + a 1 + a 0 where a n 0
More information7.5 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 information2.4 Graphing Inequalities
.4 Graphing Inequalities Why We Need This Our applications will have associated limiting values - and either we will have to be at least as big as the value or no larger than the value. Why We Need This
More informationSelf-Directed Course: Transitional Math Module 4: Algebra
Lesson #1: Solving for the Unknown with no Coefficients During this unit, we will be dealing with several terms: Variable a letter that is used to represent an unknown number Coefficient a number placed
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationReteach Simplifying Algebraic Expressions
1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order
More informationRelationships Between Planes
Relationships Between Planes Definition: consistent (system of equations) A system of equations is consistent if there exists one (or more than one) solution that satisfies the system. System 1: {, System
More informationMATH 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 informationSystems of Equations - Addition/Elimination
4.3 Systems of Equations - Addition/Elimination When solving systems we have found that graphing is very limited when solving equations. We then considered a second method known as substituion. This is
More informationChapter One: Introduction
Chapter One: Introduction Objectives 1. Understand the need for numerical methods 2. Go through the stages (mathematical modeling, solving and implementation) of solving a particular physical problem.
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are
More informationBinary Operations. Chapter Groupoids, Semigroups, Monoids
36 Chapter 5 Binary Operations In the last lecture, we introduced the residue classes Z n together with their addition and multiplication. We have also shown some properties that these two operations have.
More informationPre Algebra, Unit 1: Variables, Expression, and Integers
Syllabus Objectives (1.1) Students will evaluate variable and numerical expressions using the order of operations. (1.2) Students will compare integers. (1.3) Students will order integers (1.4) Students
More informationSystem of Linear Equations. Slide for MA1203 Business Mathematics II Week 1 & 2
System of Linear Equations Slide for MA1203 Business Mathematics II Week 1 & 2 Function A manufacturer would like to know how his company s profit is related to its production level. How does one quantity
More informationLesson 3-1: Solving Linear Systems by Graphing
For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we
More information