Section 8.1 & 8.2 Systems of Equations
|
|
- Edwina Harrington
- 5 years ago
- Views:
Transcription
1 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 a, b, c, d, f, and g are known numbers. A solution of this system is a pair of numbers x 0 and y 0 which are solutions to both equations. This pair of numbers is commonly written as a point (x 0, y 0 ) and interpreted as a point in the Euclidean plane R 2. Example 1. Are the points (4, 0) or ( 3, 2) solutions to the system of equations 2x + 7y = 8 x + y = 5 Example 2. The graph below contains both lines from the previous example. Each point on a line is a solution to the equation of that line. The point where the lines cross is the only solution to both lines. Theorem. When solving a system of two linear equations, there are three possibilities for the number of solutions. 1. There is one solution, a single point where the two lines cross. 2. There are no solutions, and the system of equations is sometimes called inconsistent. The two lines are parallel and never cross. 3. There are an infinite number of solutions (the two equations are the exact same line when graphed).
2 Math 150 c Lynch 8.1 & 8.2 Systems of Equations 2 of 5 Example 3. The three following examples demonstrate all three cases. x y = 1 x + y = 1 There is one solution where the lines cross, the point (1, 0). 2x + y = 1 2x + y = 3 There are no solutions because the lines never cross. 3x + 2y = 6 6x + 4y = 12 The two lines are exactly the same. Every point on the line solves both equations, and there are infinitely many solutions. Example 4. Graph the solution set of the system, and from your plot describe the solution set to this system. x + y = 2 2x 3y = 3 x y = 4. Algebraic Methods Substitution Substitution Method: To solve a system of linear equations by substitution, perform the following steps: 1. Solve one of the equations for one of the unknowns in terms of the other unknown (i.e., if the unknowns are x and y, solve one of the equations for y.). 2. Substitute the expression for the unknown in the remaining equations. 3. Repeat this process until one of the equations has been reduced to an equation in only one unknown. 4. Solve this equation for the unknown. 5. Use this value to determine the values of the other unknowns.
3 Math 150 c Lynch 8.1 & 8.2 Systems of Equations 3 of 5 Example 5. Solve the following equations 1. x 3y = 6 2x + 5y = x 3y = 5 8x 6y = 10 Algebraic Methods Elimination Elimination To solve a system of linear equations by elimination, perform the following steps: 1. Multiply one of your equations by a number so that two equations have the opposite number of the same variable (i.e., if one equation has 6x, and the second has 2x, multiply the second equation by 3. ) 2. Add the two equations together (use the new equation after multiplying by a constant). 3. Now you have a new equation with one less variable. 4. Continue this process eliminating variables, until you have an equation with only one variable left. 5. Solve this equation for the variable. 6. Use this value to determine the value of the other variables. Example 6. Solve the following systems of equations: 1. 3x 5y = 16 x + y = x 6y = 5 x + 3y = 4
4 Math 150 c Lynch 8.1 & 8.2 Systems of Equations 4 of 5 Example 7. Suppose a plane flies a round trip between two cities. The flight from the first city is into a strong headwind and takes 1 hr and 30 minutes. The return flight is with the wind and takes 50 minutes. If the cities are 100 miles apart, what is the aircraft s speed and the winds speed. Assume that both the aircraft s and wind s speeds are constant. Example 8. Suppose a lab worker needs to make a 15% acid solution, but the lab only has 10% and 45% acid solutions. How many milliliters of the 10% and 45% solutions should the worker use to get 350 ml of a 15% solution?
5 Math 150 c Lynch 8.1 & 8.2 Systems of Equations 5 of 5 Non-linear Systems of Equations To solve a non-linear system of equations, we use similar methods as before. Try to use either the substitution method or elimination method. Example 9. Solve the following non-linear systems of equations. 1. (x 2) 2 + y 2 = 9 y (x 2) 2 = 3 2. x 2 + y 2 = 5 x 2 6 x 6 + y2 = 4
Chapter 7A - Systems of Linear Equations
- Chapter 7A Chapter 7A - Systems of Linear Equations Geometry of Solutions In an earlier chapter we learned how to solve a single equation in one unknown. The general form of such an equation has the
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 informationA. 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 informationThe Method of Substitution. Linear and Nonlinear Systems of Equations. The Method of Substitution. The Method of Substitution. Example 2.
The Method of Substitution Linear and Nonlinear Systems of Equations Precalculus 7.1 Here is an example of a system of two equations in two unknowns. Equation 1 x + y = 5 Equation 3x y = 4 A solution of
More informationb. 3x + 2y = 4 2x + 3y = 6
To review solving systems of linear equations using non-matrix methods, watch the following set of YouTube videos. They are followed by several practice problems for you to try, covering all the basic
More informationJust as in the previous lesson, all of these application problems should result in a system of equations with two equations and two variables:
The two methods we have used to solve systems of equations are substitution and elimination. Either method is acceptable for solving the systems of equations that we will be working with in this lesson.
More informationUnit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
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
More informationCHAPTER 5 LINEAR SYSTEMS
CHAPTER 5 LINEAR SYSTEMS Systems of Linear equations have either one solution (independent), no solutions (inconsistent), or infinitely many solutions (dependent). An independent system is the case when
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Algebraic Concepts Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the inequality. ) - - 0x - -x - ) A) x > -0 B) x < -0 C) x 0 D) x
More informationDetermine the least common denominator (LCD) of the rational expressions. Solve each equation. List any restrictions and check your answer.
Algebra II - Chapter 8 Review Determine the least common denominator (LCD) of the rational expressions 1 2 Add, subtract, multiply, or divide as indicated List any restrictions for the variable(s) and
More information3. Solve the following inequalities and express your answer in interval notation.
Youngstown State University College Algebra Final Exam Review (Math 50). Find all Real solutions for the following: a) x 2 + 5x = 6 b) 9 x2 x 8 = 0 c) (x 2) 2 = 6 d) 4x = 8 x 2 e) x 2 + 4x = 5 f) 36x 3
More informationWorking with equations for speed and velocity
Working with equations for speed and velocity Objectives Interpret symbolic relationships. Describe motion using equations for speed and average velocity. Solve speed and velocity problems mathematically.
More informationLesson 30: Linear Systems in Three Variables
Lesson 30: Linear Systems in Three Variables Student Outcomes Students solve linear systems in three variables algebraically. Lesson Notes Students solved systems of linear equations in two variables using
More informationPartial Fraction Decomposition
Partial Fraction Decomposition As algebra students we have learned how to add and subtract fractions such as the one show below, but we probably have not been taught how to break the answer back apart
More informationBasic Algebra: Unit 5 Systems of Linear Equations and Inequalities. Solving Systems of linear equations in two unknown variables using algebra
Solving Systems of linear equations in two unknown variables using algebra Problems of this type look like: Solve the system of equations 3x + 67y = 12 You will have one of three possibilities when solving
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 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 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 information8-6 Solving Rational Equations and Inequalities. Solve each equation. Check your solution. ANSWER: 11 ANSWER: 9 ANSWER: 7 ANSWER: 3 ANSWER: 8
Solve each equation. Check your solution. 1. 11 2. 9 3. 7 4. 3 5. 8 6. 5 esolutions Manual - Powered by Cognero Page 1 7. 14 8. 14 9. CCSS STRUCTURE Sara has 10 pounds of dried fruit selling for $6.25
More informationMath-1010 Lesson 1-6. Textbook 1-11 (Systems of Linear Equations)
Math-1010 Lesson 1-6 Textbook 1-11 (Systems of Linear Equations) College Finals are over. You re moving back home for the summer. You need to rent a truck to move your possessions from the dorm back to
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 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 informationSTUDY GUIDE FOR TEST 2. Project 3: Fill in the blanks and do the assigned questions. 11/6/07
STUDY GUIDE FOR TEST 2 Name: Project 3: Fill in the blanks and do the assigned questions. 11/6/07 Quadrant II Quadrant I ORDERED PAIR: The first number in the ordered pair is the -coordinate and the second
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 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 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 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 informationa) b) 1 c) 19 d) e) none of these 2.) 80 0 a) undefined b) 1 c) 80 d) 0 e) none of these Evaluate the expression 3.) a) b) c) d) e) none of these
Math 5 Review Practice Questions *Note 1: The actual Math 5 final exam may include, but is not limited to, the problems in this handout. Study your notes, past homework assignments, quizzes, and tests.
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 information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
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 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 informationFranklin Math Bowl 2010 Group Problem Solving Test Grade 6
Group Problem Solving Test Grade 6 1. Carrie lives 10 miles from work. She leaves in the morning before traffic is heavy and averages 30 miles per hour. When she goes home at the end of the day, traffic
More information7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems. Main Ideas:
7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems Toolkit: graphing lines rearranging equations substitution Main Ideas: Definitions: Linear
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
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 informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 370 Exam 4 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of equations by the substitution method. 1) y = 4x - 5 y
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 informationAlgebra I / Integrated Math I 2012
Algebra I / Integrated Math I 0 Sponsored by the Indiana Council of Teachers of Mathematics Indiana State Mathematics Contest This test was prepared by faculty at University of Southern Indiana ICTM Website
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 informationy in both equations.
Syllabus Objective: 3.1 The student will solve systems of linear equations in two or three variables using graphing, substitution, and linear combinations. System of Two Linear Equations: a set of two
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 informationSystems of Linear Equations: Solving by Adding
8.2 Systems of Linear Equations: Solving by Adding 8.2 OBJECTIVES 1. Solve systems using the addition method 2. Solve applications of systems of equations The graphical method of solving equations, shown
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 informationTom Robbins WW Prob Lib1 Math , fall 2004
Tom Robbins WW Prob Lib Math -, fall 4 WeBWorK assignment due 9/9/4 at :59 PM..( pt) set/p.pg Is 3 a solution to 5 4 9?.( pt) set/p.pg Solve: 7 separated by commas (e.g.,, ). If there are no solutions,
More informationSystems of Equations and Inequalities
1 Systems of Equations and Inequalities 2015 03 24 2 Table of Contents Solving Systems by Graphing Solving Systems by Substitution Solve Systems by Elimination Choosing your Strategy Solving Systems of
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 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 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 informationSection 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra
Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent
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 informationMath Analysis Notes Mrs. Atkinson 1
Name: Math Analysis Chapter 7 Notes Day 6: Section 7-1 Solving Systems of Equations with Two Variables; Sections 7-1: Solving Systems of Equations with Two Variables Solving Systems of equations with two
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 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 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 informationUnit 4 Rational Expressions. Mrs. Valen+ne Math III
Unit 4 Rational Expressions Mrs. Valen+ne Math III 4.1 Simplifying Rational Expressions Simplifying Rational Expressions Expression in the form Simplifying a rational expression is like simplifying any
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 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 informationAlgebra 2H: Quadratic Equations Practice test
Algebra 2H: Quadratic Equations Practice test 1. There are total of 22 questions in this test. 2. Each of the first 20 worth 2 points: a. The first 16 relate directly to the present chapter. b. The last
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 informationAPPLICATIONS OF INTEGRATION
6 APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6.5 Average Value of a Function In this section, we will learn about: Applying integration to find out the average value of a function. AVERAGE
More informationSystems of Linear Equations in Two Variables. Break Even. Example. 240x x This is when total cost equals total revenue.
Systems of Linear Equations in Two Variables 1 Break Even This is when total cost equals total revenue C(x) = R(x) A company breaks even when the profit is zero P(x) = R(x) C(x) = 0 2 R x 565x C x 6000
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 information8 Wyner Honors Algebra II Fall 2013
8 Wyner Honors Algebra II Fall 2013 CHAPTER THREE: SOLVING EQUATIONS AND SYSTEMS Summary Terms Objectives The cornerstone of algebra is solving algebraic equations. This can be done with algebraic techniques,
More informationMATH 110: FINAL EXAM REVIEW
MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set
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 informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationMath 060/Final Exam Review Guide/ / College of the Canyons
Math 060/Final Exam Review Guide/ 010-011/ College of the Canyons General Information: The final exam is a -hour timed exam. There will be approximately 40 questions. There will be no calculators or notes
More informationThe line that passes through the point A ( and parallel to the vector v = (a, b, c) has parametric equations:,,
Vectors: Lines in Space A straight line can be determined by any two points in space. A line can also be determined by specifying a point on it and a direction. The direction would be a non-zero parallel
More informationAssignment #1 MAT121 Summer 2015 NAME:
Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
More informationOBJECTIVE 5 SOLVING SYSTEMS 5/19/2016 SOLVING SYSTEMS OF TWO EQUATIONS BY SUBSTITUTION:
/9/ OBJECTIVE Sstems & Matrices SOLVING SYSTEMS OF TWO EQUATIONS BY SUBSTITUTION:. Solve one of the equations for one of the variables in terms of the other.. Substitute this epression into the nd equation,
More informationLesson
Lesson 28 Zeros of a function: - the inputs that make the function equal to zero (same values as the x- coordinates of the x-intercepts) o if ( ), ( ) when o zeros are 2 and - when a function is graphed,
More informationMath Departmental Exit Assessment Review (Student Version)
Math 008 - Departmental Eit Assessment Review (Student Version) Solve the equation. (Section.) ) ( + ) - 8 = 6-80 - 0 ) + - - 7 = 0-60 - 0 ) 8 + 9 = 9 - - ) - = 60 0-0 -60 ) 0.0 + 0.0(000 - ) = 0.0 0 6000
More informationAssuming the Earth is a sphere with radius miles, answer the following questions. Round all answers to the nearest whole number.
G-MG Satellite Alignments to Content Standards: G-MG.A.3 Task A satellite orbiting the earth uses radar to communicate with two control stations on the earth's surface. The satellite is in a geostationary
More information3-1 Solving Systems of Equations. Solve each system of equations by using a table. 1. ANSWER: (3, 5) ANSWER: (2, 7)
Solve each system of equations by using a table. 1. 9. CCSS MODELING Refer to the table below. (3, 5) 2. (2, 7) Solve each system of equations by graphing. 3. a. Write equations that represent the cost
More informationSolving Systems of Linear Equations
Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes
More informationChapter 2 Describing Motion: Kinematics in One Dimension
Chapter 2 Describing Motion: Kinematics in One Dimension Units of Chapter 2 Reference Frames and Displacement Average Velocity Instantaneous Velocity Acceleration Motion at Constant Acceleration Solving
More informationBe sure to show all work! Use pencil Write an equation to support your answer.
Name: Intermediate Algebra Be sure to show all work! Use pencil. PROBLEMS Solve the equation. z + 7 + 1 z 1 = z + 8 Date Due: Chapter -B Homework ANSWERS. Write an equation to support your answer.. The
More informationMath 3 Variable Manipulation Part 1 Algebraic Systems
Math 3 Variable Manipulation Part 1 Algebraic Systems 1 PRE ALGEBRA REVIEW OF INTEGERS (NEGATIVE NUMBERS) Concept Example Adding positive numbers is just simple addition 2 + 3 = 5 Subtracting positive
More informationb) Rectangular box: length L, width W, height H, volume: V = LWH, cube of side s, V = s 3
Basic Math Review for PHYS 100 - Physics of Everyday Experience ----------------------------------------------------------------------------------------------------- Basic Algebra a) If x = y + z, then:
More informationSystems and Matrices CHAPTER 7
CHAPTER 7 Systems and Matrices 7.1 Solving Systems of Two Equations 7.2 Matrix Algebra 7.3 Multivariate Linear Systems and Row Operations 7.4 Partial Fractions 7.5 Systems of Inequalities in Two Variables
More information2) (9, 1), (3, 9) 2) A) C) 5 6. Use the vertical line test to determine whether the graph is the graph of a function. 4) 4)
Test 2 Name (please print) Find the slope of the line that goes through the given points. 1) (-4, 2), (-3, 2) 1) A) 4 B) - 4 7 C) 0 D) undefined 2) (9, 1), (3, 9) 2) A) - 4 3 B) 4 3 C) 5 6 D) - 3 4 Determine
More informationCHAPTER 9: Systems of Equations and Matrices
From Section 1.4: Equations of Lines and Modeling Use a graphing calculator to model the data with a linear function. MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9:
More informationScientific Notation. exploration. 1. Complete the table of values for the powers of ten M8N1.j. 110 Holt Mathematics
exploration Georgia Performance Standards M8N1.j 1. Complete the table of values for the powers of ten. Exponent 6 10 6 5 10 5 4 10 4 Power 3 10 3 2 10 2 1 1 0 2 1 0.01 10 10 1 10 1 1 1 0 1 1 0.1 10 0
More informationFor problems 1 4, evaluate each expression, if possible. Write answers as integers or simplified fractions
/ MATH 05 TEST REVIEW SHEET TO THE STUDENT: This Review Sheet gives you an outline of the topics covered on Test as well as practice problems. Answers are at the end of the Review Sheet. I. EXPRESSIONS
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 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 information2-2. Learn to translate between words and math. Course 1
Learn to translate between words and math. In word problems, you may need to translate words to math. Action Put together or combine Operation Add Find how much more or less Subtract Put together groups
More informationKansas City Area Teachers of Mathematics 2011 KCATM Math Competition ALGEBRA GRADES 7-8
Kansas City Area Teachers of Mathematics 2011 KCATM Math Competition ALGEBRA GRADES 7-8 INSTRUCTIONS Do not open this booklet until instructed to do so. Time limit: 20 minutes You may NOT use calculators.
More informationAlgebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix
Algebra & Trig. I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural
More informationSolutions of Linear system, vector and matrix equation
Goals: Solutions of Linear system, vector and matrix equation Solutions of linear system. Vectors, vector equation. Matrix equation. Math 112, Week 2 Suggested Textbook Readings: Sections 1.3, 1.4, 1.5
More informationMath x + 3y 5z = 14 3x 2y + 3z = 17 4x + 3y 2z = 1
Math 210 1. Solve the system: x + y + z = 1 2x + 3y + 4z = 5 (a z = 2, y = 1 and x = 0 (b z =any value, y = 3 2z and x = z 2 (c z =any value, y = 3 2z and x = z + 2 (d z =any value, y = 3 + 2z and x =
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 informationA. 16 B. 16 C. 4 D What is the solution set of 4x + 8 > 16?
Algebra II Honors Summer Math Packet 2017 Name: Date: 1. Solve for x: x + 6 = 5x + 12 2. What is the value of p in the equation 8p + 2 = p 10? F. 1 G. 1 H. J.. Solve for x: 15x (x + ) = 6 11. Solve for
More informationINTERMEDIATE ALGEBRA REVIEW FOR TEST 3
INTERMEDIATE ALGEBRA REVIEW FOR TEST 3 Evaluate the epression. ) a) 73 (-4)2-44 d) 4-3 e) (-)0 f) -90 g) 23 2-4 h) (-2)4 80 i) (-2)5 (-2)-7 j) 5-6 k) 3-2 l) 5-2 Simplify the epression. Write your answer
More informationSolving Systems Algebraically
3-2 Solving Systems Algebraically TEKS FOCUS VOCABULARY Equivalent systems Equivalent Foundational to TEKS (3)(A) Formulate systems of equations, including systems consisting of three linear equations
More informationLesson 5: Solving Linear Systems Problem Solving Assignment solutions
Write inequalities to represent the following problem, and then solve to answer the question. 1. The Rent-A-Lemon Car Rental Company charges $60 a day to rent a car and an additional $0.40 per mile. Alex
More informationMath Studio College Algebra
Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationPage 24 Monday August 03, 2015
Page Monday August 0, 05 Convert with-in the metric system Practice: How many. Practice: How many.. Centimeters in a meter?. Grams in Kilogram?. Liters in Kiloliter?. Meters in Kilometer? 5. Millimeters
More information