Steps For Solving By Substitution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Steps For Solving By Substitution"

Transcription

1 6.1 Systems of Linear Equations An equation is a linear equation in two variables x and y. Similarly, is a linear equation in three variables x, y, and z. We may also consider linear equations in four, five, or any number of variables. A system of linear equations is collection of two or more linear equations each containing one or more variables. In this section we shall consider only systems of two linear equations in two variables. Systems involving more than two variables are discussed in the next section. We can view the problem of solving a system of two linear equation containing two variables as a geometry problem. The graph of each equation in such as system is a line. So, a system of two equations containing two variables represents a pair of lines. 1. intersecting lines; system 2. Parallel lines; system. Lines coincide; system has exactly one solution has no solution. has infinitely many (system is inconsistent) solutions (system is dependent) If the system has infinitely many solutions (dependent system) the system is solved by finding the general form of the solution set. The two methods of solving a system of linear equations that will be discussed in this section are the substitution method and the elimination method. Steps For Solving By Substitution 1. Pick one of the equations and solve for one of the variables in terms of the other variable. 2. Substitute this quantity in the remaining equation and solve. Find the value of the remaining variable by back Solve by Substitution Solution: Step 1 The easiest to solve is y in equation Step 2. Substitute 56 in 6 5 equation

2 Step 2. Substitute 56 in equation The solution is 1, Step 1 The easiest to solve is x in equation Step 2. Substitute 5 in equation The solution is 1, Step 1 The easiest to solve is y in equation Step 2. Substitute 42 in equation The System has no solution. The system is inconsistent.

3 Steps For Solving By Elimination 1. Multiply both sides of one (or both) equation(s) by the appropriate nonzero numbers so that when the equations are added together one of the variables will be eliminated. 2. Solve this equation for the remaining variable.. Find the value of the remaining variable by back substitution. Solve by Elimination Solution: We multiply both sides of equation (2) by 2 so that the coefficient of x in the two Now find the value of the remaining variable by back substitution The solution is 2, Solution: The smallest common multiple between 6 and 4 is 12, so we multiply both sides of equation (1) by 2 and both sides of equation (2) by so that the coefficient of y in the two Now find the value of the remaining variable by back substitution The Solution to the system is 1,1 6.

4 Solution: We multiply both sides of equation (1) by 2 so that the coefficient of x in the two The system is a dependent system of linear equations and thus has infinitely many solutions. To determine the general form of the solution set, we solve for one of the variables in either equation The General Form of the solution set is 4 2,: 7. The population y in year x of Long Beach and new Orleans is approximated by the equations : : Where x = 0 corresponds to 1980 and y in thousands. In what year do the two cities have the same population? Solution: We multiply both sides of equation (2) by 2 so that the coefficient of y in the two The year = 2002 the two cities will have the same population. 8. An apparel shop sells skirts for $45 and blouses for $5. Its entire stock is worth $51,750, but sales are slow and only half the skirts and two thirds of the blouses are sold, for a total of $0,600. How many skirts and blouses are left in the store? x = number of skirts in the store y = number of blouses in the store , , , ,600

5 Multiply equation (1) by , , , ,50 51, , Half the skirts are left (they sold half) and one third of the blouses are left (they sold two thirds) Therefore, there are 260 skirts left and 270 blouses left in the store. 9. A company purchases two models of bicycles: model 201 and model 01. Model 201 requires 2 hours of assembly time, and model 01 requires hours of assembly time. The parts for model 201 cost $25 per bike, and the parts for model 01 cost $0 per bike. If the company has a total of 4 hours of assembly time and $65 available per day for these two models, how many of each can be made in a day? x = number of 201 models y = number of 01 models models and 8 01 models can be produced in one day.

5 Systems of Equations

5 Systems of Equations Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

More information

10.1 Systems of Linear Equations: Substitution and Elimination

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

More information

Systems of Linear Equations and Inequalities

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

More information

SYSTEMS OF LINEAR EQUATIONS

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

More information

Systems of Linear Equations in Three Variables

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

More information

Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent.

Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 2x 1 y = 2x + 3 The lines y = 2x 1 and y = 2x + 3 intersect at exactly one point

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

More information

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Solving Systems of Two Equations Algebraically

Solving Systems of Two Equations Algebraically 8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

More information

3. Solve the equation containing only one variable for that variable.

3. Solve the equation containing only one variable for that variable. Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated

More information

Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price Three functions of importance in business are cost functions, revenue functions and profit functions. Cost functions

More information

What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases Lesson Objective:

What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases Lesson Objective: What if systems are not in y = mx + b form? Strategies for Solving Systems and Special Cases Lesson Objective: Length of Activity: Students will continue work with solving systems of equations using the

More information

4.3-4.4 Systems of Equations

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

More information

Solving Linear Equations in One Variable. Worked Examples

Solving Linear Equations in One Variable. Worked Examples Solving Linear Equations in One Variable Worked Examples Solve the equation 30 x 1 22x Solve the equation 30 x 1 22x Our goal is to isolate the x on one side. We ll do that by adding (or subtracting) quantities

More information

Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve

Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by

More information

Systems of Linear Equations

Systems of Linear Equations DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

COST-VOLUME-PROFIT RELATIONSHIPS

COST-VOLUME-PROFIT RELATIONSHIPS TM 5-1 COST-VOLUME-PROFIT RELATIONSHIPS Cost-volume-profit (CVP) analysis is concerned with the effects on net operating income of: Selling prices. Sales volume. Unit variable costs. Total fixed costs.

More information

3.3 Applications of Linear Functions

3.3 Applications of Linear Functions 3.3 Applications of Linear Functions A function f is a linear function if The graph of a linear function is a line with slope m and y-intercept b. The rate of change of a linear function is the slope m.

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Question 2: How do you solve a linear programming problem with a graph?

Question 2: How do you solve a linear programming problem with a graph? Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

MathQuest: Linear Algebra. 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7

MathQuest: Linear Algebra. 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7 MathQuest: Linear Algebra Systems of Equations 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7 (a) x = 4 and y = 5 (b) x = 4 and y = 5 (c) x = 2 and y = 1 (d) x = 2 and

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Introduction to Diophantine Equations

Introduction to Diophantine Equations Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

More information

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

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

More information

Definition 1 Let a and b be positive integers. A linear combination of a and b is any number n = ax + by, (1) where x and y are whole numbers.

Definition 1 Let a and b be positive integers. A linear combination of a and b is any number n = ax + by, (1) where x and y are whole numbers. Greatest Common Divisors and Linear Combinations Let a and b be positive integers The greatest common divisor of a and b ( gcd(a, b) ) has a close and very useful connection to things called linear combinations

More information

price quantity q The Supply Function price quantity q

price quantity q The Supply Function price quantity q Shown below is another demand function for price of a pizza p as a function of the quantity of pizzas sold per week. This function models the behavior of consumers with respect to price and quantity. 3

More information

1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems 1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

More information

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN TWO VARIABLES 66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

More information

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

More information

SECTION 8-1 Systems of Linear Equations and Augmented Matrices

SECTION 8-1 Systems of Linear Equations and Augmented Matrices 86 8 Systems of Equations and Inequalities In this chapter we move from the standard methods of solving two linear equations with two variables to a method that can be used to solve linear systems with

More information

Solving systems by elimination

Solving systems by elimination December 1, 2008 Solving systems by elimination page 1 Solving systems by elimination Here is another method for solving a system of two equations. Sometimes this method is easier than either the graphing

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

Solving Systems of Linear Equations by Elimination

Solving Systems of Linear Equations by Elimination 5.3 Solving Systems of Linear Equations by Elimination How can you use elimination to solve a system of linear equations? ACTIVITY: Using Elimination to Solve a System Work with a partner. Solve each system

More information

Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes Standard equations for lines in space Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

More information

Slope-Intercept Equation. Example

Slope-Intercept Equation. Example 1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

More information

Solving Systems of Equations

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

More information

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

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

More information

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables

Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables Plan Pacing: 1 class period Chunking

More information

Math 113 Review for Exam I

Math 113 Review for Exam I Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular

More information

Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

More information

6-3 Solving Systems by Elimination

6-3 Solving Systems by Elimination Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables

More information

Chapter 6: Break-Even & CVP Analysis

Chapter 6: Break-Even & CVP Analysis HOSP 1107 (Business Math) Learning Centre Chapter 6: Break-Even & CVP Analysis One of the main concerns in running a business is achieving a desired level of profitability. Cost-volume profit analysis

More information

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

More information

Semester Exam Review ANSWERS. b. The total amount of money earned by selling sodas in a day was at least $1,000. 800 4F 200 F

Semester Exam Review ANSWERS. b. The total amount of money earned by selling sodas in a day was at least $1,000. 800 4F 200 F Unit 1, Topic 1 P 2 1 1 W L or P2 L or P L or P L 2 2 2 2 1. 2. A. 5F 160 C 9 3. B. The equation is always true, because both sides are identical.. A. There is one solution, and it is x 30. 5. C. The equation

More information

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution 4.2 Solving Systems of Linear Equations by Substitution How can you use substitution to solve a system of linear equations? 1 ACTIVITY: Using Substitution to Solve a System Work with a partner. Solve each

More information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

More information

Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

More information

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

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

More information

No Solution Equations Let s look at the following equation: 2 +3=2 +7

No 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 information

Where Do We Meet? Students will represent and analyze algebraically a wide variety of problem solving situations.

Where Do We Meet? Students will represent and analyze algebraically a wide variety of problem solving situations. Beth Yancey MAED 591 Where Do We Meet? Introduction: This lesson covers objectives in the algebra and geometry strands of the New York State standards for Algebra I. The students will use the graphs of

More information

Quadratic Modeling Business 10 Profits

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

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

More information

5.5. Solving linear systems by the elimination method

5.5. Solving linear systems by the elimination method 55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

More information

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

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

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

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

Notes from February 11

Notes from February 11 Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear 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, b are given real numbers. For example, with x and

More information

Let s explore the content and skills assessed by Heart of Algebra questions.

Let s explore the content and skills assessed by Heart of Algebra questions. Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

More information

The application of linear programming to management accounting

The application of linear programming to management accounting The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per

More information

Math 215 HW #1 Solutions

Math 215 HW #1 Solutions Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What

More information

IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa. IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for

More information

Lesson 22: Solution Sets to Simultaneous Equations

Lesson 22: Solution Sets to Simultaneous Equations Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

More information

Cost-Volume-Profit Analysis

Cost-Volume-Profit Analysis HOSP 2110 (Management Acct) Learning Centre Cost-Volume-Profit Analysis The basic principles of CVP analysis were covered in business math. CVP analysis can be done both graphically, through plotting the

More information

Systems of Linear Equations: Two Variables

Systems of Linear Equations: Two Variables OpenStax-CNX module: m49420 1 Systems of Linear Equations: Two Variables OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,

More information

CHAPTER 5: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)...

CHAPTER 5: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)... Table of Contents CHAPTER 5: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)... 11 5.0 ANCHOR PROBLEM: CHICKENS AND PIGS... 14 SECTION 5.1: UNDERSTAND SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS... 15 5.1a Class

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

Solutions of Equations in Two Variables

Solutions of Equations in Two Variables 6.1 Solutions of Equations in Two Variables 6.1 OBJECTIVES 1. Find solutions for an equation in two variables 2. Use ordered pair notation to write solutions for equations in two variables We discussed

More information

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

More information

Solving Systems of Equations Introduction

Solving Systems of Equations Introduction Solving Systems of Equations Introduction Outcome (learning objective) Students will write simple systems of equations and become familiar with systems of equations vocabulary terms. Student/Class Goal

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

A. Factoring Method - Some, but not all quadratic equations can be solved by factoring.

A. Factoring Method - Some, but not all quadratic equations can be solved by factoring. DETAILED SOLUTIONS AND CONCEPTS - QUADRATIC EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Answers to Text Questions and Problems. Chapter 22. Answers to Review Questions

Answers to Text Questions and Problems. Chapter 22. Answers to Review Questions Answers to Text Questions and Problems Chapter 22 Answers to Review Questions 3. In general, producers of durable goods are affected most by recessions while producers of nondurables (like food) and services

More information

Systems of Equations and Inequalities

Systems of Equations and Inequalities Economic Outcomes Systems of Equations and Inequalities Annual Earnings of Young Adults For both males and females, earnings increase with education: full-time workers with at least a bachelor s degree

More information

Math Review Large Print (18 point) Edition Chapter 2: Algebra

Math Review Large Print (18 point) Edition Chapter 2: Algebra GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,

More information

Elasticity. I. What is Elasticity?

Elasticity. I. What is Elasticity? Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

More information

Linear Programming. April 12, 2005

Linear Programming. April 12, 2005 Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

More information

Linear Programming. Solving LP Models Using MS Excel, 18

Linear Programming. Solving LP Models Using MS Excel, 18 SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

More information

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 7 C7 2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

More information

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

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

More information

Recitation #5 Week 02/08/2009 to 02/14/2009. Chapter 6 - Elasticity

Recitation #5 Week 02/08/2009 to 02/14/2009. Chapter 6 - Elasticity Recitation #5 Week 02/08/2009 to 02/14/2009 Chapter 6 - Elasticity 1. This problem explores the midpoint method of calculating percentages and why this method is the preferred method when calculating price

More information

Systems of Equations Involving Circles and Lines

Systems of Equations Involving Circles and Lines Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

More information

2. System of linear equations can be solved by graphing, substitution, or eliminating a variable.

2. System of linear equations can be solved by graphing, substitution, or eliminating a variable. 1 Subject: Algebra 1 Grade Level: 9 th Unit Plan #: 6 UNIT BACKGROUND Unit Title: Systems of Equations and Inequalities Grade Level: 9 Subject/Topic: Algebra 1 Key Words: Graphing, Substitution, Elimination

More information

Multiplying Polynomials 5

Multiplying Polynomials 5 Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

More information

A Detailed Price Discrimination Example

A Detailed Price Discrimination Example A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include

More information