# Chapter 4: Systems of Equations and Inequalities

Size: px
Start display at page:

Transcription

1 Chapter 4: Systems of Equations and Inequalities

2 4.1 Systems of Equations A system of two linear equations in two variables x and y consist of two equations of the following form: Equation 1: ax + by = c Equation 2: dx + ey = f where the solution (x,y) satisfies both equations. Checking Solutions of a Linear System: 3x 2y = 2 x + 2y = 6 1.) Is (2,2) a solution of the above system of equations? 2.) Is (0,-1) a solution of the above system of equations? Solving a System Graphically :

3 Examples: 1.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 2x 2y = -8 2x + 2y = 4 Check Algebraically.. 2.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 3x 2y = 6 3x 2y = 2 3.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator. 2x 2y = -8-2x + 2y = 8

4 Solving a System by Substitution Solve each system below by the method of Substitution. y = 3x 3 1) y = x + 5 2) x + y = 3 3x + y = 1 Solve the system below by the method of Substitution, demonstrating that there is no solution. 3) 2x 2y = 0 x y = 1 What does the graph of this system look like? Solve the system below by the method of Substitution, demonstrating that there are infinitely many solutions. 4) x + y = 7 2x = 14 2y What does the graph of this system look like?

5 Applications Use your graphing calculator to graph the system of equations for each application below and to answer related questions. Create a sketch of each graph, labeling the axes with appropriate scales. 1) You are checking out cell phone plans and discover that Talk Anytime Wireless charges \$50.00 per month for the first phone line and charges \$20.00 per additional phone line. Text Away Wireless charges \$80.00 per month for the first phone line and \$5.00 per additional phone line. Determine the number of additional phone lines for which it would be cheaper to use Talk Anytime verses Text Away. 2) James and Zach began saving money from their part-time jobs. James started with \$50 in his savings and earns \$10 per hour at his job. Zach started with \$225 in his savings and earns \$7.50 per hour. If both boys save all of their earnings (and we disregard tax) when will they have the same amount of savings? 3) You are choosing between two movie rental services. Company A charges \$2.99 per movie plus a \$20 monthly fee. Company B charges \$4.99 per movie with no monthly fee. How many movies could you rent and get charged the same monthly bill? If you only rent, on average, 8 movies per month, which is the better deal for you?

6 Check for Understanding 1) You are checking a solution of a system of linear equations. How can you tell if the solution is valid or not? 2) Describe how the graph of a system of linear equations looks when a. There is not solution. b. There is exactly one solution. c. There are infinitely many solutions.

7 4.2 Linear Systems in Two Variables Solving a system by the method of Elimination 1. 3x + 2y = 4 5x 2y = 8 4x 5y = x y = x + 9y = 8 2x + 6y = 7

8 Applications 1. A bus station 15 miles from the airport runs a shuttle service to and from the airport. The 9:00 a.m. bus leaves for the airport traveling 30 mph. The 9:05 a.m. bus leaves for the airport traveling 40 mph. Write a system of linear equations to represent distance as a function of time for each bus. How far from the airport will the 9:05 a.m. bus catch up to the 9:00 a.m. bus? D = 30t D = 40 t The school yearbook staff is purchasing a digital camera. Recently the staff received two ads in the mail. The ad for store #1 states that all digital cameras are 15% off. The ad for store #2 gives a \$300 coupon to use when purchasing any digital camera. Assume that the lowest priced digital camera is \$700. When could you get the same deal at either store? Let C = the cost of a camera after the discount Let x = the original cost of a camera

9 3. You are starting a business selling boxes of hand-painted greeting cards. To get started, you spend \$36 on paint and paintbrushes that you need. You buy boxes of plain cards for \$3.50 per box, paint the cards, and then sell them for \$5 per box. How many boxes must you sell for your earnings to equal your expenses? What will your earnings and expenses equal when you break even? (Write an equation to represent Total Expenses and another equation to represent Total Earnings.) 4. You commute to center city 5 days per week on a SEPTA train. You can purchase a monthly pass for \$140 per month or purchase a round trip ticket each day that you commute for \$9.50 per ticket. What is the number of days that you must ride to begin saving money by using the monthly pass? C = the cost in \$ x = the number of days commuting

10 5. A soccer league offers two options for membership plans. Option A: an initial fee of \$40 and then you pay \$5 for each game that you play. Option B: you have no initial fee but must pay \$10 for each game that you play. After how many games will the total cost of the two options be the same?

11 4.3 Linear Systems in Three Variables In addition to systems of two equations, it is sometimes necessary to solve a system of 3, 4 or more equations in 3, 4 or more variables. In this lesson we will learn to solve such systems algebraically. Later in the chapter we will use a matrix and our graphing calculator to solve such systems. Back Substitution This example has a reasonably straightforward set up allowing us to use simple back substitution to solve. x 2y + 2z = 9 y + 2z = 5 z = 3 Method of Elimination This example requires that we eliminate x by combining Equations 1 and 2, and also eliminate x by combining Equations 2 and 3. We can now use the Elimination method to solve the resulting equations for y and z, and then back substitute to solve for x. Example 1: x 2y + 2z = 9 Equation 1 x + 3y = 4 Equation 2 2x 5y + z = 10 Equation 3

12 Depending on the set up of the system, you may wish to eliminate y or z from the original pairs of equations. Example 2: 4x + y 3z = 11 2x 3y + 2z = 9 x + y + z = 3 How many solutions are possible?? The graph of a system of 3 linear equations in 3 variables consists of 3 planes. The planes may intersect in one point, in one line, in one plane or not at all.

13 An Inconsistent System: x 3y + z = 1 2x y 2z = 2 x + 2y 3z = 1 A System with Infinitely Many Solutions: x + y 3z = 1 y z = 0 x + 2y = 1

14 Let s look at these applications from your textbook.

15 4.4 Matrices and Linear Systems and 4.5 Determinants and Linear Systems (Day 1) Matrix Operations Always read a matrix ROW by COLUMN # Rows: Dimension: # Columns: Numbers in the matrix are called entries. What is the entry in the 2 nd row and 3 rd column for the matrix above? Different Types of Matrices Name Example Dimensions Row Matrix x 4 Column Matrix x 1 Square Matrix x 3 What are the dimensions of each matrix below?

16 Matrix Addition and Subtraction: Matrices must have the SAME dimensions. Add or subtract the corresponding entries. Example 1: Dimension of each matrix: Dimension of the answer matrix: Example 2: = Dimension of each matrix: Dimension of the answer matrix: Scalar Multiplication: Multiply the constant OUTSIDE the matrix to EACH entry inside the matrix. Example 3: = Dimension of the answer matrix: Scalar Multiplication combined with Addition or Subtraction: Example 4: = Dimension of each matrix: Dimension of the answer matrix: Solve the following matrix for x and y Corresponding entries are equal Example 5: 2 3x y =

17 4.4, 4.5 Homework Day 1: Matrix operations Perform the indicated operation if possible. If not possible, state the reason = = = = Solve the matrix equation for x and y x 10 = y y = 3 7 x y x 2 = y x 3 =

18 (Day 2) Matrix Multiplication: The number of columns in the first matrix must match the number of rows in the second matrix. If [A] has dimensions m x n If [B] has dimensions n x p The product of [A]x[B] will have dimensions m x p A: 2 X 3 B: 3 X 4 A: 3 X 2 B: 3 X 4 Dimension of [A]x[B]: Dimension of [A]x[B]: Example 6: Find AB A = B = Dim of [A]: Dim of [B]: Product Dim: Example 7: Find BA A = B = Dim of [A]: Dim of [B]: Product Dim: Example 8: Find AB + BC A = , B = , and C =

19 Use your calculator to add, subtract, multiply with matrices. To enter a Matrix in your calculator: 2 nd MATRIX EDIT ENTER (enter the dimensions of the matrix and the entries) To call up a Matrix in your calculator from the home screen: 2 nd MATRIX (highlight the matrix) ENTER A = , B = , and C = ) B (A + C) 2.) BA + BC Application of Matrices: A health club offers three different membership plans. With Plan X, you can use all club facilities: the pool, fitness center, and racket club. With Plan Y, you can use the pool and fitness center. With Plan Z, you can only use the racket club facilities. The matrices below show the annual cost for a Single and a Family membership for the years 2012 through [A] [B] [C] single family single family single family plan X plan Y plan Z plan X plan Y plan Z plan X plan Y plan Z ) Determine a matrix that gives the price increase from 2012 to 2014 for each of the plans. 2) Determine a matrix that gives the total cost for all three years for each of the plans. 3) The health club offered a 3-year membership based on the 2012 rates. How much money does the 3-year membership save for each plan compared to paying the regular membership rate for each of the 3 years?

20 Homework 4.4, 4.5 Day 2: Multiplying matrices For the matrices with the given dimensions, what are the dimensions of the product? If the product is undefined, explain why. 1. A: 2 X 5 B: 5 X 3 2. A: 6 X 2 B: 3 X 1 Dimension of [A]x[B]: Dimension of [A]x[B]: 3. A: 3 X 1 B: 1 X 2 4. A: 1 X 6 B: 6 X 1 Dimension of [A]x[B]: Dimension of [A]x[B]: Write the product. If it is not defined, state the reason = = = = Given matrices A, B and C, determine the products. If the product is not defined, state the reason B = 5 A = C = [A][B] = 10. [A][C] = 11. [C][B] = 12. [B][C] = 13. [C][A] = 14. [B][A] =

21 (Day 3) Use a matrix and a graphing calculator to solve a linear system 2 nd MATRIX EDIT ENTER (edit matrix) 2 nd MATRIX MATH B rref( 2 nd MATRIX (select the matrix that you edited) 1) 2x y + 4z = 48 x + 2y + 2z = 6 x 3y + 4z = 54 Use the matrix: Solution matrix: x y z 2) x + y 2z = 9 2x + y + z = 0 x 2y + 6z = 21

22 Homework 4.4, 4.5 Day 3: Use a matrix to solve a system Solve the system of equations using a matrix. 1. 9x + 8y = -6 -x y = 1 2. x 3y = -2 5x + 3y = x y 4z = 3 -x + 3y z = -1 x y + 5z = x + 10y z = -3 11x + 28y 4z = 1-6x 15y + 2z = x 3y + 5z = -1 3x + 2y + 4z = 11 2x y + 3z = 4

23 (Day 4) Determinants Determinant of a 2 x 2 matrix: a b a b det = c d = ad bc c d Determinant of a 3 x 3 matrix: a b c det d e f = (aei + bfg + cdh) ( ceg + afh + bdi) g h i Evaluate a determinant in your calculator: 1) Enter the determinant as a matrix: 2 nd MATRIX EDIT ENTER 2 nd QUIT (enter the dimensions of the matrix and the entries) 2) Evaluate the determinant: 2 nd MATRIX MATH 1:det( 2 nd MATRIX Select the matrix that you edited. ENTER Check the value of the determinants above by using your calculator.

24 Homework 4.4, 4.5 Day 4: Determinants and systems Evaluate the determinant = = Use Matrices to solve the system of equations x + y z = 3 2x 3y + 4z = 23 3x + y 2z = 15 3x + 3y + 4z = 1 3x + 5y + 9z = 2 5x + 9y +17z = 4 5x + 3y 2z = 4 2x + 2y + 2z = 0 3x + 2y +1z = 1 2x 4y + 5z = 33 4x y = 5 2x + 2y 3z = 19 Applications: 1. Claire and Dale shopped at the same store. Claire bought 5 kg of apples and 2 kg of bananas and paid altogether \$22. Dale bought 4 kg of apples and 6 kg of bananas and paid altogether \$33. Use matrices to find the cost of 1 kg of bananas.

25 2. Ann and Billy both entered a quiz. The quiz had twenty questions and points were allocated as follows: P points were added for each correctly answered question Q points were deducted for each incorrect (or unanswered) question Ann got 15 questions correct and scored 65 points. Billy got 11 questions correct and scored 37 points. Use matrices to find the value of Q. 3. A community relief fund receives a large donation of \$2800. The foundation agrees to spend the money on \$20 school bags, \$25 sweaters, and \$5 notebooks. They want to buy 200 items and send them to schools in earthquake-hit areas. They must order as many notebooks as school bags and sweaters combined. How many of each item should they order? 4. An ultimate Frisbee team has to order jerseys, shorts, and hats. They have a budget of \$1350 to spend on \$50 jerseys, \$20 shorts, and \$15 hats. They want to buy 40 items in preparation for the oncoming season and must order as many jerseys as shorts and hats combined. How many of each item should they order?

26 4.6 Systems of Linear Inequalities Graph the following systems of inequalities and label the vertex/vertices: 10 1) y 3x 1 y < x ) x 0 y 0 x y

27 3.) x < y x + 3y < 9 x ) x + 2y 10 2x + y 8 2x 5y <

28 Write the system of inequalities that correspond with the shaded region.

29 4.6 HOMEWORK Graph the system of linear inequalities. 1) y > 2 y 1 2) y > 5x x 5y ) x y > 7 2x + y < 8 4) y < 4 x > 3 y > x

30 5) 2x 3y > 6 5x 3y < 3 x + 3y > 3 6) y < 5 y > 6 2x + y 1 y x Challenge. Write a system of linear inequalities for the region.

31 Review Worksheet for Chapter 4 Test Complete the following problems from the e-book: p (9, 11, 15, 17, 23, 27, 31, 33, 37, 39, 41, 43, 71) Complete the following problems with matrices.

32

### 4.1 Matrix Operations

MATRICES 4.1 Matrix Operations Always read a matrix ROW by COLUMN 6 2 1 2 0 5 # Rows: Dimension: # Columns: Numbers in the matrix are called entries. What is the entry in the 2 nd row and 3 rd column for

### 3.1 NOTES Solving Systems of Linear Equations Graphically

3.1 NOTES Solving Systems of Linear Equations Graphically A system of two linear equations in two variables x and y consist of two equations of the following form: Ax + By = C Equation 1 Dx + Ey = F Equation

### ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6

ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6 FALL 2014 0 1 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations System of Equations: a set of two or more equations with the same variables, graphed in the

### 28 (Late Start) 7.2a Substitution. 7.1b Graphing with technology Feb 2. 4 (Late Start) Applications/ Choosing a method

Unit 7: Systems of Linear Equations NAME: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and

### Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by David Rennie. Adapted from the Georgia Department of Education Frameworks

Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by David Rennie Adapted from the Georgia Department of Education Frameworks Georgia Common Core GPS Coordinate Algebra Supplement: Unit 2 by

### To determine the slope or rate of change of a linear function, use m =, positive slopes, rises from left to right, negative

Common Core Regents Review Linear Functions The standard form for a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. To determine the slope or rate of change

### ALGEBRA 1. Unit 3 Chapter 6. This book belongs to: Teacher:

ALGEBRA 1 Teacher: Unit 3 Chapter 6 This book belongs to: UPDATED FALL 2016 1 2 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations Day 1 Warm-Up 1. Graph y = 3x 1 on a coordinate plane. 2. Check

Foundations of Math Chapter 3 Packet Name: Table of Contents Notes #43 Solving Systems by Graphing Pg. 1-4 Notes #44 Solving Systems by Substitution Pg. 5-6 Notes #45 Solving by Graphing & Substitution

### SOLVING LINEAR INEQUALITIES

Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > 7. 1. List five numbers that make the inequality

### Introduction to Systems of Equations

Systems of Equations 1 Introduction to Systems of Equations Remember, we are finding a point of intersection x 2y 5 2x y 4 1. A golfer scored only 4 s and 5 s in a round of 18 holes. His score was 80.

### Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and

### UNIT 5 INEQUALITIES CCM6+/7+ Name: Math Teacher:

UNIT 5 INEQUALITIES 2015-2016 CCM6+/7+ Name: Math Teacher: Topic(s) Page(s) Unit 5 Vocabulary 2 Writing and Graphing Inequalities 3 8 Solving One-Step Inequalities 9 15 Solving Multi-Step Inequalities

### Section 2.2 Objectives

Section 2.2 Objectives Solve multi-step equations using algebra properties of equality. Solve equations that have no solution and equations that have infinitely many solutions. Solve equations with rational

### Name: Systems 2.1. Ready Topic: Determine if given value is a solution and solve systems of equations

Name: Systems 2.1 Ready, Set, Go! Ready Topic: Determine if given value is a solution and solve systems of equations TE-16 1. Graph both equations on the same axes. Then determine which ordered pair is

### 2-4. Warm Up Lesson Presentation Lesson Quiz

Warm Up Lesson Presentation Lesson Quiz Holt Algebra McDougal 1 Algebra 1 Warm Up Solve each equation. 1. 2x 5 = 17 6 2. 14 Solve each inequality and graph the solutions. 3. 5 < t + 9 t > 4 4. a 8 Objective

### ALGEBRA UNIT 5 LINEAR SYSTEMS SOLVING SYSTEMS: GRAPHICALLY (Day 1)

ALGEBRA UNIT 5 LINEAR SYSTEMS SOLVING SYSTEMS: GRAPHICALLY (Day 1) System: Solution to Systems: Number Solutions Exactly one Infinite No solution Terminology Consistent and Consistent and Inconsistent

### 3-1 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

### Name Algebra 1 Midterm Review Period. = 10 4x e) x ) Solve for y: a) 6x 3y = 12 b) 4y 8x = 16

Name Algebra 1 Date Midterm Review Period 1) Solve each equation: a) x 2x + 2 = 3 b) 5 5 + 9 = 13 c) 64 = 9x +1 d) x 7 2 = 10 4x e) x + 2 3 = 3x 2) Solve for y: a) 6x 3y = 12 b) 4y 8x = 16 3) Solve and

### Coordinate Algebra A Final Exam Review

Class: Date: Coordinate Algebra A Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. Do NOT write on the test. You may use your calculator.

### Algebra 1R REVIEW (midterm)

Algebra 1R Algebra 1R REVIEW (midterm) Short Answer 1. Find the x- and y-intercepts. 2. Tara creates a budget for her weekly expenses. The graph shows how much money is in the account at different times.

### Final Exam Study Guide

Algebra 2 Alei - Desert Academy 2011-12 Name: Date: Block: Final Exam Study Guide 1. Which of the properties of real numbers is illustrated below? a + b = b + a 2. Convert 6 yards to inches. 3. How long

### Unit 5 Test Review Systems of Linear Equations Name Class Date

Unit 5 Test Review Systems of Linear Equations Name Class Date Find the mistake - The following problems have been solved HOWEVER there could be a mistake. Each question is worth 3 points: 1pt the mistake,

### Name Class Date. You can use the properties of equality to solve equations. Subtraction is the inverse of addition.

2-1 Reteaching Solving One-Step Equations You can use the properties of equality to solve equations. Subtraction is the inverse of addition. What is the solution of + 5 =? In the equation, + 5 =, 5 is

### Math 1 Variable Manipulation Part 4 Word Problems

Math 1 Variable Manipulation Part 4 Word Problems 1 TRANSLATING FROM ENGLISH INTO ALGEBRA (PLUG IN) The next part of variable manipulation problems is to figure out the problem from real life situations.

### 8 th Grade Domain 2: Algebra and Functions (40%) Sara

8 th Grade Domain 2: Algebra and Functions (40%) 1. Tara creates a budget for her weekly expenses. The graph shows how much money is in the account at different times. Find the slope of the line and tell

### Unit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles

Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and

### Intensive Math-Algebra I Mini-Lesson MA.912.A.3.1

Intensive Math-Algebra I Mini-Lesson MA.912.A.3.1 Summer 2013 Solving Linear Equations Student Packet Day 3 Name: Date: Benchmark MA.912.A.3.1 Solve linear equations in one variable that include simplifying

### Name: Class: Date: ID: A

Name: Class: Date: 8th Grade Advanced Topic III, Linear Equations and Systems of Linear Equations, MA.8.A.1.1, MA.8.1.1.2, MA.8.A.1.3, *MA.8.A.1.4, MA.8.A.1.5, MA.8.A.1.6 Multiple Choice Identify the choice

### September 23, Chp 3.notebook. 3Linear Systems. and Matrices. 3.1 Solve Linear Systems by Graphing

3Linear Systems and Matrices 3.1 Solve Linear Systems by Graphing 1 Find the solution of the systems by looking at the graphs 2 Decide whether the ordered pair is a solution of the system of linear equations:

### Algebra I Practice Exam

Algebra I This practice assessment represents selected TEKS student expectations for each reporting category. These questions do not represent all the student expectations eligible for assessment. Copyright

### 2.2 Creating and Solving Equations

Name Class Date 2.2 Creating and Solving Equations Essential Question: How do you use an equation to model and solve a real-world problem? Explore Creating Equations from Verbal Descriptions You can use

### 4-A5: Mid-Chapter 4 Review

-A: Mid-Chapter Review Alg H Write the equations for the horizontal and vertical lines that pass through the given point.. (, 0) Horiz. Vert.. (0, 8) Horiz. Vert. Use the slope formula to find the slope

### 2-1 Writing Equations

Translate each sentence into an equation. 1. Three times r less than 15 equals 6. Rewrite the verbal sentence so it is easier to translate. Three times r less than 15 equals 6 is the same as 15 minus 3

### Unit 6 Systems of Equations

1 Unit 6 Systems of Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations Specific Outcomes: 6.1 Solve problems that involve systems of linear equations in

### Summer Prep Work for Students Entering Geometry

Summer Prep Work for Students Entering Geometry Operations, Expressions, and Equations 4 1. Evaluate when a =, b = 0.5, c =, d = (cd) + ab. The expression x(x + ) is the same as: a.) x + b.) x + c.) x

### Algebra I Final Study Guide

2011-2012 Algebra I Final Study Guide Short Answer Source: www.cityoforlando.net/public_works/stormwater/rain/rainfall.htm 1. For which one month period was the rate of change in rainfall amounts in Orlando

### Unit Test Linear equations and Inequalities

Unit Test Linear equations and Inequalities Name: Date: Directions: Select the best answer for the following questions. (2 points each) 7L 1. The steps for solving are: 1) Read the problem and label variables,

### Solve Linear Systems Algebraically

TEKS 3.2 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Linear Systems Algebraically Before You solved linear systems graphically. Now You will solve linear systems algebraically. Why? So you can model guitar sales,

### UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES. Solving Equations and Inequalities in One Variable

UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES This unit investigates linear equations and inequalities. Students create linear equations and inequalities and use them to solve problems. They

### Systems 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

Algebra QBA 1 Review Short Answer 1. Juan scored 26 points in the first half of the basketball game, and he scored n points in the second half of the game. Write an expression to determine the number of

### 2.2 Creating and Solving Equations

Name Class Date 2.2 Creating and Solving Equations Essential Question: How do you use an equation to model and solve a real-world problem? Explore Creating Equations from Verbal Descriptions You can use

### Writing and Solving Equations

Writing and Solving Equations Melody s Music Solution Lesson 6-1 Modeling and Writing Two-Step Equations ACTIVITY 6 Learning Targets: Use variables to represent quantities in real-world problems. Model

### Chapter 6: Systems of Linear Equations and Inequalities

Lesson 6-1: Graphing Sstems of Equations Date: Eample 1: Use the graph to determine whether each sstem is consistent or inconsistent and if it is independent or dependent. a. = 1 and = + 1 b. = 1 and =

### and 5-4 Solving Compound Inequalities Solve each compound inequality. Then graph the solution set. p 8 and p

Solve each compound inequality. Then graph the solution set. p 8 and p and The solution set is {p p To graph the solution set, graph p and graph p. Then find the intersection. r + 6 < 8 or r 3 > 10 or

### 9.1 - 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

### Algebra 1 Unit 3 Practice

Lesson 1-1 Use the table for Items 1 and. Canoe Rental Days Cost (\$) 1 5 3 78 5 1 7 13 1. Use function notation to write a linear function that gives the cost C in dollars of renting a canoe for t days.

### Pre-Algebra Semester 1 Practice Exam A

. Evaluate xy when x 0 and y 6. 6 80. Which expression is equivalent to x x x xxx x x xxx x x?. In math class, we follow the order of operations when evaluating expressions. Which is the second operation

### Pre-Algebra 8 Semester 1 Practice Exam

. Evaluate xy when x = 0 and y = 6. 6 80. Which expression is equivalent to x + x x + x+ x+ x+ x x x x x x x?. In math class, we follow the order of operations when evaluating expressions. Which is the

### Topic 1. Solving Equations and Inequalities 1. Solve the following equation

Topic 1. Solving Equations and Inequalities 1. Solve the following equation Algebraically 2( x 3) = 12 Graphically 2( x 3) = 12 2. Solve the following equations algebraically a. 5w 15 2w = 2(w 5) b. 1

### Solving and Graphing Linear Inequalities 66.1 Solve Inequalities Using Addition and Subtraction

Solving and Graphing Linear Inequalities 66.1 Solve Inequalities Using Addition and Subtraction 6.2 Solve Inequalities Using Multiplication and Division 6.3 Solve Multi-Step Inequalities 6.4 Solve Compound

### Math 135 Intermediate Algebra. Homework 3 Solutions

Math Intermediate Algebra Homework Solutions October 6, 007.: Problems,, 7-. On the coordinate plane, plot the following coordinates.. Next to each point, write its coordinates Clock-wise from upper left:

### Math 7 Homework # 46 M3 L1

Name Date Math 7 Homework # 46 M3 L1 Lesson Summary Terms that contain exactly the same variable symbol can be combined by addition or subtraction because the variable represents the same number. Any order,

### 8 th Grade Algebra 5 Day Lesson Plan. *Computer* *Pan Scale* *Algebra Tiles* *Equation Mat* *TI-83 Plus/ TI-73* Karen Kmiotek

8 th Grade Algebra 5 Day Lesson Plan *Computer* *Pan Scale* *Algebra Tiles* *Equation Mat* *TI-83 Plus/ TI-73* Karen Kmiotek Objectives Students will be able to solve equations by using algebra and the

### and 5-4 Solving Compound Inequalities Solve each compound inequality. Then graph the solution set p 8 and p 14 2 SOLUTION:

Solve each compound inequality. Then graph the solution set. 1. 4 p 8 and p 14 2 and The solution set is {p 12 p 16}. To graph the solution set, graph 12 p and graph p 16. Then find the intersection. 2.

### Name. Check with teacher. equation: a. Can you find. a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) a. (-2, 6) b. (-1, 1) c. (1, 3) d. (0, 0) Explain why

7.1 Solving Systems of Equations: Graphing Name Part I - Warm Up with ONE EQUATION: a. Which of the following is a solution to the equation: y 3x 1? a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) Partt II

### Math 8 Unit 4: A Study of Linear Equations

Math 8 Unit 4: A Study of Linear Equations By the end of this unit, students should be able to: 1) Show that the slope of a line can be calculated as rise/run. Explain why the slope is the same between

### UNIT 2 SOLVING EQUATIONS

UNIT 2 SOLVING EQUATIONS NAME: GRADE: TEACHER: Ms. Schmidt _ Solving One and Two Step Equations The goal of solving equations is to. We do so by using. *Remember, whatever you to do one side of an equation.

### REVIEW: HSPA Skills 2 Final Exam June a) y = x + 4 b) y = 2x + 5 c) y = 3x +2 d) y = 2x + 3

Part I- Multiple Choice: 2 points each: Select the best possible answer. 1) The nutrition label of cookies states that there are 20 servings in a box and that one serving contains 1.5 grams of fat. Kyle

### Evaluate algebraic expressions and use exponents. Translate verbal phrases into expressions.

Algebra 1 Notes Section 1.1: Evaluate Expressions Section 1.3: Write Expressions Name: Hour: Objectives: Section 1.1: (The "NOW" green box) Section 1.3: Evaluate algebraic expressions and use exponents.

### Pre-Test. 1. Determine the solution to each system of equations. a. 3x 2 y 5 5 2x 1 7y b. 22x 5 210y x 1 8y 5 5

Pre-Test Name Date 1. Determine the solution to each system of equations. a. 3x 2 y 5 5 2x 1 7y 5 212 b. 22x 5 210y 2 2 2x 1 8y 5 5 2. Determine the number of solutions for each system of equations. 4y

### ALGEBRA I SEMESTER EXAMS PRACTICE MATERIALS SEMESTER Use the diagram below. 9.3 cm. A = (9.3 cm) (6.2 cm) = cm 2. 6.

1. Use the diagram below. 9.3 cm A = (9.3 cm) (6.2 cm) = 57.66 cm 2 6.2 cm A rectangle s sides are measured to be 6.2 cm and 9.3 cm. What is the rectangle s area rounded to the correct number of significant

### Algebra II Notes Unit Four: Matrices and Determinants

Syllabus Objectives: 4. The student will organize data using matrices. 4.2 The student will simplify matrix expressions using the properties of matrices. Matrix: a rectangular arrangement of numbers (called

### Foundations for Algebra. Introduction to Algebra I

Foundations for Algebra Introduction to Algebra I Variables and Expressions Objective: To write algebraic expressions. Objectives 1. I can write an algebraic expression for addition, subtraction, multiplication,

### 4. The table shows the number of toll booths driven through compared to the cost of using a Toll Tag.

ALGEBRA 1 Fall 2016 Semester Exam Review Name 1. According to the data shown below, which would be the best prediction of the average cost of a -bedroom house in Georgetown in the year 2018? Year Average

### This is a review packet for the entire fall semester of Algebra I at Harrison.

HARRISON HIGH SCHOOL ALGEBRA I Fall Semester Review Packet This is a review packet for the entire fall semester of Algebra I at Harrison. You are receiving it now so that: you will have plenty of time

### Pre-Algebra Semester 1 Practice Exam A

. Evaluate y when = 0 and y = 6. 5 6 80. Which epression is equivalent to 5 5 + + + + + 5?. In math class, we follow the order of operations when evaluating epressions. Which is the second operation a

### and 5-4 Solving Compound Inequalities Solve each compound inequality. Then graph the solution set p 8 and p 14 2 SOLUTION:

Solve each compound inequality. Then graph the solution set. 1. 4 p 8 and p 14 2 and The solution set is {p 12 p 16}. To graph the solution set, graph 12 p and graph p 16. Then find the intersection. {p

### Pre-Test Unit 5: Solving Equations KEY

Pre-Test Unit 5: Solving Equations KEY No calculator necessary. Please do not use a calculator. Solve the following equations for the given variable. There may be a single solution, infinite solutions,

### Systems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1

Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you

### Unit 1 Study Guide [MGSE9-12.N.Q.1-3, MGSE9-12.A.CED.1]

Name: Class: Date: Unit 1 Study Guide [MGSE9-12.N.Q.1-3, MGSE9-12.A.CED.1] Matching a. algebraic expression f. variable b. numerical expression g. constant c. like terms h. solution of an equation d. absolute

### Algebra I Notes Linear Inequalities in One Variable and Unit 3 Absolute Value Equations and Inequalities

PREREQUISITE SKILLS: students must have a clear understanding of signed numbers and their operations students must understand meaning of operations and how they relate to one another students must be able

### Name Class Date. Essential question: How do you interpret, evaluate and write algebraic expressions that model real-world situations?

Name Class Date 1-1 1 Variables and Expressions Going Deeper Essential question: How do you interpret, evaluate and write algebraic expressions that model real-world situations? A-SSE.1.1a ENGAGE Interpreting

### MATH 410 Notes Simplifying Algebraic Expressions

MATH 410 Notes 2016 1.9 - Simplifying Algebraic Expressions Commutative Property: a + b = b + a and a b = b a Associative Property: a + (b + c) = (a + b) + c and a (b c) = (a b) c Distributive Property:

### Why? Speed Skating Tracks offi cial track short track

Applying Systems of Linear Equations Then You solved systems of equations by using substitution and elimination. (Lessons 6-2, 6-3, and 6-4) Now 1Determine the best method for solving systems of 2Apply

### Due for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

MTH 209 Week 1 Due for this week Homework 1 (on MyMathLab via the Materials Link) Monday night at 6pm. Read Chapter 6.1-6.4, 7.1-7.4,10.1-10.3,10.6 Do the MyMathLab Self-Check for week 1. Learning team

### What You ll Learn Solve two-step equations. Solve real-world problems involving two-step equations.

Lesson 8- Solving Two-Step Equations Essential Question How are equations and inequalities used to describe and solve multi-step problems? Common Core State Standards Content Standards 7.EE.4, 7.EE.4a,

### 1. The sum of four consecutive even numbers is 52. What is the largest of these numbers?

1. The sum of four consecutive even numbers is 52. What is the largest of these numbers? 26 22 C 16 10 2. In a high school basketball game, Sarah scored 10 points in the first half of the game. In the

### The Top 11 Keystones of Algebra 1

The Top 11 Keystones of Algebra 1 The Top Eleven Keystones of Algebra 1 You should be able to 1) Simplify a radical expression. 2) Solve an equation. 3) Solve and graph an inequality on a number line.

### 7 = 8 (Type a simplified fraction.)

Student: Date: Assignment: Exponential and Radical Equations 1. Perform the indicated computation. Write the answer in scientific notation. 3. 10 6 10. 3. 4. 3. 10 6 10 = (Use the multiplication symbol

### Oregon Focus on Linear Equations Lesson 1 Answers

Lesson 1 Answers 1. a. Nathan; multiplication b. Subtraction 2. 30 3. 28 4. 40 5. 17 6. 29 7. 21 8. 7 9. 4 10. 33 11. 8 12. 1 13. 5 14. 19 15. 12 16. 15 17. a. 130 5 + 40 8 b. \$970 18. a. (11 + 8 + 13)

### H.Alg 2 Notes: Day1: Solving Systems of Equations (Sections ) Activity: Text p. 116

H.Alg 2 Notes: Day: Solving Systems of Equations (Sections 3.-3.3) Activity: Text p. 6 Systems of Equations: A set of or more equations using the same. The graph of each equation is a line. Solutions of

### Systems of Equations Unit Five ONE NONE INFINITE

Systems of Equations Unit Five ONE NONE INFINITE Standards: 8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables

### = \$ m. Telephone Company B charges \$11.50 per month plus five cents per minute. Writing that mathematically, we have c B. = \$

Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you

### Mathematics 10 Exercise and Homework Book MHR 307. Method 2: Isolate the variable y in the equation 5x + y = 2.

Chapter 9 Solving Systems of Linear Equations Algebraically 9.1 Solving Systems of Linear Equations by Substitution 1. a) x = 13.5 and y = 11.5 b) x = 11 and y = c) x = 1 and y = 0 d) x = and y = 3 e)

### Midterm Review Packet

Algebra 1 CHAPTER 1 Midterm Review Packet Name Date Match the following with the appropriate property. 1. x y y x A. Distributive Property. 6 u v 6u 1v B. Commutative Property of Multiplication. m n 5

### Use the Equal Values method to solve each system.

4.2 Equal Values Method Name: Recall Writing and solving equations involving rate of change and initial value a. In a jumping frog race, your frog received a 6 in. head start and jumps 3 in. every 2 seconds.

### Evaluate and Simplify Algebraic Expressions

TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So

### Unit 4 Linear Functions

Algebra I: Unit 4 Revised 10/16 Unit 4 Linear Functions Name: 1 P a g e CONTENTS 3.4 Direct Variation 3.5 Arithmetic Sequences 2.3 Consecutive Numbers Unit 4 Assessment #1 (3.4, 3.5, 2.3) 4.1 Graphing

### ALGEBRA 1 SEMESTER 1 INSTRUCTIONAL MATERIALS Courses: Algebra 1 S1 (#2201) and Foundations in Algebra 1 S1 (#7769)

Multiple Choice: Identify the choice that best completes the statement or answers the question. 1. Ramal goes to the grocery store and buys pounds of apples and pounds of bananas. Apples cost dollars per

### Grade 8. Functions 8.F.1-3. Student Pages

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Functions 8.F.1-3 Student Pages 2012 2012 COMMON CORE CORE STATE STATE STANDARDS ALIGNED ALIGNED MODULES Grade 8 - Lesson 1 Introductory Task

### Math 803. Unit 1: Solving Equations in One Variable (8.EE.7) Part 2

Math 803 Unit 1: Solving Equations in One Variable (8.EE.7) Part 2 1.4 Variables on both sides (2.4 text) 1.5 Solve multi-step equations (2.5 text) Name: Period: Teacher s Name: 1 Lesson 1.4 Equations

### EQUATIONS WITH MULTIPLE VARIABLES 5.1.1

EQUATIONS WITH MULTIPLE VARIABLES 5.1.1 Solving equations with more than one variable uses the same process as solving an equation with one variable. The only difference is that instead of the answer always

### Expressions & Equations Chapter Questions. 6. What are two different ways to solve equations with fractional distributive property?

Expressions & Equations Chapter Questions 1. Explain how distribution can simplify a problem. 2. What are like terms? 3. How do you combine like terms? 4. What are inverse operations? Name them. 5. How

### Quarter 2. Review. Calculator Inactive: NO calculator Look on the back of the book to make sure you complete the gridded response correctly.

7 th Grade Quarter 2 Review Calculator Inactive: NO calculator Look on the back of the book to make sure you complete the gridded response correctly. Name Teacher Adapted from SchoolNet and CMapp 1 1.

### Elimination Exploring Linear Systems QUIZ ( ) Solving Problems with Systems of Equations. Distance/Velocity/Time Problems WS 1.

UNIT 1 SYSTEMS OF LINEAR EQUATIONS Lesson TOPIC Homework Sept. 4 1.0 Sept. 5 1.1 1.1 Sept. 6 1.2 1.3 Sept. 7 1.3 1.4 Sept. 10 Sept. 11 Sept. 12 Sept. 13 Sept. 14 Sept. 17 Sept. 18 Sept. 20 1.4 1.6 1.5

### Solve Systems of Linear Equations in Three Variables

TEKS 3.4 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Systems of Linear Equations in Three Variables Before You solved systems of equations in two variables. Now You will solve systems of equations in three variables.

### Algebra 1 Enriched- Midterm Review

Algebra 1 Enriched- Midterm Review Know all vocabulary, pay attention to the highlighted words in the text, and understand the various types of directions in each of the sections of the textbook. Practice