Enhanced Instructional Transition Guide

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1 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Unit 04: Systems of Linear Equations (1 days) Possible Lesson 01 (1 days) POSSIBLE LESSON 01 (1 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students define systems of equations and identify their characteristics. Students solve linear systems of equations using tables, graphs, algebraic methods, and matrices. Students formulate linear systems of equations to represent problem situations, solve the systems by a method of choice, and justify the solutions in terms of the problem situation. TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at A. Foundations for functions.. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: A.A Use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations. Supporting Standard A. Foundations for functions.. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to: A.A A.B A.C Analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems. Readiness Standard Use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities. Readiness Standard Interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts. Readiness Standard page 1 of 17

2 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Performance Indicator(s): High School Mathematics AlgebraII Unit04 PI01 Analyze problem situations that can be represented algebraically by linear systems of equations in two or more unknowns such as the following: In the textbook office, there is a box of new math books, some for Algebra 1, some for Geometry, and some for. A label on the box reads: Contents: books, Weight: 9 lbs. Each Algebra 1 book weighs 4 pounds, each Geometry book weighs pounds, and each book weighs 5 pounds. The number of Geometry books and books combined is one less than the number of Algebra 1 books. How many Algebra 1, Geometry, and books were in the box? Two taxi services use different methods to determine the fare (or charge) customers pay when using their drivers. Terri s charges $ plus $1.50 per mile, and Rick s costs $.5 plus $1.60 per mile. Compute the charges for riding 1, 5, and 10 miles with each taxi service. For what mileage is the cost the same with both companies? Create a graphic organizer for each problem situation that includes formulating a representative system of equations, solving the system of equations by an appropriate method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.A, A.A, A.B, A.C ELPS ELPS.c.1C, ELPS.c.5B page of 17

3 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days High School Mathematics AlgebraII Unit04 PI0 Formulate systems of inequalities in two variables to model problem situations such as the following: The local television station is hosting a telethon for a local children s charity. They have asked students and sponsors from high school clubs and organizations to volunteer to answer the phones. When people call in to pledge money or make a donation, the volunteers must take each person s name, address, phone number, and pledge and enter the information into a computer. However, the TV station does have some guidelines: (1) the station needs at least 0 total volunteers, which includes students as well as their adult sponsors; () there must be at least one adult sponsor for every students who volunteer; and () the group of volunteers must be able to process at least 10 calls per hour. From previous telethons, the station estimates that each high school student can process 9 calls per hour, but adult sponsors usually only handle about 4 calls per hour. To show appreciation for their volunteers, the TV station usually provides pizza. They order pieces for every student and pieces for each adult sponsor. What is the minimum number of pieces of pizza the station could order? Solve the system of inequalities using various methods including graphs, tables, algebraic methods, and technology to interpret and determine the reasonableness of the solution. Display all results using a graphic organizer. Standard(s): A.A, A.A, A.B, A.C ELPS ELPS.c.1C, ELPS.c.5B, ELPS.c.5G Key Understanding(s): Linear systems of equations can be written to represent problem situations and solved using a variety of methods with and without technology. The reasonableness of the solution can be justified in terms of the problem situation. Linear systems of inequalities can be written to represent problem situations and solved using a selected method. The reasonableness of the solution can be justified in terms of the problem situation. Vocabulary of Instruction: augmented matrix boundary coinciding column consistent constraint inconsistent independent inequality infinitely many solutions intersection inverse matrix no solutions ordered pair ordered triple parallel reciprocal reduced row echelon form (rref) page of 17

4 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days dependent dimension of a matrix element feasible region greater than greater than or equal to identity matrix less than less than or equal to linear equation linear programming matrix maximize minimize row slope standard form of a system system of linear equations vertices of a region y-intercept Materials List: cardstock (1 sheet per students) cardstock (multicolored) (1 sheet per 4 students) graphing calculator (1 per student) graphing calculator with display (1 per teacher) plastic zip bag (quart size) (1 per students) plastic zip bag (quart size) (1 per teacher) scissors (1 per teacher) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Restaurant Receipts Cards Restaurant Receipts KEY Restaurant Receipts Systems of Equations KEY page 4 of 17

5 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Systems of Equations Plug into the System KEY Plug into the System Graphing x Systems KEY Graphing x Systems Electrical Problems KEY Electrical Problems Systems by Algebraic Methods KEY Systems by Algebraic Methods Pick a Method Cards Pick a Method KEY Pick a Method Solving x Systems of Linear Equations KEY Solving x Systems of Linear Equations Solving Systems with Technology and Matrices KEY Solving Systems with Technology and Matrices Quadratic Quest KEY Quadratic Quest Movies, Kids, and Calculators KEY page 5 of 17

6 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Movies, Kids, and Calculators Situation Problems Involving Systems KEY Situation Problems Involving Systems Mixing It Up KEY Mixing It Up Systems Situations KEY Systems Situations Graphing Inequalities KEY Graphing Inequalities Linear Programming Basics KEY Linear Programming Basics Linear Programming Problems KEY Linear Programming Problems Evaluation Systems of Equations KEY Evaluation System of Equations PI Evaluation Systems of Inequalities KEY Evaluation System of Inequalities PI GETTING READY FOR INSTRUCTION page 6 of 17

7 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1 Topics: ATTACHMENTS Engage 1 Introduction to systems of linear equations Students use trial and error to solve a system of linear equations in a problem situation. Instructional Procedures: 1. Prior to instruction, create card set: Restaurant Receipts Cards by copying on cardstock, laminating, cutting apart, and placing all cards in a plastic zip bag. One set of cards will be needed for each group of 4 students. To distinguish the card sets by group, they could be created on different colors of cardstock, or symbols could be put on the back of each card to designate the group sets (such as numbers, hearts, diamonds, triangles, circles, squares, and stars).. As students enter the room, randomly distribute a card from card sets: Restaurant Receipts Cards to each student. Instruct students to Find the students who have the same color card, or Find the students who have the same number or symbol drawn on the back of their cards.. Distribute handout: Restaurant Receipts to each student. Facilitate a class discussion of the problem involved, reminding students that it is a trial and error problem. Ask: Card Set: Restaurant Receipts Cards (1 set per 4 students) Teacher Resource: Restaurant Receipts KEY (1 per teacher) Teacher Resource: Restaurant Receipts (1 per teacher) Handout: Restaurant Receipts (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) cardstock (multicolored) (1 sheet per 4 students) scissors (1 per teacher) plastic zip bag (quart size) (1 per teacher) What are we asked to find in this problem? (The cost of each item sold at the diner.) How many items are for sale? (four) page 7 of 17

8 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures What are their possible prices? (either $, $, $4, or $5) Notes for Teacher 4. Instruct students to solve the problem of the restaurant receipts by first independently recording any information from their card on their handout, then sharing and discussing the problem with group members. Allow students time to complete the problem, and monitor to check for student understanding. Display teacher resource: Restaurant Receipts Cards, and facilitate a class discussion of student responses, clarifying any misconceptions. Discuss any problem-solving strategies that surfaced as students completed the problem. Inform students that this type of problem with many unknowns is called a system of equations; however, there are more efficient ways to solve them besides trial and error. Topics: ATTACHMENTS Systems of linear equations Solutions to systems of linear equations Explore/Explain 1 Students review x systems of linear equations from Algebra 1. Students are also introduced to vocabulary and x systems of linear equations. Instructional Procedures: 1. Place students in pairs. Distribute handout: Systems of Equations to each student. Refer students to the top of page 1. Display teacher resource: Systems of Equations. Instruct students to formulate equations for the verbal statements in the table. Allow students time to formulate the representative equations, and compare results with their partner. Teacher Resource: Systems of Equations KEY (1 per teacher) Teacher Resource: Systems of Equations (1 per teacher) Handout: Systems of Equations (1 per student) Teacher Resource: Plug into the System KEY (1 per teacher) Handout: Plug into the System (1 per student) MATERIALS page 8 of 17

9 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher. Using teacher resource: Systems of Equations, facilitate a class discussion of student results, clarifying any misconceptions. Model Sample problems 1, discussing systems of equations and their solutions. Caution students that, in order to be a solution, the set of numbers must make every equation in the system true. If an ordered pair checks in one equation but not the other then it cannot be a solution.. Instruct students to work with their partner to complete Sample problem on handout: Systems of Equations. Allow students time to complete the problem, and monitor to check for student understanding. Using teacher resource: Systems of Equations, facilitate a class discussion of student results, clarifying any misconceptions. 4. Distribute handout: Plug into the System to each student. Instruct students to continue working with their partner to complete the handout. This may be assigned as homework, if necessary. graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE This may be the students first introduction to the term ordered triple. Point out to students that an ordered triple represents a three-dimensional point and lists (in order) values for x, y, and z. In other words, (, 4, -) means that x, y 4, and z -. Compare this to the two-dimensional point represented by an ordered pair (4, 9), where x 4 and y 9. Topics: Systems of linear equations Solving by graphs and tables Explore/Explain Students solve systems of linear equations using graphs and tables. Instructional Procedures: ATTACHMENTS Teacher Resource: Graphing x Systems KEY (1 per teacher) Teacher Resource: Graphing x Systems (1 per teacher) Handout: Graphing x Systems (1 per student) Teacher Resource: Electrical Problems page 9 of 17

10 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher 1. Place students in pairs. Distribute handout: Graphing x Systems to each student. Instruct students to work independently to complete the Getting Started section. Allow students time to answer the questions, and compare answers with their partner. Display teacher resource: Graphing x Systems, and facilitate a class discussion of student responses, clarifying any misconceptions. Ask: What is the y-intercept of the first equation? () At what point could you begin to graph the line? (0, ) What is the slope? (- or, -/1) How does this help me generate the other points on the line? Go down, to the right 1; repeat. What is the y-intercept of the second equation? (-) At what point could you begin to graph the line? (0,-) What is the slope? (1/) How does this help me generate the other points on the line? Go up 1, to the right ; repeat. At what point do the two lines intersect on the graph? (, -1) What point is common in the two tables? (, -1), the same point Can you solve a system of equations from both a table and a graph? Explain. Yes. Sometimes it is easier to look at the graph because the point may not show in the table. Sometimes it is easier to look at the table because fractional values are harder to read on a graph. No matter how the point of intersection is determined, the point should satisfy both equations. KEY (1 per teacher) Handout: Electrical Problems (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE When going from standard form to solving for slope intercept form, there is no need to simplify to lowest terms if you are using a calculator.. Refer students to the Important Information Box on handout: Graphing x Systems. Using teacher resource: Graphing x Systems, facilitate a class discussion of the statement and explanation of why the point of intersection is the solution to the system, clarifying any misconceptions.. Instruct students to work with their partner to complete problems 1 6 on handout: Graphing x Systems TEACHER NOTE The word consistent means that the lines come together. This happens when they intersect or coincide. Otherwise, when lines are parallel, they do not come together and are referenced as page 10 of 17

11 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher without a graphing calculator. Allow students time to complete the problem, and monitor to check for student understanding. Facilitate a class discussion of student results, clarifying any misconceptions. 4. Refer students to the Sample problem on page of handout: Graphing x Systems. Using teacher resource: Graphing x Systems, model the Sample problem using the graphing calculator to determine the solution to a system of equations. Ask: inconsistent. TEACHER NOTE Parallel lines have the same slopes but different y- intercepts. Coinciding lines have the same slope and the same y-intercepts. Why can the equation x + 4y 8 not be typed into a calculator as it is? Answers may vary. It is not in slope intercept form, not solved for y ; etc. How do we write the equation in Y or, slope intercept form? Answers may vary. Solve for y, by adding -x to both sides, then dividing both sides by 4; etc. 5. Instruct students to work with their partner to complete problems 7 10 on handout: Graphing x Systems. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Graphing x Systems, facilitate a class discussion of student responses, clarifying any misconceptions. 6. Refer students to the systems vocabulary table on page 4 of handout: Graphing x Systems. Using teacher resource: Graphing x Systems, facilitate a class discussion of the vocabulary of systems of equations and their solutions, clarifying any misconceptions. Instruct students to work with their partner to complete problems Allow students time to complete the problem, and monitor to check for student understanding. Using teacher resource: Graphing x Systems, facilitate a class discussion of student responses, clarifying any misconceptions. 7. Distribute handout: Electrical Problems to each student. Instruct students to work independently to complete the handout. This may be assigned as homework, if necessary. page 11 of 17

12 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day 4 Topics: Suggested Instructional Procedures ATTACHMENTS Notes for Teacher Systems of x linear equations Solving by algebraic methods Explore/Explain Students compare and contrast methods to solve systems of linear equations using algebraic methods including substitution and elimination. This is a review of methods used in Algebra 1 and Geometry. Teacher Resource: Systems by Algebraic Methods KEY (1 per teacher) Teacher Resource: Systems by Algebraic Methods (1 per teacher) Handout: Systems by Algebraic Methods (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief handout: Electrical Problems by having students share methods used to solve the problem and the results.. Place students in pairs. Distribute handout: Systems by Algebraic Methods to students. Refer students to Part A. Display teacher resource: Systems by Algebraic Methods, and facilitate a class discussion on solving systems of equations by substitution, using problem 1 as a model. Instruct students to work with their partner to complete problems 6. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Systems by Algebraic Methods, facilitate a class discussion of student responses, clarifying any misconceptions. Discuss results on problems 5 6, and summarize special cases using the table under the problems.. Refer students to Part B on handout: Systems by Algebraic Methods. Display teacher resource: Systems by Algebraic Methods, and facilitate a class discussion on solving systems of equations by elimination. Instruct students to work with their partner to complete problems Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Systems by Algebraic Methods, facilitate a class discussion of student responses, clarifying any misconceptions. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE On handout: Systems by Algebraic Methods, some of the problems have three variables. These problems are to challenge students to solve by extending their knowledge of substitution from two equations to three equations. 4. Instruct students to work independently to complete problems 7 10, This may be assigned as page 1 of 17

13 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day homework, if necessary. Suggested Instructional Procedures Notes for Teacher Topics: ATTACHMENTS Systems of x linear equations Solving by graphs and tables Solving by algebraic methods Elaborate 1 Students compare and contrast methods to solve systems of linear equations. Instructional Procedures: Card Set: Pick a Method Cards (1 set per students) Teacher Resource: Pick a Method KEY (1 per teacher) Teacher Resource: Pick a Method (1 per teacher) Handout: Pick a Method (1 per student) 1. Prior to instruction, create card set: Pick a Method Cards by copying on cardstock, laminating, cutting apart, and placing in plastic zip bags. One set of cards will be needed for each pair of students.. Place students in pairs. Distribute card set: Pick a Method Cards to each student. Instruct students to work with their partner to sort cards into stacks according to whether it would be easiest to solve the system using Graphing, Substitution, or Combination/Elimination. Allow students time to sort their card set.. Distribute handout: Pick a Method to each student. Instruct students to fill in the Venn diagram according to their discussion and card set stacks, and fill in the table to summarize why they chose each of the methods. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Pick a Method, facilitate a class discussion of student responses, clarifying any misconceptions. Stress to students that there is no correct method for solving any of these equations, and any system can be solved with any method. However, often a method is preferred depending on how the system is written. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) cardstock (1 sheet per students) scissors (1 per teacher) plastic zip bag (quart size) (1 per students) 4. Instruct students to work independently to solve each of the equations A K on handout: Pick a Method. This may be assigned as homework, if necessary. page 1 of 17

14 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day 5 Topics: Suggested Instructional Procedures ATTACHMENTS Notes for Teacher Systems of x linear equations Solving by algebraic methods Explore/Explain 4 Students solve x systems of linear equations using substitution and elimination. Teacher Resource: Solving x Systems of Linear Equations KEY (1 per teacher) Handout: Solving x Systems of Linear Equations (1 per student) Instructional Procedures: 1. Distribute handout: Solving x Systems of Linear Equations to students. Facilitate a class discussion on solving x systems of linear equations by substitution and combination/elimination, modeling example problems. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher). Instruct students to work with their partner to complete the Practice Problems on handout: Solving x Systems of Linear Equations. This may be assigned as homework, if necessary. 6 7 Topics: Systems of linear equations Solving by matrices Explore/Explain 5 Students solve systems of linear equations using technology and matrices. Instructional Procedures: ATTACHMENTS Teacher Resource: Solving Systems with Technology and Matrices KEY (1 per teacher) Teacher Resource: Solving Systems with Technology and Matrices (1 per teacher) Handout: Solving Systems with Technology and Matrices (1 per student) 1. Place students in pairs. Distribute handout: Solving Systems with Technology and Matrices to each student. page 14 of 17

15 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher Display teacher resource: Solving Systems with Technology and Matrices, and facilitate a class discussion of solving systems using matrices.. Instruct students to work with their partner to complete Practice Problems 1 4. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Solving Systems with Technology and Matrices, facilitate a class discussion of student results, clarifying any misconceptions. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE A matrix is an arrangement of numbers in rows and columns. Point out to students that the plural of matrix is the word matrices. Any number in the matrix is called an element. The dimensions of a matrix are given as rows x columns (or, RxC). TEACHER NOTE When solving a matrix multiplication equation (such as [A][X] [C]), students may want to divide by [A]. However, division is not a defined operation with matrices. Instead to divide you multiply by the inverse. Topics: Systems of x linear equations Solving by matrices Quadratic functions ATTACHMENTS Teacher Resource: Quadratic Quest KEY (1 per teacher) Teacher Resource: Quadratic Quest (1 page 15 of 17

16 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Elaborate Suggested Instructional Procedures Students use x systems of linear equations to determine the representative equation of a quadratic function given points. Notes for Teacher per teacher) Handout: Quadratic Quest (1 per student) Instructional Procedures: 1. Place students in pairs. Distribute handout: Quadratic Quest to each student. Refer students to page 1 of the handout. Display teacher resource: Quadratic Quest, and facilitate a class discussion of applying x systems to determine coefficients and constants for non-linear functions, modeling the Example problem. Instruct students to work with their partner to complete problem 1. Allow students time to complete the problem, and monitor to check for student understanding. Using teacher resource: Quadratic Quest, facilitate a class discussion of student results, clarifying any misconceptions.. Instruct students to continue working with their partner to complete problems on handout: Quadratic Quest. This may be assigned as homework, if necessary. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) 8 9 Topics: Systems of linear equations Applications of systems of linear equations Explore/Explain 6 Students formulate, solve, and justify solutions of systems of linear equations in terms of problem situations. Instructional Procedures: ATTACHMENTS Teacher Resource: Movies, Kids, and Calculators KEY (1 per teacher) Teacher Resource: Movies, Kids, and Calculators (1 per teacher) Handout: Movies, Kids, and Calculators (1 per student) Teacher Resource: Situation Problems page 16 of 17

17 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher 1. Distribute handout: Movies, Kids, and Calculators to each student. Instruct students to work with their partner to complete the handout. Allow students time to complete the problems, and monitor to check for student understanding. Display teacher resource: Movies, Kids, and Calculators, and facilitate a class discussion of student responses, clarifying any misconceptions.. Distribute handout: Situation Problems Involving Systems to each student. Instruct students to work independently to complete the handout. This may be assigned as homework, if necessary. Involving Systems KEY (1 per teacher) Handout: Situation Problems Involving Systems (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) Topics: ATTACHMENTS Systems of linear equations Applications of systems of linear equations Elaborate Students continue to formulate, solve, and justify solutions of systems of linear equations in terms of problem situations. Instructional Procedures: 1. Place students in pairs. Distribute handout: Mixing It Up. Instruct students to work independently to complete the Warm Up problems. Allow students time to complete the problems. Display teacher resource: Mixing It Up, and facilitate a class discussion of student responses. Teacher Resource: Mixing It Up KEY (1 per teacher) Teacher Resource: Mixing It Up (1 per teacher) Handout: Mixing It Up (1 per student) Teacher Resource: Systems Situations KEY (1 per teacher) Handout: Systems Situations (1 per student) MATERIALS page 17 of 17

18 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher. Instruct students to work with their partner to complete problems 1 8 on handout: Mixing It Up. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Mixing It Up, facilitate a class discussion of student results, clarifying any misconceptions.. Distribute handout: Systems Situations to each student. Instruct students to work independently to complete the problems. This may be assigned as homework, if necessary. graphing calculator (1 per student) graphing calculator with display (1 per teacher) Topics: Systems of linear inequalities Solutions to systems of linear inequalities Explore/Explain 7 Students solve and represent solutions to systems of linear inequalities. Instructional Procedures: 1. Place students in pairs. Distribute handout: Graphing Inequalities. Refer students to page 1 of the handout. Display teacher resource: Graphing Inequalities, and facilitate a class discussion of representations of linear inequalities in two variables, modeling problem 1. Instruct students to work independently to complete problems 5, and compare answers with their partner. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Graphing Inequalities, facilitate a class discussion of student results, clarifying any misconceptions.. Refer students to page of handout: Graphing Inequalities. Using teacher resource: Graphing Inequalities, facilitate a class discussion of representing solutions to systems of linear inequalities in two variables, modeling problems 6 7. Instruct students to work with their partner to complete problems Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Graphing Inequalities, facilitate a class discussion of student results, clarifying any misconceptions. ATTACHMENTS Teacher Resource: Graphing Inequalities KEY (1 per teacher) Teacher Resource: Graphing Inequalities (1 per teacher) Handout: Graphing Inequalities (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE To graph inequalities in standard form, the suggestion is to rewrite in slope-intercept form. page 18 of 17

19 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher. Instruct students to work independently to complete problems on handout: Graphing Inequalities. This may be assigned as homework, if necessary. However, the intercepts method may also be used by obtaining the two intercept points when x 0 and y 0. To determine shading, you can test a point on either side of the boundary. A good choice is (0,0). In this case, since (0,0) makes the original inequality true, shade this side of the boundary line. Topics: ATTACHMENTS Systems of linear inequalities Applications of systems of linear inequalities Linear programming Elaborate 4 Students formulate, solve, and justify solutions for problem situations involving linear programming. Instructional Procedures: Teacher Resource: Linear Programming Basics KEY (1 per teacher) Teacher Resource: Linear Programming Basics (1 per teacher) Handout: Linear Programming Basics (1 per student) Teacher Resource: Linear Programming Problems KEY (1 per teacher) page 19 of 17

20 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher 1. Place students in pairs. Distribute handout: Linear Programming Basics to each student. Refer students to page 1 of the handout. Display teacher resource: Linear Programming Basics, and facilitate a class discussion of linear programming, modeling problems 1. Instruct students to work with their partner to complete problems 4. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Linear Programming Basics, facilitate a class discussion of student results, clarifying any misconceptions.. Distribute handout: Linear Programming Problems to each student. Instruct students to work with their partner to complete the handout. Refer students to page 1 of the handout. Display teacher resource: Linear Programming Problems, and facilitate a class discussion of applying linear programming to problem situations by modeling problem 1. Instruct students to work with their partner to complete problem. Allow students time to complete the problem, and monitor to check for student understanding. Using teacher resource: Linear Programming Problems, facilitate a class discussion of student results, clarifying any misconceptions.. Instruct students to work independently on problems 4 on handout: Linear Programming Problems. This may be assigned as homework, if necessary. Teacher Resource: Linear Programming Problems (1 per teacher) Handout: Linear Programming Problems (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) State Resources TMT Student Lesson 1 Systems of Equations and Linear Programming may be used as alternate activities. TEXTEAMS: Part II: Algebra II I Systems;. Linear Programming,.1 Applications of Linear Programming, Act. 1 (Cotton vs. Corn), St. Act. (Linear Programming Applications) may be used as alternate activities. 1 Evaluate 1 ATTACHMENTS page 0 of 17

21 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher Instructional Procedures: 1. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. Performance Indicator(s): Teacher Resource (optional): Evaluation Systems of Equations KEY (1 per teacher) Handout (optional): Evaluation Systems of Equations PI (1 per student) Teacher Resource (optional): Evaluation Systems of Inequalities KEY (1 per page 1 of 17

22 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher High School Mathematics AlgebraII Unit04 PI01 Analyze problem situations that can be represented algebraically by linear systems of equations in two or more unknowns such as the following: In the textbook office, there is a box of new math books, some for Algebra 1, some for Geometry, and some for. A label on the box reads: Contents: books, Weight: 9 lbs. Each Algebra 1 book weighs 4 pounds, each Geometry book weighs pounds, and each book weighs 5 pounds. The number of Geometry books and books combined is one less than the number of Algebra 1 books. How many Algebra 1, Geometry, and books were in the box? Two taxi services use different methods to determine the fare (or charge) customers pay when using their drivers. Terri s charges $ plus $1.50 per mile, and Rick s costs $.5 plus $1.60 per mile. Compute the charges for riding 1, 5, and 10 miles with each taxi service. For what mileage is the cost the same with both companies? teacher) Handout (optional): Evaluation Systems of Inequalities PI (1 per student) MATERIALS graphing calculator (1 per student) TEACHER NOTE As an optional assessment tool, use a combination of handouts (optional): Evaluation Systems of Equations PI and Evaluation Systems of Inequalities PI. Create a graphic organizer for each problem situation that includes formulating a representative system of equations, solving the system of equations by an appropriate method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.A, A.A, A.B, A.C ELPS ELPS.c.1C, ELPS.c.5B page of 17

23 Enhanced Instructional Transition Guide High School Courses/Algebra II Unit 04: Suggested Duration: 1 days Suggested Day Suggested Instructional Procedures Notes for Teacher High School Mathematics AlgebraII Unit04 PI0 Formulate systems of inequalities in two variables to model problem situations such as the following: The local television station is hosting a telethon for a local children s charity. They have asked students and sponsors from high school clubs and organizations to volunteer to answer the phones. When people call in to pledge money or make a donation, the volunteers must take each person s name, address, phone number, and pledge and enter the information into a computer. However, the TV station does have some guidelines: (1) the station needs at least 0 total volunteers, which includes students as well as their adult sponsors; () there must be at least one adult sponsor for every students who volunteer; and () the group of volunteers must be able to process at least 10 calls per hour. From previous telethons, the station estimates that each high school student can process 9 calls per hour, but adult sponsors usually only handle about 4 calls per hour. To show appreciation for their volunteers, the TV station usually provides pizza. They order pieces for every student and pieces for each adult sponsor. What is the minimum number of pieces of pizza the station could order? Solve the system of inequalities using various methods including graphs, tables, algebraic methods, and technology to interpret and determine the reasonableness of the solution. Display all results using a graphic organizer. Standard(s): A.A, A.A, A.B, A.C ELPS ELPS.c.1C, ELPS.c.5B, ELPS.c.5G 04/6/1 page of 17

24 Restaurant Receipts Cards The following four cards serve as a group that can be used collectively to answer the question. Alice s Diner sells only four items, which have individual prices of $, $, $4, or $5. Alice s Diner sells only four items, which have individual prices of $, $, $4, or $5. CUSTOMER RECEIPT CUSTOMER RECEIPT Qty Item Super Sodas Big Burgers Qty 5 6 Item Super Sodas Big Burgers TOTAL: $18 TOTAL: $9 THANK YOU!! THANK YOU!! After dining there, Group A got this receipt. What is the cost of each individual item? After dining there, Group C got this receipt. What is the cost of each individual item? Alice s Diner sells only four items, which have individual prices of $, $, $4, or $5. Alice s Diner sells only four items, which have individual prices of $, $, $4, or $5. CUSTOMER RECEIPT CUSTOMER RECEIPT Qty Item Qty Item Big Burgers Monster Nachos Fast Fries Big Burgers Super Sodas Fast Fries 4 Super Sodas TOTAL: $18 TOTAL: $8 THANK YOU!! THANK YOU!! After dining there, Group B got this receipt. What is the cost of each individual item? After dining there, Group D got this receipt. What is the cost of each individual item? 01, TESCCC 07/18/1 page 1 of 1

25 Restaurant Receipts KEY Fill in or complete the information on this page as you work through the problem. What information is found on each receipt? Receipt A sodas + burgers $18 Receipt B burgers + fries + sodas $18 Receipt C 5 sodas + 6 burgers $9 Receipt D nachos + burgers + fries + 4 sodas $8 What prices are possible for each item? List possible item prices Sample A: Maybe 1 soda $5 Sample B: Maybe 1 soda $ Sample C: Maybe 1 burger $4, 1 fries $, and 1 soda $ Check Do the prices work? sodas + burgers $18 (5) + B 18 Or, B 8, so B 8/, which is not a possible price. So, the soda does not cost $5. 5 sodas + 6 burgers $9 5() + 6B 9 Or, 6B 9, so B 9/6, which is not a possible price. So, the soda does not cost $. burgers + fries + sodas $18 (4)+()+ ()18 These values are a possibility, but may be in the wrong order. What prices can be eliminated for certain items? Write the names of the items in the first column. Write the possible costs for each item in the top row. 01, TESCCC 07/18/1 page 1 of 1

26 Restaurant Receipts Fill in or complete the information on this page as you work through the problem. What information is found on each receipt? Receipt A Receipt C Receipt B Receipt D What prices are possible for each item? List possible item prices Check Do the prices work? What prices can be eliminated for certain items? Write the names of the items in the first column. Write the possible costs for each item in the top row. Eliminate possibilities by placing X s in the appropriate boxes. 01, TESCCC 07/18/1 page 1 of 1

27 Systems of Equations KEY Using B for the cost of a burger, F for the cost of fries, N for the cost of nachos, and S for the cost of a soda, write equations for each statement below. sodas and burgers cost $18. S + B 18 5 sodas and 6 burgers cost $9. 5S + 6B 9 burgers, fries, and sodas cost $18. B + F + S 18 nachos, burgers, fries, and 4 sodas cost $8. N + B + F + 4S 8 DEFINITION: DEFINITION: When used together, these form a system of equations, which is a set of two or more equations with two or more unknowns (or variables). The solution to a system lists the values that can be used for each variable to make the equations in the system true (or, check ). Samples: 1) x y 4z x z 4 x y z 4 This is a x system of equations because it uses variables and has equations. Which of the following would be the solution to this system? C A) x 4, y 1, z B) x 9.5, y 0, z 5. 5 C) x 5, y, z 1 ) x 7y 9 y 1 x This is a x system of equations because it uses variables and has equations. Which ordered pair shows the solution to this system? B A) (15, ) B) (8, 1) C) (, -) Below, circle the correct solution to each equation. How can you determine the solution? Varies ) 4) 5x y z 16 y x 8 The answer is (A) x y 5 z The answer is (B) x 4y 1 8x 5y z 5 A) x, y 1 A) (1, -0., 5.8) These answers are B) x 7, y 9 B) (-, 4, 7) called ordered triples C) x 7, y 9 C) (1, 1, 4) 01, TESCCC 07/18/1 page 1 of 1

28 Systems of Equations Using B for the cost of a burger, F for the cost of fries, N for the cost of nachos, and S for the cost of a soda, write equations for each statement below. sodas and burgers cost $18. 5 sodas and 6 burgers cost $9. burgers, fries, and sodas cost $18. nachos, burgers, fries, and 4 sodas cost $8. DEFINITION: DEFINITION: When used together, these form a of equations, which is a set of two or more equations with two or more unknowns (or variables). The to a system lists the values that can be used for each variable to make the equations in the system true (or, check ). Samples: 1) x y 4z x z 4 x y z 4 This is a of equations because it uses and has. Which of the following would be the solution to this system? A) x 4, y 1, z B) x 9.5, y 0, z 5. 5 C) x 5, y, z 1 ) x 7y 9 y 1 x This is a of equations because it uses and has. Which ordered pair shows the solution to this system? A) (15, ) B) (8, 1) C) (, -) Below, circle the correct solution to each equation. How can you determine the solution? ) 4) 5x y z 16 y x 8 x y 5 z x 4y 1 8x 5y z 5 A) x, y 1 A) (1, -0., 5.8) B) x 7, y 9 B) (-, 4, 7) C) x 7, y 9 C) (1, 1, 4) These answers are called ordered triples 01, TESCCC 07/18/1 page 1 of 1

29 Plug into the System KEY Evaluate each equation for the given values in the ordered triple. If the values check in the equation, then shade in the corresponding box. When all items are complete, the remaining boxes (which have not been shaded) will spell out a quote. Be sure to check each solution set for each equation. The equation will have more than one set that works. 1) System of Equations: (-,, -) (, 6, -4) (, 4, -9) (1, 5, 1) x y z 14 IFT HER HEY EAS x y 5z ONT ONY GIV EYO 4x y z 18 URU LIN GER LED Which ordered triple is the solution to the system? (, 6, -4) ) System of Equations: (,9,-) (-, 4, 5) (1, 5, -4) (5, 1, 1) x y 4z SYS PAPE TEMP RWR 9x 4z 7 ITET HERL UTNA HEO x 6y 4z 15 THE RWAY TYH OUSE Which ordered triple is the solution to the system? (1, 5, -4) Place the remaining letters (from left to right) in the blanks. IF THEY GIVE YOU RULED PAPER, WRITE THE OTHER WAY. Juan Ramon Jimenez 01, TESCCC 07/18/1 page 1 of 1

30 Plug into the System Evaluate each equation for the given values in the ordered triple. If the values check in the equation, then shade in the corresponding box. When all items are complete, the remaining boxes (which have not been shaded) will spell out a quote. Be sure to check each solution set for each equation. The equation will have more than one set that works. 1) System of Equations: (-,, -) (, 6, -4) (, 4, -9) (1, 5, 1) x y z 14 IFT HER HEY EAS x y 5z ONT ONY GIV EYO 4x y z 18 URU LIN GER LED Which ordered triple is the solution to the system? ) System of Equations: (,9,-) (-, 4, 5) (1, 5, -4) (5, 1, 1) x y 4z SYS PAPE TEMP RWR 9x 4z 7 ITET HERL UTNA HEO x 6y 4z 15 THE RWAY THY OUSE Which ordered triple is the solution to the system? Place the remaining letters (from left to right) in the blanks.. Juan Ramon Jimenez 01, TESCCC 7/18/1 page 1 of 1

31 Graphing Systems KEY Getting Started Consider the system: y x y x 1 For each equation, make a table of ordered pairs (x, y). Then graph each line. Which ordered pair appears in both tables? (, -1) y - x + x y y x - x y What do we call this point, in terms of the system? What do we call this point in terms of the graph? The solution to the system. The point of intersection. IMPORTANT: When the equations in a system can be graphed, then the solution to the system can be described using the coordinates of the point of intersection. Why? Sample: This point is the only pair of x- and y-values that works (or checks ) in both equations! y -x + y 1 x -1 -() () Use the slope and intercept of each linear equation to generate the graphs. Then state the solution to the system. y x y x7 1) System: ) System: y x1 y x Solution: (,5) Solution: (6,) 01, TESCCC 07/18/1 page 1 of 4

32 Graphing Systems KEY Continue graphing to find the solution to each system. ) System: y y 4 x x 4) System: y y - 1 x + 1 x + 5 Solution: (-4, -) Solution: (-, 4) 5) System: y y 5-1 x + 6) System: y y x x Solution: (-6, 5) Solution: No solution 01, TESCCC 07/18/1 page of 4

33 Graphing Systems KEY Systems of equations can also be solved using the CALC 5: intersect command on a graphing calculator. However, before equations can be entered, you must solve each equation for y. The table check is not included in sample!!! System Equations in Y form Sketch the Graph Solution Sample - x + y x + 4y 9 8 Y 1 0.5x Y (- 4 )x + (-,.5) 7) System Equations in Y form Sketch the Graph Table Check Solution x + y 4x + y 5 1 Y 1 -x + 5 Y -4x + 1 x Y1 Y (.5, -) 8) System Equations in Y form Sketch the Graph Table Check Solution 4x + 7y 14 8y + 8 5x 4 Y 1 (- 7 )x + Y ( 5 8 )x 1 x Y1 Y Approx. (.5, 0.6) Or 168, 8 9) System Equations in Y form Sketch the Graph Table Check Solution x 5y 0 6x 0 10y Y 1 ( 5 )x 6 Y ( 5 )x + x Y1 Y Parallel lines No solution 10) System Equations in Y form Sketch the Graph Table Check Solution x y x y Y 1 x + 5 Y x + 5 x Y1 Y Same line Infinite solutions 01, TESCCC 07/18/1 page of 4

34 Graphing Systems KEY When solving systems (such as by graphing), three different cases can occur. What type of lines? The lines are parallel. The lines intersect. The lines coincide. What type of system? Inconsistent Consistent Consistent What type of solutions? NO solutions ONE solution (represented by the point of intersection). An infinite number of solutions (or, every point on the line) Other Vocabulary Sometimes, the Empty Set is used to describe the solution to such a system. Symbol: If the two equations in the system graph into lines which intersect at a single point, then they are called independent equations. If the two equations in the system have the same graph, then they are called dependent equations. On graph paper or on a calculator, solve each system graphically. Then describe: The type of lines parallel, intersecting, or coincident (coinciding). The type of system inconsistent, consistent/independent, or consistent/dependent. The type of solutions one solution, no solutions, or infinitely many solutions. If the solution exists, state it as an ordered pair. y x ) y 1 x - 1 Intersecting Consistent/independent One solution: (-, -) y - x ) 1x + 4y 0 Coinciding Consistent/dependent Infinitely many solutions y - x + 1 1) y - x + 7 Intersecting Consistent/independent One solution: (4, -5) 4x - y ) - 6x + y 1 Parallel Inconsistent No solution 1) y y 5 x - 1 x - Intersecting Consistent/independent One solution: (7.5, ) y 0.8x + 16) 5x - 4y 8 Intersecting Consistent/independent One solution: (8.9, 9.1) 01, TESCCC 07/18/1 page 4 of 4

35 Graphing Systems Getting Started Consider the system: y x y x 1 For each equation, make a table of ordered pairs (x,y). Then graph each line. Which ordered pair appears in both tables? y - x + x y y x - x y What do we call this point, in terms of the system? What do we call this point in terms of the graph? IMPORTANT: When the equations in a system can be, then the to the system can be described using the of the of. Why? Use the slope and intercept of each linear equation to generate the graphs. Then state the solution to the system. y x y x 7 1) System: ) System: y x 1 y x Solution: Solution: 01, TESCCC 07/18/1 page 1 of 4

36 Graphing Systems Continue graphing to find the solution to each system. ) System: y y 4 x x 4) System: y y - 1 x + 1 x + 5 Solution: Solution: 5) System: y y 5-1 x + 6) System: y y x x Solution: Solution: 01, TESCCC 07/18/1 page of 4

37 Graphing Systems Systems of equations can also be solved using the CALC 5: intersect command on a graphing calculator. However, before equations can be entered, you must solve each equation for y. Sample System Equations in Y form Sketch the Graph Table Check Solution - x + y x + 4y 9 8 Y 1 0.5x Y (- 4 )x + x Y1 Y (-,.5) 7) System Equations in Y form Sketch the Graph Table Check Solution x + y 4x + y 5 1 x Y1 Y Y 1 Y 8) System Equations in Y form Sketch the Graph Table Check Solution 4x + 7y 14 8y + 8 5x x Y1 Y Y 1 Y 9) System Equations in Y form Sketch the Graph Table Check Solution x 5y 0 6x 0 10y x Y1 Y Y 1 Y 10) System Equations in Y form Sketch the Graph Table Check Solution x y x y x Y1 Y Y 1 Y 01, TESCCC 07/18/1 page of 4

38 Graphing Systems When solving systems (such as by graphing), three different cases can occur. What type of lines? What type of system? The lines are The lines The lines What type of solutions? solutions solution (represented by the An of solutions (or, every Other Vocabulary Sometimes, the is used to describe the solution to such a system. Symbol: ). If the two equations in the system graph into lines which intersect at a single point, then they are called equations. on the line). If the two equations in the system have the same graph, then they are called equations. On graph paper or on a calculator, solve each system graphically. Then describe: The type of lines parallel, intersecting, or coincident (coinciding). The type of system inconsistent, consistent/independent, or consistent/dependent. The type of solutions one solution, no solutions, or infinitely many solutions. If the solution exists, state it as an ordered pair. 11) y y x x - 1 1) y y - x x + 7 1) y y 5 x - 1 x - 14) y - x + 5 1x + 4y 0 15) 4x - y x + y 1 16) y 0.8x + 5x - 4y 8 01, TESCCC 07/18/1 page 4 of 4

39 Electrical Problems KEY While doing some renovations to their home, a family needs to hire an electrician. After looking through the phone directory, they find advertisements for two different companies. Which company seems to offer the better deal? Why? Electrical Services SAVE $$$ We Charge NO Diagnostic Fees! *Labor charges: $4.50/hr 1. For each company, write a function rule that determines the total charge for an electrician to do x hours of work. Larson Electric Power Planners L (x) 6.50x + 50 P(x) 4.50x. When electricians have to replace a blown fuse, they typically charge for 0 minutes of labor (plus any applicable fees). Which company would charge less for this service? How much less? L(0.5) $68.5, P(0.5) $1.75. Power Planners is cheaper by $ Installing additional electrical outlets in a newly remodeled room requires much more intensive work and is considered to be a 1-hour job. Which company would charge less for this service? How much less? L(1) $488, P(1) $5. Larson Electric is cheaper by $4. 4. There is a certain number of hours of labor for which the companies charge the same amount. Use appropriate tables, graphs, or computations to estimate this value. Hours Larson Power x L (x) P (x) Cost 0 P(x) Number of Hours L(x) Intersection: ( , ) At approximately 7 hours and 8 1 minutes the companies both charge about $ Use at least two complete sentences and your answer from item #4 to answer the following question: Which company offers the better deal? Power Planners is cheaper up to approximately 7 hours. For more time, Larson is cheaper. 01, TESCCC 07/18/1 page 1 of

40 Electrical Problems KEY Cost 0 P(x) Number of Hours L(x) Which company offers the better deal? Answering this question requires finding the graphical point of intersection, which is also known as the solution to the system. How can this point be found algebraically (or, without the graph) and exactly (or, without having to estimate)? We must first turn the TWO equations into ONE. Cost at Larson Cost at Power Planners L ( x) P( x) Set the cost functions equal to one another. 6.50x x Rewrite in terms of x x -6.50x 50 7x Solve x 1 5 y 4.50(7 ) To find the corresponding y-value, substitute the solution back into either original equation. Use a similar method to solve these systems algebraically. y x - y - x y 4x + 7 y 9x - 15 x 4x + 7 -x + 8 9x 15 x -5 x 1 y -1 y 9 4 Or (-5, -1) Or (, 9 ) y x y x x x x 5.65 y 0.15 Or (5.65, 0.15) How do you think the following systems can be solved? Find the solutions. y 7x - 0 x y x + y x + y - 1 8x + 7x 0 5 x 17 y 99 Or (17, 99) 9x - 5y + z 6 y +4 + y -1 y -1 x z 1 Or (, -1, 1) 01, TESCCC 07/18/1 page of

41 Electrical Problems While doing some renovations to their home, a family needs to hire an electrician. After looking through the phone directory, they find advertisements for two different companies. Which company seems to offer the better deal? Why? Electrical Services SAVE $$$ We Charge NO Diagnostic Fees! *Labor charges: $4.50/hr 1. For each company, write a function rule that determines the total charge for an electrician to do x hours of work. Larson Electric Power Planners L (x) P (x). When electricians have to replace a blown fuse, they typically charge for 0 minutes of labor (plus any applicable fees). Which company would charge less for this service? How much less?. Installing additional electrical outlets in a newly remodeled room requires much more intensive work and is considered to be a 1-hour job. Which company would charge less for this service? How much less? 4. There is a certain number of hours of labor for which the companies charge the same amount. Use appropriate tables, graphs, or computations to estimate this value. Hours Larson Power x L (x) P (x) 0Cost Number of Hours 5. Use at least two complete sentences and your answer from item #4 to answer the following question: Which company offers the better deal? 01, TESCCC 07/18/1 page 1 of

42 Electrical Problems Cost 0 P(x) Number of Hours L(x) Which company offers the better deal? Answering this question requires finding the graphical point of intersection, which is also known as the to the. How can this point be found algebraically (or, without the graph) and exactly (or, without having to estimate)? We must first turn the equations into. Cost at Larson Cost at Power Planners L ( x) P( x) Set the cost functions equal to one another. Rewrite in terms of x. Solve. To find the corresponding y-value, substitute the solution back into either original equation. Use a similar method to solve these systems algebraically. y x - y - x y 4x + 7 y 9x y y x x How do you think the following systems can be solved? Find the solutions. y 7x - 0 x y x + y x + y - 1 9x - 5y + z 6 01, TESCCC 07/18/1 page of

43 Part A Systems by Algebraic Methods KEY The substitution method can be used to solve systems of equations algebraically. x - y -11 7x + 4y - This is the system. +y +y Solve for a variable in one of the equations. x y Suggestion: Solve for x in the first equation x (5) x -6 7(-11 + y) + 4y y + 4y y - 11y 55 y 5 Solution: (-6, 5) In the other equation, replace this variable with its equivalent expression. Solve (for the numerical value of the variable). Substitute this answer into another equation to find the value of the other variable. Where would you start? For each system, without solving, identify which variable in which equation it would be easiest to solve for first. Explain why this variable is easiest to solve for. Answers will vary. 4 x + y 7 x + y 4 9 x + 11y 7y x x - y 90 - x + 7y 11 x + 9y 51 x + 4y 18 Solve for y in the first equation Solve for x in the second equation Solve for y in the first equation Probably, solve for x in either equation Now solve the systems. Show your work. 1) 4 x + y 7 x + y 4 Solution: (, -5) ) 9 x + 11y 7y x - 0 Solution: (6, -) ) 8 x - y 90 - x + 7y 11 Solution: (14, ) 4) x + 9y 51 x + 4y 18 Solution: (-7, 8) 01, TESCCC 07/18/1 page 1 of 4

44 Systems by Algebraic Methods KEY Sometimes, solving systems with the substitution method can yield strange results Try to solve these systems. 5) 5 x + y 10 x + y 9 6) - x + 9y -4 x - 6y 16 y 5x 10x + ( 5x) 9 x 6y x x 9 -(y + 8) + 9y -4 x y ??? -9y y When solving a system algebraically If you end up with an equation that is never true, 4 9 the system has no solutions (or, it is inconsistent) or, graphically, the equations lines would be parallel. If you end up with an equation that is always true, -4-4 the system has an infinite number of solutions (or, it is consistent/dependent) or, graphically, the equations lines would be the same (coincide). Continue solving these systems algebraically using the substitution method. 7) - x + y x + 7y 5 Solution: (-40, 5) 8) - 7x + y - x + 4y 4 Solution: (,9) 9) - x + 1y 40 x - 6 y 19 No solutions (system is inconsistent) 10) 9 x - y 6 x + 5y 4 Solution: (1.5, 7.5) 01, TESCCC 07/18/1 page of 4

45 Part B Systems by Algebraic Methods KEY The elimination or combination methods can also be used to solve systems of equations. When given a system of equations, you can create other important equations by x + 5y 19 4x + y x + 0y 1 x + 9y 7 Add Subtract x + 9y 1y 4 y For each equation select a constant to multiply by that will make the absolute value of the coefficient of one of the variables equal. Multiply both sides of each equation by the selected constants. (Multiplication of equal value on both sides of an equation) Combine the equations together by addition or subtraction. Which eliminates one variable? (Addition or subtraction of equal values on both sides of an equation) Combining the equations to eliminate a variable, is another method for solving systems. Ex. x + y x - 11 y 7 x 18 x 6 (6) + y 11 y -1 Solution: (6, -1) 11) a + b 19 a - b 7 a 6 a b 19 b 6 Solution: (1, 6) 1) 5x + y 1-5x + y 4 5y 5 y 7 5x +(7)1 x Solution: (, 7) 1) 9x - y 17 x + y -7 10x 10 x 1 (1) + y -7 y -4 Solution: (1, -4) Other times, to eliminate a variable through combination, one equation must first be multiplied by a constant. Use this method on the systems below. x + 5y 19 ( ) - 4x - 10y -8 x + y (4) _1x+8y 9_ Ex. 14) 4x + y 4 4x + y 4 x - 8y 6 x - 8y 6 x 5() 19 x x 9 x 4.5 Solution: (4.5, ) 7y 14 y 14x 154 (11)+y x 11 +y y -10 y -5 Solution: (11, -5) 15) x + y 4 x + 5y 6 Multiply bottom equation by (-) Solution: (-8, 14) 16) x - y -10 4x - 8y 8 Multiply top equation by (-4) Solution: (-6, -4) 01, TESCCC 07/18/1 page of 4

46 Systems by Algebraic Methods KEY Finally, you can multiply BOTH equations by a constant, so that when they are combined, a variable will be eliminated. Then solve the system. Ex. 8x + y 41 (5) 6x + 5y 9 ( ) 6(4) + 5y 9 x 4 4+5y 9 5y 15 y 40x + 15y 05 MULTIPLY the equations by constants so -18x + -15y -117 that something will cancel ADD (or COMBINE) x 88 the equations together SOLVE the resulting equation PLUG the value back into an original equation to find the other variable Solution: (4, ) 17) 4x + 5y x - y ) 1x - 7y 59 8x + 11y -9 Solution: (10, 6) Solution: (, -5) 19) 10x + 7y 15x + 8y ) 6x - 7y x + 5y 47-1 Solution: (, -) Solution: (, -5) 01, TESCCC 07/18/1 page 4 of 4

47 Systems by Algebraic Methods Part A The substitution method can be used to solve systems of equations algebraically. x - y -11 7x + 4y - This is the system. Solve for a variable in one of the equations. In the other equation, replace this variable with its equivalent expression. Solve (for the numerical value of the variable). Substitute this answer into another equation to find the value of the other variable. Where would you start? For each system, without solving, identify which variable in which equation it would be easiest to solve for first. Explain why this variable is easiest to solve for. 4 x + y 7 x + y 4 9 x + 11y 7y x x - y 90 - x + 7y 11 x + 9y 51 x + 4y 18 Now solve the systems. Show your work. 1) 4 x + y 7 x + y 4 ) 9 x + 11y 7y x - 0 ) 8 x - y 90 - x + 7y 11 4) x + 9y 51 x + 4y 18 01, TESCCC 07/18/1 page 1 of 4

48 Systems by Algebraic Methods Sometimes, solving systems with the substitution method can yield strange results Try to solve these systems. 5) 5 x + y 10 x + y 9 6) - x + 9y -4 x - 6y 16 When solving a system algebraically If you end up with an equation that is true, 4 9 the system has (or, it is ) or, graphically, the equations lines would be. If you end up with an equation that is true, -4-4 the system has (or, it is ) or, graphically, the equations lines would be. Continue solving these systems algebraically using the substitution method. 7) - x + y x + 7y 5 8) - 7x + y - x + 4y 4 9) - x + 1y 40 x - 6 y 19 10) 9 x - y 6 x + 5y 4 01, TESCCC 07/18/1 page of 4

49 Part B Systems by Algebraic Methods The elimination or combination methods can also be used to solve systems of equations. When given a system of equations, you can create other important equations by x + 5y 4x + y x + 0y 1 x + 9y 4 x + 9y 4 y For each equation select a constant to multiply by that will make the absolute value of the coefficient of one of the variables equal. Multiply both sides of each equation by the selected constants. (Multiplication of equal value on both sides of an equation) Combine the equations together by addition or subtraction. Which eliminates one variable? (Addition or subtraction of equal values on both sides of an equation) Combining the equations to eliminate a variable, is another method for solving systems. Ex. x + y x - y ) a + b 19 a - b 7 1) - 5x + y 5x + y 1 4 1) 9x - y x + y 17-7 x 18 x 6 (6) + y 11 y -1 Solution: (6, -1) Other times, to eliminate a variable through combination, one equation must first be multiplied by a constant. Use this method on the systems below. x + 5y 19 ( ) - 4x - 10y -8 x + y (4) Ex. 14) 4x + y 4 4x + y 4 x - 8y 6 x - 8y 6 x 5() 19 x x 9 x 4.5 Solution: (4.5, ) 7y 14 y 15) x + y x + 5y ) x - 4x - y 8y , TESCCC 07/18/1 page of 4

50 Systems by Algebraic Methods Finally, you can multiply BOTH equations by a constant, so that when they are combined, a variable will be eliminated. Then solve the system. 8x + y 41 Ex. 6x + 5y 9 + (5) ( ) MULTIPLY the equations by constants so that something will cancel ADD (or COMBINE) the equations together SOLVE the resulting equation PLUG the value back into an original equation to find the other variable Solution: (4, ) 17) 4x + 5y x - y ) 1x - 7y 59 8x + 11y -9 19) 10x + 7y 15x + 8y ) 6x 7y x 5y , TESCCC 07/18/1 page 4 of 4

51 Pick a Method Cards A) y x F) y x y 9x 14 9x 7y 6 B) x y 1 G) x 4y 11 x y y x 4 5 C) 5x y 11 H) x y 8 y 5 x y x 1 D) y y x 1 x 7 J) 6x 7y 47 x 5y 1 E) 9x 7y 5 K) 10x y 16 x 9y 51 x y 1 01, TESCCC 07/18/1 page 1 of 1

52 Pick a Method KEY Take the systems cards and categorize them according to which method you would use. Graphing Combination / Elimination D) y y x 1 x 7 G) x y 5 y x 4 C) 5x y x y 11 8 J) 6x 7y x 5y 47 1 H) y y 5 x 1 x A) y y x x K) x 9y x y 51 1 E) 9x 7y 5 10x y 16 B) x y 1 x 4y 11 F) y 9x 14 9x 7y 6 Substitution Explain how you identify when to use each method. Answers will vary. Samples are given. Graphing Substitution Combination/Elimination Equations are already in y form. Answers are integral. Equations have fraction slopes, which are easy to graph by hand but troublesome to solve with (algebraically). One equation is already solved for a variable. OR It would be easy (one step) to solve for a variable in one of the equations. The equations are lined up in standard form, which looks like: x + y x + y 01, TESCCC 07/18/1 page 1 of

53 Pick a Method KEY Solve each system by the method selected. A) y x y x F) y 9x+ 14 9x+ 7y 6 (5.65, 0.15) (-1, 5) B) x y-1 x+ 4y 11 G) x+ y 5 y x- 4 (1.8, 1.4) (4, 1) C) 5x+ y 11 x+ y 8 H) y x-1 5 y x- (1, ) (7.5, ) D) y - x+ 1 y - x+ 7 J) 6x- 7y 47 x+ 5y -1 (4, -5) (, -5) E) 9x- 7y 5 10x+ y -16 K) x- 9y 51 x+ y 1 (-1, -) (6, -5) 01, TESCCC 07/18/1 page of

54 Pick a Method Take the systems cards and categorize them according to which method you would use. Graphing Combination / Elimination Substitution Explain how you identify when to use each method. Graphing Substitution Combination/Elimination 01, TESCCC 07/18/1 page 1 of

55 Pick a Method Solve each system by the method selected. A) y x y x F) y 9x+ 14 9x+ 7y 6 B) x y-1 x+ 4y 11 G) x+ y 5 y x- 4 C) 5x+ y 11 x+ y 8 H) y x-1 5 y x- D) y - x+ 1 y - x+ 7 J) 6x- 7y 47 x+ 5y -1 E) 9x- 7y 5 10x+ y -16 K) x- 9y 51 x+ y 1 01, TESCCC 07/18/1 page of

56 Solving x Systems of Linear Equations KEY Study the situation below. Mary found an online store selling her favorite style of shirt, capri pants, and skirt. She placed three separate online orders: shirts and 1 pair of capri pants for $40 One skirt and two pair of capri pants would cost for $ One shirt and pair of capri pants for $5 Which item costs the most? What methods could be used to determine the values of the variables? Sample answers: Guess and check, logic table The problem can be solved algebraically. Solving x systems of linear equations using substitution Answers may vary. Sample answers given. Identify three variables in the problem. x shirt y pair of capri pants z skirt Use the first three rows to set up three equations using the variables identified above. x + y 40 z + y x + y 5 Which two equations contain the same two variables? First and third x + y 40 x + y 5 What are the equations? 01, TESCCC 04/7/1 page 1 of 4

57 Solving x Systems of Linear Equations KEY What method can be used to solve this system of two equations? Students may choose graphing, substitution, or elimination. x + y 40 x + y 5 Solve the equations x 5 y (5 y) + y 40 y 10, x 15 Use the two values of the variables to solve for the third variable in the remaining equation. z + y z + (10) z + 0 z 1 Answer the question in terms of the problem situation. A shirt costs $15; a pair of capri pants costs $10; and a skirt costs $1. Therefore, a shirt costs the most. Solving x systems of linear equations using combination/elimination For some x systems substitution does not work. These x systems can be solved using combination/elimination. Example: x 4y z 5 x 6y z 6x y z 4 What are the variables in the system? x, y, and z 01, TESCCC 04/7/1 page of 4

58 Solving x Systems of Linear Equations KEY Select one equation and write it down twice. x 6y z x 6y z Place the other two equations below the first. x 6y z x 4y z 5 x 6y z 6x y z 4 Select and cancel one variable in each system. The same variable must be canceled in each. Canceling the x variable in each x 6y z x 4y z 5 x 18y 6z 9 x 4y z 5 x 6y z 6x y z 4 6x 6y 1z 18 6x y z 4 14y 9z 4 4y 9z 14 Collect the remaining two equations and cancel for one of the remaining variables. 14y 9z 4 4y 9z 14 14y 9z 4 4y 9z 14 y 0y 10 1 Take the one variable solved for and substitute backwards to find the other variables. z 14y 9z ( ) 9z 4 7 9z 4 9z 1 x 6y z 1 1 6( ) ( ) Write the final answer in point form, (x, y, z), and check the solution in all equations. 1 1,, x x x 01, TESCCC 04/7/1 page of 4

59 Solving x Systems of Linear Equations KEY Practice Problems Solve each x system of linear equations by substitution or combination/elimination. Show all work and check the final solution. x y z 4 6x y z y z 1. 7x 5y z 7 y 5z 6 x 8y z 1 (1, 1, 1) (4, -7, 9) x y z 8 4x 7y z 10. y z x y 8z 9 9x 6z 1 6x y 6z (-, 8, 1) (-1.5, -0.5, 1.5) 01, TESCCC 04/7/1 page 4 of 4

60 Solving x Systems of Linear Equations Study the situation below. Mary found an online store selling her favorite style of shirt, capri pant, and skirt. She placed three separate online orders: shirts and 1 pair of capri pants for $40 One skirt and two pair of capri pant would cost for $ One shirt and pair of capri pants for $5 Which item costs the most? What methods could be used to determine the values of the variables? The problem can be solved algebraically. Solving x systems of linear equations using substitution Identify three variables in the problem. x y z Use the first three rows to set up three equations using the variables identified above Which two equations contain the same two variables? What are the equations? 01, TESCCC 04/7/1 page 1 of 4

61 Solving x Systems of Linear Equations What method can be used to solve this system of two equations? Solve the equations Use the two values of the variables to solve for the third variable in the remaining equation. Answer the question in terms of the problem situation. Solving x systems of linear equations using combination/elimination For some x systems substitution does not work. These x systems can be solved using combination/elimination. Example: x 4y z 5 x 6y z 6x y z 4 What are the variables in the system? 01, TESCCC 04/7/1 page of 4

62 Solving x Systems of Linear Equations Select one equation and write it down twice. x 6y z Place the other two equations below the first. Select and cancel one variable in each system. The same variable must be canceled in each. Collect the remaining two equations and cancel for one of the remaining variables. Take the one variable solved for and substitute backwards to find the other variables. Write the final answer in point form, (x, y, z), and check the solution in all equations. 01, TESCCC 04/7/1 page of 4

63 Solving x Systems of Linear Equations Practice Problems Solve each x system of linear equations by substitution or combination/elimination. Show all work and check the final solution. x y z 4 6x y z y z 1. 7x 5y z 7 y 5z 6 x 8y z 1 x y z 8 4x 7y z 10. y z x y 8z 9 9x 6z 1 6x y 6z 01, TESCCC 04/7/1 page 4 of 4

64 Solving Systems with Technology and Matrices KEY How can we solve problems requiring more variables and more equations using technology? First, you must make sure that the system is in standard form. This means that the variable terms are lined up on the left side of the equations, and the constants are on the right. Sample System: x + y + z 9 Looks good x + y - z Add z to both sides x - 1 4y + 5z Add 1, -4y, and -5z to both sides Standard Form: x + y + z 9 x + y + z 0 x - 4y - 5z 1 Using Inverse Matrices With this method, think of the system as matrix of coefficients 1 matrix of variables 1 matrix of constants In this case: é1 1 ù éxù é9ù 1 y 0 ê -4-5ú êzú ê1ú ë û ë û ë û Enter the coefficients in one matrix (here, [A]): Using Augmented Matrices With this method, think of the system as an augmented 4 matrix. é1 1 9ù 1 0 ê ú ë û The first three columns are coefficients The variables are not shown The last column holds the constants Enter this augmented matrix into the calculator: Enter the constants in another (here, [C]): Compute the solutions using [A] -1 [C]: From the MATRIX find the command: MATH menu, scroll down to rref( This places a matrix in reduced row echelon form. So, the solutions are: éxù é ù y 5 êzú ê-ú ë û ë û Here, solutions appear in the last column. So, the solutions are: éxù é ù y 5 êzú ê-ú ë û ë û 01, TESCCC 07/18/1 page 1 of

65 Special Cases Solving Systems with Technology and Matrices KEY When using the graphing calculator to determine solutions to special cases, the inverse matrices only give ERR: SINGULAR MAT and do not distinguish between infinite and no solutions. Therefore another method must be used to determine whether it is an infinite number of solutions or no solutions. The augmented matrices can be determined by analyzing the last row. Inverse Matrices Augmented Matrices Infinite number of solutions ERR: SINGULAR MAT Infinite number of solutions Note: In the last row all are zeros, and solves to 0 0. é ù ê ú ë û No solutions ERR: SINGULAR MAT No solutions Note: In the last row all are not zeros, and solves to 0 1. é ù ê ú ë û Practice Problems: x + y z z x + 1 y z - 6 Using Inverse Matrices Standard form Written as Matrix Equation Matrix Solution x + y z -7 -x + 0y + 4z 1 0x + y z -6 1 x y z 6 x 5 y z 4 Standard form Written as Augmented Matrix Matrix Solution Using Augmented Matrices x + y z -7 -x + 0y + 4z 1 0x + y z Values of x,y,z x 5, y -, z 4 (5, -, 4) Values of x,y,z x 5, y -, z 4 (5, -, 4) 01, TESCCC 07/18/1 page of

66 Solving Systems with Technology and Matrices KEY. Standard form Written as Matrix Equation Matrix Solution x+ y- z-7 4z y+ 1 x+ 6 z Using Inverse Matrices x + y z -7 0x y + 4z 1 x + 0y z -6 1 x y z 6 x 6. y 0.6 z 0. Standard form Written as Augmented Matrix Matrix Solution Using Augmented Matrices x + y z -7 0x y + 4z 1 x + 0y z Values of x,y,z x -6., y -0.6, z -0. (-6., -0.6, -0.) Values of x,y,z x -6., y -0.6, z -0. (-6., -0.6, -0.). x+ y+ z 10 x- y -z x+ z 5 4. x y z 1 x y 4z 4 6x y z Infinite number of solutions No solution 01, TESCCC 07/18/1 page of

67 Solving Systems with Technology and Matrices How can we solve problems requiring more variables and more equations using technology? First, you must make sure that the system is in standard form. This means that the variable terms are lined up on the left side of the equations, and the constants are on the right. Sample System: x + y + z 9 Looks good x + y - z Add z to both sides x - 1 4y + 5z Add 1, -4y, and -5z to both sides Standard Form: x + y + z 9 x + y + z 0 x - 4y - 5z 1 Using Inverse Matrices With this method, think of the system as matrix of coefficients 1 matrix of variables 1 matrix of constants In this case: é1 1 ù éxù é9ù 1 y 0 ê -4-5ú êzú ê1ú ë û ë û ë û Enter the coefficients in one matrix (here, [A]): Using Augmented Matrices With this method, think of the system as an augmented 4 matrix. é1 1 9ù 1 0 ê ú ë û The first three columns are coefficients The variables are not shown The last column holds the constants Enter this augmented matrix into the calculator: Enter the constants in another (here, [C]): Compute the solutions using [A] -1 [C]: From the MATRIX find the command: MATH menu, scroll down to rref( This places a matrix in reduced row echelon form. So, the solutions are: éxù é ù y 5 êzú ê-ú ë û ë û Here, solutions appear in the last column. So, the solutions are: éxù é ù y 5 êzú ê-ú ë û ë û 01, TESCCC 04/14/1 page 1 of

68 Solving Systems with Technology and Matrices Special Cases When using the graphing calculator to determine solutions to special cases, the inverse matrices only give ERR: SINGULAR MAT and do not distinguish between infinite and no solutions. Therefore another method must be used to determine whether it is an infinite number of solutions or no solutions. The augmented matrices can be determined by analyzing the last row. Inverse Matrices Augmented Matrices Infinite number of solutions ERR: SINGULAR MAT Infinite number of solutions Note: In the last row all are zeros, and solves to 0 0. é ù ê ú ë û No solutions ERR: SINGULAR MAT No solutions Note: In the last row all are not zeros, and solves to 0 1. é ù ê ú ë û Practice Problems: x + y z z x + 1 y z - 6 Using Inverse Matrices Using Augmented Matrices Standard form Standard form Written as Matrix Equation Written as Augmented Matrix Matrix Solution Matrix Solution Values of x,y,z Values of x,y,z 01, TESCCC 04/14/1 page of

69 Solving Systems with Technology and Matrices. x + y - z - 7 4z y + 1 x + 6 z Using Inverse Matrices Using Augmented Matrices Standard form Standard form Written as Matrix Equation Written as Augmented Matrix Matrix Solution Matrix Solution Values of x,y,z Values of x,y,z. x+ y + z 10 x- y -z x+ z 5 4. x y z 1 xy 4z 4 6xy z 01, TESCCC 04/14/1 page of

70 Quadratic Quest KEY Quadratic functions have the general equation y ax + bx + c, where a 0. If coordinates for three points on a parabola are known, then the equation can be determined using a system of equations to find a, b, and c. Example: Select three points through which the function passes Q(-1, 1.75), R(.5, 4) and D(.5,.5). ax + bx + c y For Q a(-1) + b(-1) + c 1.75 For R a(.5) + b(.5) + c 4 For D a(.5) + b(.5) + c.5 Solve the system for a, b, and c using matrices. Inverse matrix method shown; augmented matrix method would also be acceptable a 1.75 éaù b 4 b c.5 ê cú ë û.75 Quadratic Function y -0.5x + 1.5x +.75 Use this strategy to solve more problems involving quadratic functions. 1) The parabola shown has its vertex at V(0.5, -.6). A point on the left side of the parabola is located at L(-,.). What are the coordinates for R, the corresponding point on the right side? (0.5+.5,.), or (4,.) Set up a system to find the constants a, b, and c. ax + b x + c y For L a(-) + b(-) + c. For V a(0.5) + b(0.5) + c -.6 For R a(4) + b(4) + c. Inverse matrix method shown; augmented matrix method would also be acceptable. 9 1 a. éaù b.6 b c. ê cú ë û.5 Quadratic Function y 0.4x 0.4x.5 Find the values of the x-intercepts. Graph approximates values of - and. The calculator gives x and x.0495 as the x- intercepts. By the quadratic formula, these values can be expressed x , TESCCC 04/15/1 page 1 of

71 Quadratic Quest KEY ) Bart throws a water balloon off the top of a 40-ft building. From surveillance photographs, it was determined that, after he released the balloon, its height at 0.8 seconds was feet, and it was.04 feet high after 1.6 seconds. The height (h, in feet) of the water balloon is a quadratic function of time (x) in seconds. Find this function in the form h ( x) ax + bx + c. Time Height (sec) (ft) ???? hx x x ( ) Graph the function in a calculator. After precisely how many seconds did the water balloon hit the ground? The balloon hits the ground at x.1179 seconds How high did Bart throw the balloon above the height of the building? Bart threw the balloon to a height of about 4.5 feet, which is.5 feet higher than the building ) The journalism department has put together a magazine that features students poems and short stories as well as pictures of student artwork. They plan on selling the collections as a fundraiser, but are unsure what to charge. They can generate more revenue if they charge more for the magazine but if it s too expensive, no one will buy it (and they ll lose money). By surveying students in English classes, they determine that charging $4 for the books will generate $71 in revenue, and charging $6 each will create $85 in sales; but at $11 apiece, they will only bring in $57. The revenue (R, in dollars) earned from the sale of the magazines is a quadratic function of their individual cost (x). Find this function in the form R ( x) ax + bx + c. Rx ( ) 18x 50x Using a calculator, determine how much money they could make if they charged $1 for each magazine. R(1) $408 About how much should they charge to earn the most money? According to the function, what s the most they could make? If they charge around $6.94, they will make about $ , TESCCC 04/15/1 page of

72 Quadratic Quest Quadratic functions have the general equation y ax + bx+ c, where a 0. If coordinates for three points on a parabola are known, then the equation can be determined using a system of equations to find a, b, and c. Example: Select three points through which the function passes Q(-1, 1.75), R(.5, 4) and D(.5,.5). ax + bx + c y For Q a( ) + b( ) + c For R a( ) + b( ) + c For D a( ) + b( ) + c Solve the system for a, b, and c using matrices. y Quadratic Function Use this strategy to solve more problems involving quadratic functions. 1) The parabola shown has its vertex at V(0.5, -.6). A point on the left side of the parabola is located at L(-,.). What are the coordinates for R, the corresponding point on the right side? Set up and solve a system to find a, b, and c using matrices. ax + b x + c y For L a( ) + b( ) + c For V a( ) + b( ) + c For R a( ) + b( ) + c y Quadratic Function Find the values of the x-intercepts. 01, TESCCC 07/18/1 page 1 of

73 Quadratic Quest ) Bart throws a water balloon off the top of a 40-ft building. From surveillance photographs, it was determined that, after he released the balloon, its height at 0.8 seconds was feet, and it was.04 feet high after 1.6 seconds. The height (h, in feet) of the water balloon is a quadratic function of time (x) in seconds. Find this function in the form h ( x) ax + bx+ c. Time Height (sec) (ft) ???? Graph the function in a calculator. After precisely how many seconds did the water balloon hit the ground? How high did Bart throw the balloon above the height of the building? ) The journalism department has put together a magazine that features students poems and short stories as well as pictures of student artwork. They plan on selling the collections as a fundraiser, but are unsure what to charge. They can generate more revenue if they charge more for the magazine but if it s too expensive, no one will buy it (and they ll lose money). By surveying students in English classes, they determine that charging $4 for the books will generate $71 in revenue, and charging $6 each will create $85 in sales; but at $11 apiece, they will only bring in $57. The revenue (R, in dollars) earned from the sale of the magazines is a quadratic function of their individual cost (x). Find this function in the form R ( x) ax + bx+ c. Using a calculator, determine how much money they could make if they charged $1 for each magazine. About how much should they charge to earn the most money? According to the function, what s the most they could make? 01, TESCCC 07/18/1 page of

74 Movies, Kids, and Calculators KEY A local movie theater sells tickets for $6, but they also offer a reduced ticket price of $4 for children under the age of 1. Lately, the theater manager has been suspecting that teenagers are buying children s tickets to save money, even though they do not meet the age requirements. So the manager asks you to do a little research. She asks: MOVIE TICKETS Adults $6 Children (under 1) $4 What percent of the tickets being sold are children s tickets? To answer this, you must determine how many of each type were purchased for the current showing of the movie Sunshine s Revenge. Total People You walk into the back of the theater while the movie is showing. You count 46 people in the theater. Total Money Next, you go to the ticket booth. While the money-takers do not know how many of each type of ticket were sold, they do know that the showing of Sunshine s Revenge earned $8. Equations For the situations described above, if a the number of adult tickets sold and c the number of children s tickets sold, what two equations can be used to relate a and c? a + c 46 (or, c 46 a ) 6a + 4c 8 Solve the system. Then answer the original question. a 7 and c 19 So 41.% of the tickets were children s ($4). Check the reasonableness of the solution (7) + 4(19) , TESCCC 07/18/1 page 1 of

75 Movies, Kids, and Calculators KEY For some reason, in our history class, there are 14 more boys than there are girls. If a total students are in the class, how many boys and how many girls are there? Define variables for the unknowns in this situation. G girls, B boys Identify the two equations to represent the problem situation. B G + 14, B + G Solve the system and answer the question. G 9, B The class is comprised of boys and 9 girls. Check the reasonableness of the solution The school spent $4,650 last year to purchase 100 new calculators. Some were fancy graphing calculators that cost $105 each, but the rest were cheap scientific calculators that only cost $15. How many of each kind were purchased? Define variables for the unknowns in this situation. G graphing calculators S scientific calculators Identify the two equations to represent the problem situation. G + S G + 15S 4650 Solve the system and answer the question. G 5, S 65 The school purchased 5 graphing calculators and 65 scientific calculators. Check the reasonableness of the solution (105) + 65(15) , TESCCC 07/18/1 page of

76 Movies, Kids, and Calculators A local movie theater sells tickets for $6, but they also offer a reduced ticket price of $4 for children under the age of 1. Lately, the theater manager has been suspecting that teenagers are buying children s tickets to save money, even though they do not meet the age requirements. So the manager asks you to do a little research. She asks: MOVIE TICKETS Adults $6 Children (under 1) $4 What percent of the tickets being sold are children s tickets? To answer this, you must determine how many of each type were purchased for the current showing of the movie Sunshine s Revenge. Total People You walk into the back of the theater while the movie is showing. You count 46 people in the theater. Total Money Next, you go to the ticket booth. While the money-takers do not know how many of each type of ticket were sold, they do know that the showing of Sunshine s Revenge earned $8. Equations For the situations described above, if a the number of adult tickets sold and c the number of children s tickets sold, what two equations can be used to relate a and c? Solve the system. Then answer the original question. a and c So % of the were. Check the reasonableness of the solution. 01, TESCCC 07/18/1 page 1 of

77 Movies, Kids, and Calculators For some reason, in our history class, there are 14 more boys than there are girls. If a total students are in the class, how many boys and how many girls are there? Define variables for the unknowns in this situation. Identify the two equations to represent the problem situation. Solve the system and answer the question. Check the reasonableness of the solution. The school spent $4,650 last year to purchase 100 new calculators. Some were fancy graphing calculators that cost $105 each, but the rest were cheap scientific calculators that only cost $15. How many of each kind were purchased? Define variables for the unknowns in this situation. Identify the two equations to represent the problem situation. Solve the system and answer the question. Check the reasonableness of the solution. 01, TESCCC 07/18/1 page of

78 Situation Problems Involving Systems KEY 1. The parents of the Little League players are working in the concession stand selling soft drinks and popcorn. The soft drinks and popcorn are both being sold in the same size paper cup. They sold $7.5 worth of soft drinks and popcorn, and they used a total of 75 cups at the last game night. If the soft drinks sell for $1.5 per drink and popcorn sells for $.75 per cup, how many of each did they sell? System: d + p 75, 1.5d p 7.5; Solution: 1 soft drinks, 14 cups of popcorn.. Amallia is purchasing a new stove for her apartment. One brand costs $1800 and estimates $60 per month to operate. A more expensive brand costs $600, but only costs $40 per month to operate. a. Write an equation representing the cost of each stove in terms of months owned. C m, C600+40m b. Find the total cost of each after 1 year, years, and years. Time A) m B) C600+40m 1 year (1 mo.) $50 $080 years (4 mo.) $40 $560 years (6 mo.) $960 $4040 c. Plot the graph of each on the same axes. See above. d. Find the break-even point for the two stoves. When does this occur? What is this cost? At 40 months, both stoves cost $4,00 to own and operate. e. If Amallia plans on keeping this stove for five years, which stove would be the most economical? Explain. After the break-even point, the more expensive stove is more economical. At year 5 or 60 months, the more expensive brand would be more economical with ownership cost of $5000 while the other brand would cost $ At the end of the school year, the New Bryant High School mathematics department was required to run an inventory of the graphing calculators. They have two types, TI-84 and TI- Nspire. The math department had purchased 50 calculators for a total of $7,750. If TI-84 calculators cost $95 each and TI-Nspire calculators cost $15 each, how many of each type of calculator should the school have in inventory? System: T + N 50, 95T + 15N 7,750; Solution: 00 TI-84s and 150 TI-Nspires. 4. Jamar is planning on purchasing a new truck and is comparing the benefits of his two favorite trucks. The first truck has a regular engine and costs $1,500 to purchase and cents per mile to operate. The second truck has a fuel-injection engine and costs $,500 to purchase and only 9 cents per mile to operate. a. Write an equation for each truck comparing total cost in terms of miles operated. C1,500+0.m, C, m b. Calculate the cost of operating each vehicle for 1,000, 10,000, and 50,000 miles. Miles C1,500+0.m C, m 1,000 $1,80 $,790 10,000 $4,700 $5,400 50,000 $7,500 $7,000 c. Plot the graph of each on the same axes. d. Find the break-even point for the two vehicles. At what mileage does this occur? What is this cost? At, miles, both trucks cost about the same, $,166. e. If Jamar plans to drive his new truck for at least 40,000 miles, which truck would be the most economical purchase? Explain. At 40,000 miles, the truck with fuel injection would be more economical to own and operate at a cost of $4,100 while the regular engine truck would cost $4,00. 01, TESCCC 0/8/1 page 1 of 1

79 Situation Problems Involving Systems 1. The parents of the Little League players are working in the concession stand selling soft drinks and popcorn. The soft drinks and popcorn are both being sold in the same size paper cup. They sold $7.5 worth of soft drinks and popcorn, and they used a total of 75 cups at the last game night. If the soft drinks sell for $1.5 per drink and popcorn sells for $.75 per cup, how many of each did they sell?. Amallia is purchasing a new stove for her apartment. One brand costs $1800 and estimates $60 per month to operate. A more expensive brand costs $600, but only costs $40 per month to operate. a. Write an equation representing the cost of each stove in terms of months owned. b. Find the total cost of each after 1 year, years, and years. c. Plot the graph of each on the same axes. d. Find the break-even point for the two stoves. When does this occur? What is this cost? e. If Amallia plans on keeping this stove for five years, which stove would be the most economical? Explain.. At the end of the school year, the New Bryant High School mathematics department was required to run an inventory of the graphing calculators. They have two types, TI-84 and TI- Nspire. The math department had purchased 50 calculators for a total of $7,750. If TI-84 calculators cost $95 each and TI-Nspire calculators cost $15 each, how many of each type of calculator should the school have in inventory? 4. Jamar is planning on purchasing a new truck and is comparing the benefits of his two favorite trucks. The first truck has a regular engine and costs $1,500 to purchase and cents per mile to operate. The second truck has a fuel-injection engine and costs $,500 to purchase and only 9 cents per mile to operate. a. Write an equation for each truck comparing total cost in terms of miles operated. b. Calculate the cost of operating each vehicle for 1,000, 10,000, and 50,000 miles. c. Plot the graph of each on the same axes. d. Find the break-even point for the two vehicles. At what mileage does this occur? What is this cost? e. If Jamar plans to drive his new truck for at least 40,000 miles, which truck would be the most economical purchase? Explain. 01, TESCCC 0/8/1 page 1 of 1

80 Mixing It Up KEY WARM UP Compute answers to each of the following mixture problems. A) Florence took a road trip to the beach, which was several hundred miles away. As she left, she had to drive 6 mph through the city for about half an hour. Once she hit the highway, she drove 68 mph for.5 hours. Then, as she approached the beach, the traffic was so heavy she could only go 0 mph for the last 0 minutes. How many hours was Florence on the road? hours How many miles did Florence travel? 6(0.5) + 68(.5) + 0(0.5) 66 miles Over the course of her trip, what was Florence s average speed? 66 miles / 4.5 hours 59.1 miles per hour B) When you buy hydrogen peroxide, it is really a solution diluted with water. One bottle comes in a % solution, and another comes in 5% strength. Suppose you mix 40 ml of the % solution with 80 ml of the 5% solution. How many milliliters of the mixture is actually pure hydrogen peroxide? 0.0(40) (80) 4.8 ml What percent of this mixture is hydrogen peroxide? 4.8 / % C) Donna always gets really high grades on tests and quizzes (95 s every time), but she only does homework half the time (so her homework grade is a 50). Donna s teacher has a policy where tests and quizzes make up only 0% of students grades and homework makes up the other 70%. Under this system, what would Donna s class grade be? 0.0(95) (50) 6.5 A teacher down the hall uses a different policy, where tests and quizzes are 60% of the grade, and homework is only 40%. What would Donna s grade be if she had this teacher? 0.60(95) (50) 77 01, TESCCC 0/8/1 page 1 of 5

81 Mixing It Up KEY For each situation, define variables and set up a system of equations that relates them. Then, solve the system using inverse or augmented matrix methods. Inverse matrix equations are shown, but augmented matrix equations are also acceptable. 1) When traveling to see relatives, Duke drives part of the trip on the highway at 60 mph and the rest through cities at 40 mph. The 70-mile trip takes him a total of 5 hours. How much of this time does he spend on the highway? How much time through cities? Solutions: h Time spent on the highway c Time spent driving through cities System: Matrix Equation: h.5 hours c 1.5 hours h + c 5 60h + 40c h c 70 ) On the return trip, Duke encounters some road construction that slows him down. While he is still able to go 60 mph on the highway and 40 mph through cities, he can only go 0 mph past the construction. He spends twice the time passing the construction as he does on the highway, so this time the 70-mile trip takes 7.8 hours. How much of this time does he spend on the highway, in the city, and through road construction? Solutions: h Time spent on the highway c Time spent driving through cities n Time spent driving through road construction System: Matrix Equation: h.1 hours c 1.5 hours n 4. hours h + c + n h + 40c + 0n 70 n h -h + n h c n 0 01, TESCCC 0/8/1 page of 5

82 Mixing It Up KEY Continue setting up and solving systems for these situations. Inverse matrix equations are shown, but augmented matrix equations are also acceptable. ) In the school s chemistry lab, a certain acid is stored in two different dilutions. One bottle contains a weaker solution that is only 5% acid, but another has a stronger solution of 65% acid. Students must pour some from each container to make 60 ml of a solution that is 50% acid. How much from each container should be used? Solutions: w Amount to use from the weaker (5%) solution (ml) s Amount to use from the stronger (65%) solution (ml) System: Matrix Equation: w.5 ml s 7.5 ml w + s w s 0.5(60) w s 0 4) When Ebenezer won a $50,000 lottery jackpot, everyone started asking for donations. His brother Bob said, His sister Sally said, And his friend Fred said, I need a loan! I can pay you back in a year. I ll even pay you back with 4% interest! I need $0,000 more than Bob does, but I want to start my own business. And you ll earn more than 4% on this sweet deal. No way. You need to put your money in the stock market. It s doing great right now. Ebenezer split all his money among these three investments. Bob repaid his loan and interest, and the investment in Sally s business came back with 7% profit. Even though the money he put in the stock market lost 8.%, Ebenezer still came out with an extra $5,80. How much money did he place into each of these investments? Solutions: B Amount Ebenezer let Bob borrow S Amount Ebenezer invested in Sally s business M Amount Ebenezer placed in the stock market System: Matrix Equation: B $85,000 S $105,000 M $60,000 B + S + M 50,000 S B + 0,000 -B + S 0, B S 0.08M 5, B 50, S 0, M 5,80 01, TESCCC 0/8/1 page of 5

83 Mixing It Up KEY Continue setting up and solving systems for these situations. Inverse matrix equations are shown, but augmented matrix equations are also acceptable. 5) When extreme athletes compete at a motorcycle jumping competition, judges give them scores on a scale of 0-10 in three categories: use of fundamentals, degree of difficulty, and overall style. However, these categories are not weighted equally in determining the final score. Funda- Difficulty score Final Style mentals D. Howser R. Finn J. Hoban F Percent that fundamentals counts toward final score D Percent that difficulty counts toward final score S Percent that style counts toward final score System: The table shows how the category scores for three riders determined their final rating. By what percent is each category weighted? Matrix Equation: Solutions: F % D % S 0.5 5% 7.4F + 8.D + 8.8S F + 7.8D + 9.6S F + 7.6D + 8.6S F D S ) At the lake, Curtis caught three fish. He wanted to weigh the fish individually, but his scale was broken and could only read weights between 5 and 10 pounds. The large and medium fish together weighed 7.8 pounds. The large and small fish weighed 7.0 pounds, and the small and medium fish were 5.6 pounds. How much did each fish weigh individually? What was the total weight of all three fish? Solutions: S Weight of the small fish M Weight of the medium-sized fish L Weight of the large fish System: Matrix Equation: S.4 pounds M. pounds L 4.6 pounds Total: 10. lbs L + M 7.8 L + S 7 S + M S M L , TESCCC 0/8/1 page 4 of 5

84 Mixing It Up KEY Continue setting up and solving systems for these situations. Inverse matrix equations are shown, but augmented matrix equations are also acceptable. 7) On the way home from school, a family enters the drive-through at a fast food restaurant. Mom: I need root beers, junior burgers and one deluxe salad. Clerk: That will be $10.6. Please pull around. Lucy: Mom, I don t want a burger. I m a vegetarian now! Mom: OK sorry. Can I get root beers, 1 junior burger and deluxe salads, instead? Clerk: No problem, ma am. That brings your total to $11.. Mom: Oh no. I ve only got $11. Could you take off one of the root beers? Clerk Yes. Two root beers, 1 junior burger, and salads. That s $10.0. What is the cost for each item individually? Solutions: R The cost of a root beer R $1.19 B The cost of a junior burger B $.15 S $.75 S The cost of a deluxe salad System: Matrix Equation: R + B + S 10.6 R + B + S 11. R + B + S R B S ) A box in the textbook warehouse has the given label. It contains some Algebra II textbooks (each costs $84 and weighs 5. pounds), some Chemistry books (each costs $90 and weighs 4.4 pounds), and some Literature books (each costs $75 and weighs 6 pounds). How many of each type of book are in the box? A Number of Algebra II books C Number of Chemistry books L Number of Literature books System: Matrix Equation: Smart School Publishing Quantity: 17 books Weight: 84.4 pounds Value: $1,449 A 6 C 8 L Solutions: A + C + L 17 84A + 90C + 75L A + 4.4C + 6L A C L , TESCCC 0/8/1 page 5 of 5

85 Mixing It Up WARM UP Compute answers to each of the following mixture problems. A) Florence took a road trip to the beach, which was several hundred miles away. As she left, she had to drive 6 mph through the city for about half an hour. Once she hit the highway, she drove 68 mph for.5 hours. Then, as she approached the beach, the traffic was so heavy she could only go 0 mph for the last 0 minutes. How many hours was Florence on the road? How many miles did Florence travel? Over the course of her trip, what was Florence s average speed? B) When you buy hydrogen peroxide, it is really a solution diluted with water. One bottle comes in a % solution, and another comes in 5% strength. Suppose you mix 40 ml of the % solution with 80 ml of the 5% solution. How many milliliters of the mixture is actually pure hydrogen peroxide? What percent of this mixture is hydrogen peroxide? C) Donna always gets really high grades on tests and quizzes (95 s every time), but she only does homework half the time (so her homework grade is a 50). Donna s teacher has a policy where tests and quizzes make up only 0% of students grades and homework makes up the other 70%. Under this system, what would Donna s class grade be? A teacher down the hall uses a different policy, where tests and quizzes are 60% of the grade, and homework is only 40%. What would Donna s grade be if she had this teacher? 01, TESCCC 0/8/1 page 1 of 5

86 Mixing It Up For each situation, define variables and set up a system of equations that relates them. Then, solve the system using inverse or augmented matrix methods. 1) When traveling to see relatives, Duke drives part of the trip on the highway at 60 mph and the rest through cities at 40 mph. The 70-mile trip takes him a total of 5 hours. How much of this time does he spend on the highway? How much time through cities? Solutions: System: Matrix Equation: ) On the return trip, Duke encounters some road construction that slows him down. While he is still able to go 60 mph on the highway and 40 mph through cities, he can only go 0 mph past the construction. He spends twice the time passing the construction as he does on the highway, so this time the 70-mile trip takes 7.8 hours. How much of this time does he spend on the highway, in the city, and through road construction? Solutions: System: Matrix Equation: 01, TESCCC 0/8/1 page of 5

87 Mixing It Up Continue setting up and solving systems for these situations. ) In the school s chemistry lab, a certain acid is stored in two different dilutions. One bottle contains a weaker solution that is only 5% acid, but another has a stronger solution of 65% acid. Students must pour some from each container to make 60 ml of a solution that is 50% acid. How much from each container should be used? Solutions: System: Matrix Equation: 4) When Ebenezer won a $50,000 lottery jackpot, everyone started asking for donations. His brother Bob said, His sister Sally said, And his friend Fred said, I need a loan! I can pay you back in a year. I ll even pay you back with 4% interest! I need $0,000 more than Bob does, but I want to start my own business. And you ll earn more than 4% on this sweet deal. No way. You need to put your money in the stock market. It s doing great right now. Ebenezer split all his money among these three investments. Bob repaid his loan and interest, and the investment in Sally s business came back with 7% profit. Even though the money he put in the stock market lost 8.%, Ebenezer still came out with an extra $5,80. How much money did he place into each of these investments? Solutions: System: Matrix Equation: 01, TESCCC 0/8/1 page of 5

88 Mixing It Up Continue setting up and solving systems for these situations. 5) When extreme athletes compete at a motorcycle jumping competition, judges give them scores on a scale of 0-10 in three categories: use of fundamentals, degree of difficulty, and overall style. However, these categories are not weighted equally in determining the final score. Funda- Difficulty score Final Style mentals D. Howser R. Finn J. Hoban The table shows how the category scores for three riders determined their final rating. By what percent is each category weighted? Solutions: System: Matrix Equation: 6) At the lake, Curtis caught three fish. He wanted to weigh the fish individually, but his scale was broken and could only read weights between 5 and 10 pounds. The large and medium fish together weighed 7.8 pounds. The large and small fish weighed 7.0 pounds, and the small and medium fish were 5.6 pounds. How much did each fish weigh individually? What was the total weight of all three fish? Solutions: System: Matrix Equation: 01, TESCCC 0/8/1 page 4 of 5

89 Mixing It Up Continue setting up and solving systems for these situations. 7) On the way home from school, a family enters the drive-through at a fast food restaurant. Mom: I need root beers, junior burgers and one deluxe salad. Clerk: That will be $10.6. Please pull around. Lucy: Mom, I don t want a burger. I m a vegetarian now! Mom: OK sorry. Can I get root beers, 1 junior burger and deluxe salads, instead? Clerk: No problem, ma am. That brings your total to $11.. Mom: Oh no. I ve only got $11. Could you take off one of the root beers? Clerk Yes. Two root beers, 1 junior burger, and salads. That s $10.0. What is the cost for each item individually? Solutions: System: Matrix Equation: 8) A box in the textbook warehouse has the given label. It contains some Algebra II textbooks (each costs $84 and weighs 5. pounds), some Chemistry books (each costs $90 and weighs 4.4 pounds), and some Literature books (each costs $75 and weighs 6 pounds). How many of each type of book are in the box? Smart School Publishing Quantity: 17 books Weight: 84.4 pounds Value: $1,449 Solutions: System: Matrix Equation: 01, TESCCC 0/8/1 page 5 of 5

90 Systems Situations KEY For each situation, define variables and set up a system of equations. 1. Copies. A teacher has to make 159 copies of a review packet for the students in her classes. When she arrives in the workroom, she starts using both copy machines (call them X and Y). During the copying, machine X runs out of paper, so it only printed half as many packets as copier Y. How many copies does each machine print? Solution: 5 copies from Machine X, 106 copies from Machine Y Variables: System: x Number of copies made by Machine X x + y 159 (or, y 159 x) y Number of copies made by Machine Y x ( 1 )y (or, y x). Fast Food. A bunch of friends went to the Snack Shack for lunch. The first family ordered 4 hamburgers and 4 orders of fries for $9.00. The next family ordered only 1 hamburger and orders of fries for $. How much would each item cost individually? Solution: $1.50 for a hamburger, $0.75 for fries Variables: System: h The cost for one hamburger 4h + 4f 9 f The cost for one order of fries h + f. Exercise. Several times a week, Chuck goes to the gym to run and swim. When running, Chuck burns 5 calories per minute, and when he swims he burns 0 calories per minute. He has found a way to burn 70 calories after exercising for a total of minutes. How long does Chuck spend at each activity? Solution: 15 minutes of swimming, 8 minutes of running Variables: System: r Number of minutes running r + w w Number of minutes swimming 5r + 0w Coins. Donald has a bunch of nickels and dimes in his piggy bank. If there are 100 coins in the bank that make a total of $6.60 in change, how many of each type of coin does Donald have? Solution: 68 nickels and dimes Variables: System: n Number of nickels n + d 100 d Number of dimes 0.05n d 6.60 (or, 5n + 10d 660) 01, TESCCC 07/18/1 page 1 of

91 Systems Situations KEY 5. Golf. To play golf at Blackhawk Range, golfers must first pay a $75 membership fee, and must still pay $7 for every round of golf. At the Royal Estates Country Club Course, golfers only have to pay $5 per round, but their membership dues are $185. After how many rounds of golf would the total amount paid by a golfer be the same at both golf courses? Solution: After 55 rounds of golf, the total paid by golfers at both courses would be $460. Variables: System: r Number of rounds of golf p r p Total amount paid p r 6. Discs. There are 4 discs in Josh s entertainment center. However, some of them are music CDs and some of them are video DVDs. If the number of music CDs in Josh s collection is 5 less than three times the number of DVDs, how many of each type does he own? Solution: 1 DVDs and 1 music CDs Variables: System: m Number of music CDs m + v 4 v Number of video DVDs m v 5 7. Salesman. Carmen sells cars at a dealership that pays her a monthly salary of $500 plus a commission of $15 for every car she sells. A competing dealership has offered her a job that pays a greater monthly salary of $700, but only gives $100 commission on each car sold. How many cars would Carmen have to sell in a month to get the same pay from either dealership? Solution: Selling 8 cars in a month would provide the same pay ($500) at either car lot Variables: System: c Number of cars Carmen sells in a month m c m Total amount she gets paid that month m c 8. Circus. There are 18 performers in Barnum s Famous Flea Circus. However, none of them are actually fleas they are either spiders (with 8 legs) or bugs (with 6 legs). Mr. Barnum wants to provide every performer with tiny new shoes for the ends of each of their many legs. If this requires a total of 10 shoes, how many spiders and bugs are in the circus? Solution: 11 spiders and 7 bugs Variables: System: x Number of spiders in the Flea Circus x + y 18 y Number of bugs in the Flea Circus 8 x + 6 y 10 01, TESCCC 07/18/1 page of

92 Systems Situations KEY 9. Amused. Several families are in line to enter an amusement park. The first family pays $54 for two adult tickets and two child tickets. The second family pays $105 for adults and 5 children. What are the ticket prices for adults and for children? Solution: Adult tickets are $15, and children s tickets are $1 Variables: System: a Price of an adult ticket a + c 54 c Price of a children s ticket a + 5c Age. Samantha has an older brother. Right now, he is twice her age. But 8 years ago, her brother was times her age. How old is each person now? Solution: Now, Samantha is 16 and her brother is. Variables: System: s Samantha s current age b s b Her brother s current age b 8 (s 8) 11. Health. Sherri is considering joining a health club. To be a member of Roman s Gym, she would have to pay a sign-up fee of $49.99 and then $5 per month. Another club, Athens Fitness Center, charges less per month (only $18) but has a $16.99 sign-up fee. After how many months of membership would the total paid to each club be the same? Solution: After 11 months, Sherri would have paid $4.99 to either health club. Variables: System: m Number of months p m p Total amount paid p m 1. More Discs. After cleaning house, Josh took a box of 9 used discs to sell at the flea market. Some were computer games (which he sold for $ each), some were music CDs (which he sold for $ each), and the rest were video DVDs (which he sold for $4 each). If Josh sold everything (which included twice as many DVDs as computer games), and he made a total of $9, how many of each type of disc did he sell? Solution: 6 computer games, 11 music CDs, and 1 DVDs. Variables: System: c Number of computer games c + m + v 9 m Number of music CDs c + m + 4v 9 v Number of video DVDs v c 01, TESCCC 07/18/1 page of

93 Systems Situations For each situation, define variables and set up a system of equations. 1. Copies. A teacher has to make 159 copies of a review packet for the students in her classes. When she arrives in the workroom, she starts using both copy machines (call them X and Y). During the copying, machine X runs out of paper, so it only printed half as many packets as copier Y. How many copies does each machine print? Variables: x Number of copies made by Machine X System: y Number of copies made by Machine Y. Fast Food. A bunch of friends went to the Snack Shack for lunch. The first family ordered 4 hamburgers and 4 orders of fries for $9.00. The next family ordered only 1 hamburger and orders of fries for $. How much would each item cost individually? Variables: System: 4h + 4f 9 h + f. Exercise. Several times a week, Chuck goes to the gym to run and swim. When running, Chuck burns 5 calories per minute, and when he swims he burns 0 calories per minute. He has found a way to burn 70 calories after exercising for a total of minutes. How long does Chuck spend at each activity? Variables: System: 4. Coins. Donald has a bunch of nickels and dimes in his piggy bank. If there are 100 coins in the bank that make a total of $6.60 in change, how many of each type of coin does Donald have? Variables: System: 01, TESCCC 07/18/1 page 1 of

94 Systems Situations 5. Golf. To play golf at Blackhawk Range, golfers must first pay a $75 membership fee, and must still pay $7 for every round of golf. At the Royal Estates Country Club Course, golfers only have to pay $5 per round, but their membership dues are $185. After how many rounds of golf would the total amount paid by a golfer be the same at both golf courses? Variables: System: 6. Discs. There are 4 discs in Josh s entertainment center. However, some of them are music CDs and some of them are video DVDs. If the number of music CDs in Josh s collection is 5 less than three times the number of DVDs, how many of each type does he own? Variables: System: 7. Salesman. Carmen sells cars at a dealership that pays her a monthly salary of $500 plus a commission of $15 for every car she sells. A competing dealership has offered her a job that pays a greater monthly salary of $700, but only gives $100 commission on each car sold. How many cars would Carmen have to sell in a month to get the same pay from either dealership? Variables: System: 8. Circus. There are 18 performers in Barnum s Famous Flea Circus. However, none of them are actually fleas they are either spiders (with 8 legs) or bugs (with 6 legs). Mr. Barnum wants to provide every performer with tiny new shoes for the ends of each of their many legs. If this requires a total of 10 shoes, how many spiders and bugs are in the circus? Variables: System: 01, TESCCC 07/18/1 page of

95 Systems Situations 9. Amused. Several families are in line to enter an amusement park. The first family pays $54 for two adult tickets and two child tickets. The second family pays $105 for adults and 5 children. What are the ticket prices for adults and for children? Variables: System: 10. Age. Samantha has an older brother. Right now, he is twice her age. But 8 years ago, her brother was times her age. How old is each person now? Variables: System: 11. Health. Sherri is considering joining a health club. To be a member of Roman s Gym, she would have to pay a sign-up fee of $49.99 and then $5 per month. Another club, Athens Fitness Center, charges less per month (only $18) but has a $16.99 sign-up fee. After how many months of membership would the total paid to each club be the same? Variables: System: 1. More Discs. After cleaning house, Josh took a box of 9 used discs to sell at the flea market. Some were computer games (which he sold for $ each), some were music CDs (which he sold for $ each), and the rest were video DVDs (which he sold for $4 each). If Josh sold everything (which included twice as many DVDs as computer games), and he made a total of $9, how many of each type of disc did he sell? Variables: System: 01, TESCCC 07/18/1 page of

96 Graphing Inequalities KEY The symbols >, <,, tell you how to graph a linear inequality, when it is in slope-intercept form. Inequality Line Shading Illustration y > mx + b Dotted Above y < y ³ y mx + b Dotted Below mx + b Solid Above mx + b Solid Below Graph these inequalities. 1) y x- Slope: y-intercept: (0, -) ) y > - x + Slope: - y-intercept: (0, ) Line: Solid Line: Dotted Shade: Below Shade: Above Here, write the linear inequalities in slope-intercept form. Remember: When solving an inequality, If you multiply or divide by a negative number, you must switch (or reverse) the inequality sign. ) 10 x + 5y > 40 y >- x+ 8 4) x- y 1 y ³ - x 6 5) - x + 4y < -0 y < x 5 Slope: -, y-intercept: (0, 8) Line: Dotted, Shade: Above Slope:, y-intercept: (0, -6) Line: Solid, Shade: Above Slope: 1 4, y-intercept: (0, -5) Line: Dotted, Shade: Below 01, TESCCC 07/18/1 page 1 of 4

97 Graphing Inequalities KEY The solution to a system of inequalities is the region that would be included in every inequality. To graph 1. Lightly shade each individual inequality. (This shows your work. ) a solution:. Heavily shade the region where they overlap. (This is the solution.) y x-5 6) 1 y ³- x+ 7) x+ y < y x+ y ³- 8) x + y < 1 y x-1 x ³ 0 9) x y 4 y x 5 x y 10) What system of inequalities could be used to create this graph and solution? y -x + 1 y < x 1 01, TESCCC 07/18/1 page of 4

98 Graphing Inequalities KEY Graph these systems for additional practice. 11) 5x- y < 10 y <- x+ 4 1) y x + y > x+ 1 1 x+ y 10 1) x > y x+ y³- y<- x+ 4 14) y ³- x+ y ³ 4x+ 1 x- y > 8 No solution There is no region where all inequalities overlap! 01, TESCCC 07/18/1 page of 4

99 Graphing Inequalities KEY 15) Develop the system of inequalities that would make the given graph. System of Inequalities: y x7 y 0.5x 4.75 y 5x 11 y 1.75x4.5 16) Develop the system of inequalities that would make the given graph. System of Inequalities: y 7x4 y 5x 45 y x , TESCCC 07/18/1 page 4 of 4

100 Graphing Inequalities The symbols >, <,, tell you how to graph a linear inequality, when it is in slope-intercept form. Inequality Line Shading Illustration y > mx + b y < y ³ y mx + b mx + b mx + b Graph these inequalities. 1) y x- Slope: ) y >- x+ Slope: y-intercept: y-intercept: Line: Line: Shade: Shade: Here, write the linear inequalities in slope-intercept form. Remember: When solving an inequality, If you, you must. ) 10x+ 5y > 40 4) x- y 1 5) - x+ 4y <- 0 Slope:, y-intercept: Line:, Shade: Slope:, y-intercept: Line:, Shade: Slope:, y-intercept: Line:, Shade: 01, TESCCC 07/18/1 page 1 of 4

101 Graphing Inequalities The solution to a system of inequalities is the region that would be included in every inequality. To graph 1. Lightly shade each individual inequality. (This shows your work. ) a solution:. Heavily shade the region where they overlap. (This is the solution.) y x-5 6) 1 y ³- x+ 7) x+ y < y x+ y ³- 8) x+ y < 1 y x-1 x ³ 0 9) x y 4 y x5 x y 10) What system of inequalities could be used to create this graph and solution? 01, TESCCC 07/18/1 page of 4

102 Graphing Inequalities Graph these systems for additional practice. 11) 5x- y < 10 y x+ 1 y <- x+ 4 1) y > x+ 1 x+ y 10 1) x > y x+ y ³- y <- x+ 4 14) y ³- x+ y ³ 4x+ 1 x- y > 8 01, TESCCC 07/18/1 page of 4

103 Graphing Inequalities 15) Develop the system of inequalities that would make the given graph. System of Inequalities: 16) Develop the system of inequalities that would make the given graph. System of Inequalities: 01, TESCCC 07/18/1 page 4 of 4

104 Linear Programming Basics KEY What do you have? What must you do? How do you do it? You have a feasible region formed by graphing a system of inequalities. Under these restrictions, you must maximize or minimize some other function (cost, time, etc.) Use the vertices of the region. 1) y y x x 4 x 7 y Find the maximum and minimum values for C using C 8x 11y. Vertex C 8x 11y C (-6,1) 8(-6) + 11(1) -7 (-1,6) 8(-1) + 11(6) 58 (,-) 8() + 11(-) Maximum: C 58 Minimum: C -7 y x 4 1 ) y x x Find the maximum and minimum values for D using D x 7y. Vertex D x 7y D (-,8) (-) 7(8) -6 (-,-1) (-) 7(-1) 1 (4,-4) (4) 7(-4) 40 Maximum: D 40 Minimum: D -6 01, TESCCC 07/18/1 page 1 of

105 Linear Programming Basics KEY ) y x x y 6 x y 4 y 0 Find the minimum value for M using M 0x 16y. Vertex M 0x 16y M (,4) 0() + 16(4) 14 (,6) 0() + 16(6) 186 (1,0) 0(1) + 16(0) 60 (6,0) 0(6) + 16(0) 180 Minimum: M 14 4) y x 1 x y 18 x y 10 x y 0 0 Find the minimum value for T using T 19x 5y. Vertex T 19x 5y T (0,1) 19(0) + 5(1) 00 (,6) 19() + 5(6) 188 (4,) 19(4) + 5() 151 (10,0) 19(10) + 5(0) 190 Minimum: T , TESCCC 07/18/1 page of

106 Linear Programming Basics What do you have? What must you do? How do you do it? You have a formed by graphing a of. Under these restrictions, you must or some other function (cost, time, etc.) Use the of the region. 1) y y x x 4 x 7 y Find the maximum and minimum values for C using C 8x 11y. Vertex C 8x 11y C y x 4 1 ) y x x Find the maximum and minimum values for D using D x 7y. Vertex D x 7y D 01, TESCCC 07/18/1 page 1 of

107 Linear Programming Basics ) y x x y 6 x y 4 y 0 Find the minimum value for M using M 0x 16y. Vertex M 0x 16y M 4) y x 1 x y 18 x y 10 x y 0 0 Find the minimum value for T using T 19x 5y. Vertex T 19x 5y T 01, TESCCC 07/18/1 page of

108 Linear Programming Problems KEY 1) Military Rations A military unit is about to embark on an extended mission. The unit carries food in two types of servings power bars, which resemble little snack cakes, and mini-meals, which resemble small microwave dinners. These servings have different nutritional value, as shown in the chart. Carbohydrates (g) Calories The unit commanders are trying to plan how power bar many of each type of meal to carry on the mini-meal mission, according to the restrictions below. Each day, a soldier can have no more than 1 total servings. Each day, a soldier must consume at least 00 g of carbohydrates. Each day, a soldier must consume at least 015 calories. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the numbers of each type of serving a soldier can consume in a day. x number of power bars a soldier can eat in a day, y number of mini-meals a soldier can eat in a day. ì x+ y 1 ï í75x+ 15y ³ 00 ï ïî 155x+ 10y ³ 015 Or ì y - x+ 1 ï íy ³- 5x + 0 ï ïî y ³- 0.5x Vertices: (,5), (,10), (11,1) Cost for each: (,5) $.50 (,10) $40 (11,1) $17.50 The contracted food service charges the military $1.5 for each power bar and $.75 for each mini-meal. Determine the maximum and minimum costs for feeding 50 troops on a 0-day mission. Minimum cost: $17.50 per troop per day 50 troops 0 days $11,50 Maximum cost: $40.00 per troop per day 50 troops 0 days $00,000 01, TESCCC 07/18/1 page 1 of 4

109 Linear Programming Problems KEY ) Pecan Prices The Heart of Texas Sweets Shop makes pecan pies during the fall and winter months. To do so, the bakery purchases and processes native pecans from two distributors one in North Texas, and the other in South Texas. The owners are trying to figure out the most economical way to do this. They know that pies must be made under the following conditions. To meet demand, the shop must process at least 5 pounds of pecans each week. Because of limited labor and storage, the shop can process a maximum of 70 pounds a week. Each shipment of pecans contains a certain amount of waste product (such as shells and leaves) that will not be used. The shop owners want no more than 14 pounds per week of 1 such waste. They estimate that 6 of every pound of pecans from North Texas is waste, as 1 well as of each pound from South Texas. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the numbers of pounds of pecans from each distributor they can order in a week. x number of pounds of pecans from North Texas (in a week), y number pounds of pecans from South Texas (in a week). xy 70 x 0 x y 5 and x y y 0 14 Or y x70 x 0 y x 5 and y 1 0 y x 4 Vertices: (0,5), (0,4), (5,0), (56,14), and (70,0) Cost for each: (0,5) $98.00 (0,4) $ (5,0) $11.75 (56,14) $1.0 (70,0) $7.50 The North Texas distributor charges $.5/pound for pecans, and the South Texas distributor charges $.80. What are the maximum and minimum costs that Heart of Texas should budget each week for pecans? Maximum: $7.50, Minimum: $ , TESCCC 07/18/1 page of 4

110 Linear Programming Problems KEY ) Lab Staff The local university has received a grant to open up a laboratory for scientific research. The grant will fund the use of a certain amount of building space on the university grounds as well as salaried positions for each of the research team s scientists and assistants. However, the grant does have certain restrictions. The research team can only include up to 0 staff members, made up of full research scientists and assistants. The staff must include at least 4 scientists. The number of assistants must be at least half the number of scientists. Each scientist requires 400 square feet of workspace, and each assistant requires 100 square feet; but the university only has 7,00 square feet of space available. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the possible numbers of research scientists and assistants. x number of scientists, y number assistants. x y 0 x 4 1 y x 400x100y 700 Or y x0 x 4 1 y x y 4x 7 Vertices: (4,), (4,6), (14,16), and (16,8) Combined salaries for each: (4,) $8,000 (4,6) $94,000 (14,16) $1,094,000 (16,8) $95,000 The annual salary scale mandated by the grant is $45,000 for a scientist and $9,000 for an assistant. Determine the maximum and minimum amounts needed to budget for salaries for the research team. Maximum: $1,094,000, Minimum: $8,000 01, TESCCC 07/18/1 page of 4

111 Linear Programming Problems KEY 4) Farm Figures A farmer has 150 total acres on which to plant corn and cotton. Because of demand, the farmer knows to plant more corn than cotton (or at least the same amounts). Also, the farmer has budgeted $540 for seeds that cost $/acre for corn and $5/acre for cotton. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the possible numbers of acres on which to plant corn and cotton. x number of acres of corn, y number of acres of cotton. xy 150 x 0 x y and y 0 x5y 540 Or y x150 x 0 y x and y 0 y 5 x 108 Vertices: (0,0), (67.5, 67.5), (105,45) and (150,0) Combined profit for each: (0,0) $0 (67.5, 67.5) $1,500 (105, 45) $1,800 (150,0) $1,000 At harvest time, each acre of corn yields a profit of $80, and each acre of cotton yields a profit of $10. How many acres of each should the farmer plant in order to maximize profit? 105 acres of corn and 45 acres of cotton for a maximum of $1, , TESCCC 07/18/1 page 4 of 4

112 Linear Programming Problems 1) Military Rations A military unit is about to embark on an extended mission. The unit carries food in two types of servings power bars, which resemble little snack cakes, and mini-meals, which resemble small microwave dinners. These servings have different nutritional value, as shown in the chart. Carbohydrates (g) Calories The unit commanders are trying to plan how power bar many of each type of meal to carry on the mini-meal mission, according to the restrictions below. Each day, a soldier can have no more than 1 total servings. Each day, a soldier must consume at least 00 g of carbohydrates. Each day, a soldier must consume at least 015 calories. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the numbers of each type of serving a soldier can consume in a day. The contracted food service charges the military $1.5 for each power bar and $.75 for each mini-meal. Determine the maximum and minimum costs for feeding 50 troops on a 0-day mission. 01, TESCCC 04/7/1 page 1 of 4

113 Linear Programming Problems ) Pecan Prices The Heart of Texas Sweets Shop makes pecan pies during the fall and winter months. To do so, the bakery purchases and processes native pecans from two distributors one in North Texas, and the other in South Texas. The owners are trying to figure out the most economical way to do this. They know that pies must be made under the following conditions. To meet demand, the shop must process at least 5 pounds of pecans each week. Because of limited labor and storage, the shop can process a maximum of 70 pounds a week. Each shipment of pecans contains a certain amount of waste product (such as shells and leaves) that will not be used. The shop owners want no more than 14 pounds per week of such waste. They estimate that 6 1 of every pound of pecans from North Texas is waste, as well as 1 of each pound from South Texas. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the numbers of pounds of pecans from each distributor they can order in a week. The North Texas distributor charges $.5/pound for pecans, and the South Texas distributor charges $.80. What are the maximum and minimum costs that Heart of Texas should budget each week for pecans? 01, TESCCC 04/7/1 page of 4

114 Linear Programming Problems ) Lab Staff The local university has received a grant to open up a laboratory for scientific research. The grant will fund the use of a certain amount of building space on the university grounds as well as salaried positions for each of the research team s scientists and assistants. However, the grant does have certain restrictions. The research team can only include up to 0 staff members, made up of full research scientists and assistants. The staff must include at least 4 scientists. The number of assistants must be at least half the number of scientists. Each scientist requires 400 square feet of workspace, and each assistant requires 100 square feet; but the university only has 7,00 square feet of space available. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the possible numbers of research scientists and assistants. The annual salary scale mandated by the grant is $45,000 for a scientist and $9,000 for an assistant. Determine the maximum and minimum amounts needed to budget for salaries for the research team. 01, TESCCC 04/7/1 page of 4

115 Linear Programming Problems 4) Farm Figures A farmer has 150 total acres on which to plant corn and cotton. Because of demand, the farmer knows to plant more corn than cotton (or at least the same amounts). Also, the farmer has budgeted $540 for seeds that cost $/acre for corn and $5/acre for cotton. For this situation, define the variables and then write a system of inequalities to describe the constraints. Next, graph the feasible region for the possible numbers of acres on which to plant corn and cotton. At harvest time, each acre of corn yields a profit of $80, and each acre of cotton yields a profit of $10. How many acres of each should the farmer plant in order to maximize profit? 01, TESCCC 04/7/1 page 4 of 4

116 Evaluation Systems of Equations KEY Multiple choice questions C 1. Which of the following describes the solution to the given system of equations? A x 1, y C x 11, y 8 B x 1, y - D x 5, y x y 5 5x 6y 7 B. Which of the following ordered triples would be the solution to the given system of equations? A (10, 6, 4) C (4, 1, 1) B (, -, 1) D (1,, ) x y z 1 x y z x y z 4 A. The equations of two lines are given. What would be their point of intersection? A (, 0) C (1, ) x y y 4x 1 B (9, -) D (, ) A 4. Which of the following best describes the given system and its solution? x y 4x 1y 8 A Inconsistent (no solution) C Inconsistent (infinitely many solutions) B Consistent (no solution) D Consistent (infinitely many solutions) D 5. Samuel and Eduardo have collected a total of 75 stamps. Samuel has 50 stamps more than twice the number of stamps that Eduardo has. Which system of equations can be used to find s, the number of stamps that Samuel has, and e, the number of stamps that Eduardo has? s e 75 s e 75 A C s e50 e s50 B s e 75 e s50 D s e 75 s e50 01, TESCCC 07/18/1 page 1 of

117 Evaluation Systems of Equations KEY Solve each system using the method indicated. Show your work! 6) Use substitution to solve this system. 7) Use combination/elimination to solve this one. x y 5 x y Solution: (-1.4,.6) x y 4 x y 10 Solution: (, ) 8) Complete the tables for the given equations. x y 1 y y y x.9 1.1x.1 What is the solution to the system? ( 5,.4 ) 9) Graph each equation on the grid provided. x y y x 4 Solve each system using a method of your choice. Show your work! Name the solution: ( -, 5 ) 10) 7x 8y x y 1 11) x 15y 18 x 5y 4 1) x 4y 8 x 6y 10 Solution: (1, ) No solution Solution: (-4, -) 01, TESCCC 07/18/1 page of

118 Evaluation Systems of Equations KEY 1) In the textbook office is a box of new math books, some for Geometry and some for. A label on the box reads: Contents: 15 books, Weight: 61 lbs. You know that Geometry books weigh pounds each, and Algebra books weigh 5 pounds each. a) Let g the number of Geometry books in the box, and a the number of books in the box. Set up a system to find a and g. g + a 15 g + 5a 61 Geometry & CONTENTS: 15 books WEIGHT: 61 lbs. b) Solve the system. How many Geometry and books were in the box? Solution: There are 7 Geometry and 8 books in the box. 14) Two taxi services use different methods to determine the fare (or charge) customers pay when using their drivers. Terri s charges $ plus $1.50 per mile, and Rick s costs $.5 plus $1.60 per mile. a) Write two equations to represent each company s total fee (y) in terms of the number of miles (x). y x y x b) Compute the charges for riding 1, 5, and 10 miles with each taxi service. miles Terri s Taxi Rick s Rides 1 $4.50 $.85 5 $10.50 $ $18.00 $18.5 c) For what mileage is the cost the same with both companies? In this instance, what is the fee? Solution: After 7.5 miles, the cost is the same at both places ($14.5) 01, TESCCC 07/18/1 page of

119 Evaluation Systems of Equations PI Multiple choice questions 1. Which of the following describes the solution to the given system of equations? A x 1, y C x 11, y 8 B x 1, y - D x 5, y x y-5 5x- 6y 7. Which of the following ordered triples would be the solution to the given system of equations? A (10, 6, 4) C (4, 1, 1) B (, -, 1) D (1,, ) x y z 1 x y z x y z 4. The equations of two lines are given. What would be their point of intersection? A (, 0) C (1, ) B (9, -) D (, ) x+ y y 4x-1 4. Which of the following best describes the given system and its solution? x y + - 4x+ 1y 8 A Inconsistent (no solution) C Inconsistent (infinitely many solutions) B Consistent (no solution) D Consistent (infinitely many solutions) 5. Samuel and Eduardo have collected a total of 75 stamps. Samuel has 50 stamps more than twice the number of stamps that Eduardo has. Which system of equations can be used to find s, the number of stamps that Samuel has, and e, the number of stamps that Eduardo has? s e 75 s e 75 A C s e50 e s 50 B s e 75 e s 50 D s e 75 s e50 01, TESCCC 07/18/1 page 1 of

120 Evaluation Systems of Equations PI Solve each system using the method indicated. Show your work! 6) Use substitution to solve this system. 7) Use combination/elimination to solve this one. x y - 5 x+ y x- y 4 x+ y 10 8) Complete the tables for the given equations. x y 1 y y1-0.x y 1.1x-.1 1 What is the solution to the system? (, ) 7 9) Graph each equation on the grid provided. x+ y y -x- 4 Solve each system using a method of your choice. Show your work! Name the solution: (, ) 10) 7x 8y x y 1 11) x+ 15y 18 x+ 5y -4 1) x 4y + 8 x- 6y 10 01, TESCCC 07/18/1 page of

121 Evaluation Systems of Equations PI 1) In the textbook office is a box of new math books, some for Geometry and some for. A label on the box reads: Contents: 15 books, Weight: 61 lbs. You know that Geometry books weigh pounds each, and Algebra books weigh 5 pounds each. a) Let g the number of Geometry books in the box, and a the number of books in the box. Set up a system to find a and g. Geometry & CONTENTS: 15 books WEIGHT: 61 lbs. b) Solve the system. How many Geometry and books were in the box? 14) Two taxi services use different methods to determine the fare (or charge) customers pay when using their drivers. Terri s charges $ plus $1.50 per mile, and Rick s costs $.5 plus $1.60 per mile. a) Write two equations to represent each company s total fee (y) in terms of the number of miles (x). b) Compute the charges for riding 1, 5, and 10 miles with each taxi service. c) For what mileage is the cost the same with both companies? In this instance, what is the fee? 01, TESCCC 07/18/1 page of

122 Evaluation Systems of Inequalities KEY Multiple choice questions D 1) Which of the following ordered pairs would be included in the solution to the given system of inequalities? Point P (1,1) x y 5 Point Q (, -) System: y x Point R (-1, -7) A. Point Q only B. Point R only C. Points P and Q D. Points Q and R E. Points P and R C ) Which of the following ordered pairs would be included in the solution to the given system of inequalities? A. (, 7) B. (-, -) C. (-1, 7) D. (, 4) System: x y x y y 10 x 5 C ) If g the number of girls in our class, and b the number of boys, then the inequality b g can best be interpreted by which of the following sentences? 1 A. The number of boys is at most half the number of girls. B. The number of boys is at least as much as the number of girls. C. The number of girls is at most twice the number of boys. D. The number of girls is less than or equal to the number of boys. A 4) Which gives the best description of how to graph the inequality x y? A. The y-intercept is (0,-10), and the slope is 1.5. Shade above the solid line. B. The y-intercept is (0,-10), and the slope is Shade below the solid line. C. The y-intercept is (0,-10), and the slope is 1.5. Shade below the solid line. D. The y-intercept is (0,8), and the slope is 0.8. Shade above the solid line. E. The y-intercept is (0,8), and the slope is 0.8. Shade below the solid line. F. The y-intercept is (0,-10), and the slope is 1.5. Shade above the dotted line. G. The y-intercept is (0,-10), and the slope is Shade below the dotted line. H. The y-intercept is (0,-10), and the slope is 1.5. Shade below the dotted line. I. The y-intercept is (0,8), and the slope is 0.8. Shade above the dotted line. J. The y-intercept is (0,8), and the slope is 0.8. Shade below the dotted line. 01, TESCCC 07/18/1 page 1 of

123 Evaluation Systems of Inequalities KEY 5) On the graph provided, shade the solution to the system of inequalities. x y 8 x y 8 6) Write the system of inequalities that would have the given solution. y < y x 1 y -x 5 7) Graph the feasible region given by the system x y 7 x y 16 x y Find values for M using M 10x 8y. Vertex M 10x 8y M (,5) 10() + 8(5) 60 (,7) 10() + 8(7) 76 (4,) 10(4) + 8() 64 (10,) 10(10) + 8() 14 What is the minimum value for M? 60 01, TESCCC 07/18/1 page of

124 Evaluation Systems of Inequalities KEY 8) Volunteers The local television station is hosting a telethon for a local children s charity. They have asked students and sponsors from high school clubs and organizations to volunteer to answer the phones. When people call in to pledge money or make a donation, the volunteers must take each person s name, address, phone number, and pledge and enter the information into a computer. However, the TV station does have some guidelines: The station needs at least 0 total volunteers, which includes students as well as their adult sponsors. There must be at least one adult sponsor for every students who volunteer. The group of volunteers must be able to process at least 10 calls per hour. From previous telethons, the station estimates that each high school student can process 9 calls per hour, but adult sponsors usually only handle about 4 calls per hour. For this situation, let x number of students, and y number of adult sponsors who volunteer. Write a system of inequalities to describe the constraints. Then, graph the feasible region. Inequalities: x y 0 y 1 x 9x4y 10 x 0 y 0 What are the vertices of the feasible region? Vertices: (0, 0) (8, 1) (15, 5) To show appreciation for their volunteers, the TV station usually provides pizza. They order pieces for every student and pieces for each adult sponsor. What is the minimum number of pieces of pizza the station could order? Minimum: 48 pieces of pizza (from 8 students and 1 adult sponsors) 01, TESCCC 07/18/1 page of

125 Evaluation Systems of Inequalities PI Multiple choice questions 1) Which of the following ordered pairs would be included in the solution to the given system of inequalities? Point P (1,1) ì Point Q (, -) System: ïx+ y < 5 í ï y x- Point R (-1, -7) ïî A. Point Q only B. Point R only C. Points P and Q D. Points Q and R E. Points P and R ) Which of the following ordered pairs would be included in the solution to the given system of inequalities? A. (, 7) B. (-, -) C. (-1, 7) D. (, 4) System: ì x+ y < 10 ïy > x- í x ³-5 ï ïî y ³ ) If g the number of girls in our class, and b the number of boys, then the inequality 1 b³ g can best be interpreted by which of the following sentences? A. The number of boys is at most half the number of girls. B. The number of boys is at least as much as the number of girls. C. The number of girls is at most twice the number of boys. D. The number of girls is less than or equal to the number of boys. 4) Which gives the best description of how to graph the inequality 5x- 4y 40? A. The y-intercept is (0,-10), and the slope is 1.5. Shade above the solid line. B. The y-intercept is (0,-10), and the slope is Shade below the solid line. C. The y-intercept is (0,-10), and the slope is 1.5. Shade below the solid line. D. The y-intercept is (0,8), and the slope is 0.8. Shade above the solid line. E. The y-intercept is (0,8), and the slope is 0.8. Shade below the solid line. F. The y-intercept is (0,-10), and the slope is 1.5. Shade above the dotted line. G. The y-intercept is (0,-10), and the slope is Shade below the dotted line. H. The y-intercept is (0,-10), and the slope is 1.5. Shade below the dotted line. I. The y-intercept is (0,8), and the slope is 0.8. Shade above the dotted line. J. The y-intercept is (0,8), and the slope is 0.8. Shade below the dotted line. 01, TESCCC 07/18/1 page 1 of

126 Evaluation Systems of Inequalities PI 5) On the graph provided, shade the solution to the system of inequalities. x+ y < 8 x- y 8 6) Write the system of inequalities that would have the given solution. 7) Graph the feasible region given by the system ì x+ y ³ 7 ïx+ y 16 í x ³ ï ïî y ³ Find values for M usingm 10x+ 8y. Vertex M 10x+ 8y M What is the minimum value for M? 01, TESCCC 07/18/1 page of

127 Evaluation: Systems of Inequalities PI 8) Volunteers The local television station is hosting a telethon for a local children s charity. They have asked students and sponsors from high school clubs and organizations to volunteer to answer the phones. When people call in to pledge money or make a donation, the volunteers must take each person s name, address, phone number, and pledge and enter the information into a computer. However, the TV station does have some guidelines: The station needs at least 0 total volunteers, which includes students as well as their adult sponsors. There must be at least one adult sponsor for every students who volunteer. The group of volunteers must be able to process at least 10 calls per hour. From previous telethons, the station estimates that each high school student can process 9 calls per hour, but adult sponsors usually only handle about 4 calls per hour. For this situation, let x number of students, and y number of adult sponsors who volunteer. Write a system of inequalities to describe the constraints. Then, graph the feasible region. Inequalities: What are the vertices of the feasible region? To show appreciation for their volunteers, the TV station usually provides pizza. They order pieces for every student and pieces for each adult sponsor. What is the minimum number of pieces of pizza the station could order? 01, TESCCC 07/18/1 page of

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