Mathematics Grade 8 Part 1

Teacher’s Note and Reminders 2. 3x ≥ 12 – 6y a. (1, -1) b. (4, 0) c. (6, 3) d. (0, 5) e. (-2, 8) 3. 3y ≥ 2x – 6 5. 2x + y > 3 a. (0, 0) b. (3, -4) c. (0, -2) d. (-9, -1) e. (-5, 6) Don’t 4. -4y < 2x - 12Forget! a. (2, 4) b. (-4, 5) c. (-2, -2) d. (8.2, 5.5) e. (4, 1 ) 2 250

Teacher’s Note and Reminders 5. 2x + y > 3 a. (1 1 , 0) 2 b. (7, 1) c. (0, 0) d. (2, -12) e. (-10, -8) QU ?E S T I ONS a. How did you determine if the given coordinates of points on the graph satisfy the inequality? b. What did you do to justify your answer? Don’t Were you able to determine if the given coordinates of points on the graph satisfy Forget! the inequality? In the next activity, you will shade the part of the plane divider where the solutions of the inequality are found. Activity 10 COLOR ME! Directions: Shade the part of the plane divider where the solutions of the inequality is found. 1. y < x + 3 2. y – x > – 5 Answer KeyActivity 101. right side of the plane divider2. left side of the plane divider3. left side of the plane divider4. right side of the plane divider5. left side of the plane divider 251

Teacher’s Note and Reminders 3. x ≤ y – 4 5. 2x + y < 2 4. x + y ≥ 1 QU ?E S T I ONS a. How did you determine the part of the plane to be shaded? b. Suppose a point is located on the plane where the graph of a linear inequality is drawn. How do you know if the coordinates of this point is a solution of the inequality? c. Give at least 5 solutions for each linear inequality. Don’t From the activity done, you were able to shade the part of the plane divider whereForget! the solutions of the inequality are found. In the next activity, you will draw and describe the graph of linear inequalities. 252

Ask the students to draw and describe the graphs of linear inequalities. Let Activity 11 GRAPH AND TELL…them perform Activity 11. Emphasize that one of the half-planes contain thesolutions of the linear inequality. Use solid line if the symbol ≥ or ≤ is used Directions: Show the graph and describe the solutions of each of the following inequalities.and broken line if the symbol used is > or <. If math software like GeoGebra Use the Cartesian coordinate plane below.is available, ask the students to make use of this. GeoGebra is a dynamicmathematics software that can be used to visualize and understand concepts 1. y > 4x in algebra, geometry, calculus, and statistics. 2. y > x + 2 Answer KeyActivity 111. 4. 3. 3x + y ≤ 5 4. y < 1 x 3 5. x – y < -22. 5. QU ?E S T I ONS a. How did you graph each of the linear inequalities? b. How do you describe the graphs of linear inequalities in two variables? c. Give at least 3 solutions for each linear inequality. d. How did you determine the solutions of the linear inequalities?3. Were you able to draw and describe the graph of linear inequalities? Were you able to give at least 3 solutions for each linear inequality? In the next activity, you will determine the linear inequality whose graph is described by the shaded region. 253

Let the students determine the linear inequality whose graph is described Activity 12 NAME THAT GRAPH!by the shaded region. Ask them to perform Activity 12. Encourage them touse different ways of finding the linear inequality. In this activity, one possible Directions: Write a linear inequality whose graph is described by the shaded region.error that students might commit is the wrong use of inequality symbol. 1. 4.Let them check their own errors by testing some ordered pairs against theinequality they have formulated. Emphasize to them also the meanings ofthe broken and solid lines.Answer Key 2. 5. Activity 121. y > 2x + 32. x + 3y ≤ 13. y < 2x + 24. y + x ≥ 45. 5 < 3x + y Teacher’s Note and Reminders Don’t 3.Forget! 254

Teacher’s Note and Reminders ?E S T I O QU NS a. How did you determine the linear inequality given its graph? b. What mathematics concepts or principles did you apply to come up with the inequality? c. When will you use the symbol >, <, ≥, or ≤ in a linear inequality? From the activity done, you were able to determine the linear inequality whose graph is described by the shaded region. In the succeeding activity, you will translate real-life situations into linear inequalities in two variables. Don’t Activity 13 TRANSLATE ME! Forget! Directions: Write each statement as linear inequality in two variables.In Activity 13, let the students translate real-life situations into linearinequalities in two variables. Give emphasis on the meanings of the phrases 1. The sum of 20-peso bills (t) and fifty peso bills (f) is greater than Php 420.“less than”, “more than”, “greater than”, “at most” and “at least”. Let thestudents differentiate also “less than” and “is less than” and “more than” 2. The difference between the weight of Diana (d) and Princess (p) is atand “is more than”. Provide examples on how these are used for students to least 26.understand their differences. 3. Five times the length of a ruler (r) increased by two inches is less than theAnswer Key height of Daniel (h).Activity 13 6. 12s +6p ≤ 960 4. In a month, the total amount the family spends for food (f) and educational1. t + f > 420 7. p – q ≥ 30 expenses (e) is at most Php 8, 000.2. d – p ≥ 26 8. 3r < b3. 5r + 2 < h 9. 2p + a > 24 5. The price of a motorcycle (m) less Php 36,000 is less than or equal to the4. f + e ≤ 8000 10. 2b + 3s ≤ 1150 price of a bicycle (b).5. m – 36 000 ≤ b 6. A dozen of short pants (s) added to half a dozen of pajamas (p) has a total cost of not greater than Php 960. 7. The difference of the number of 300-peso tickets (p) and 200-peso tickets (q) is not less than 30. 8. Thrice the number of red balls (r) is less than the number of blue balls (b). 9. The number of apples (a) more than twice the number of ponkans (p) is greater than 24. 10. Nicole bought 2 blouses (b) and 3 shirts (s) and paid not more than Php 1,150. 255

Let students broaden their understanding of linear inequalities in two QU ?E S T I ONS a. How did you translate the given situations into linear inequalities?variables as to how they are used in solving real-life problems. Ask them to b. When do we use the term “at most”? How about “at least”?perform Activity 14. Encourage them to use different ways of arriving at the c. What other terms are similar to “at most”? How about “at least”?solutions to the problems. More importantly, provide them the opportunities d. Give at least two statements that make use of these terms.to choose the most convenient way of solving each problem. e. In what real-life situations are the terms such as “at most” and “at least” used?Answer Key Were you able to translate real-life situations into linear inequalities in two variables? In the next activity, you will find out how linear inequalities in two variablesActivity 14 are used in real-life situations and in solving problems.1. a. c – j ≤ 1.5; c represents Connie’s height and j Janel’s height b. Connie Activity 14 MAKE IT REAL! c. 3 ft and 9 inches and below Directions: Answer the following questions. Give your Complete solutions or explanations.2. a. y ≥4 - x 20 1. The difference between Connie’s height and Janel’s height is not more than 1.5 ft. b. about 2 liters a. What mathematical statement represents the difference in heights of Connie and Janel? Define the variables used. c. yes b. Based on the mathematical statement you have given, who is taller? Why?3. a. 5x + 2y < 400 c. Suppose Connie’s height is 5 ft and 3 in, what could be the height of b. Php 109 Janel? Explain your answer. c. Php 116 2. A motorcycle has a reserved fuel of 0.5 liter which can be used if its 3-liter4. a. x + y ≤ 270 fuel tank is about to be emptied. The motorcycle consumes at most 0.5 liters b. possible answers: car = 65 km/hr and bus = 55 km/hr car = 70 km/hr of fuel for every 20 km of travel. and bus = 65 km/hr a. What mathematical statement represents the amount of fuel that would be left in the motorcycle’s fuel tank after travelling a certain c. 65 km/hr distance if its tank is full at the start of travel? d. possible b. Suppose the motorcycle’s tank is full and it travels a distance of 55 km, e. not possible about how much fuel would be left in its tank? c. If the motorcycle travels a distance of 130 km with its tank full, is the Teacher’s Note and Reminders amount of fuel in its tank be enough to cover the given distance? Explain your answer. Don’t Forget! 3. The total amount Jurene paid for 5 kilos of rice and 2 kilos of fish is less than Php 600. a. What mathematical statement represents the total amount Jurene paid? Define the variables used. b. Suppose a kilo of rice costs Php 35. What could be the greatest cost of a kilo of fish to the nearest pesos? c. Suppose Jurene paid more than Php 600 and each kilo of rice costs Php 34. What could be the least amount she will pay for 2 kilos of fish to the nearest pesos? 256

Teacher’s Note and Reminders 4. A bus and a car left a place at the same time traveling in opposite directions. After two hours, the distance between them is at most 350 km. Don’t a. What mathematical statement represents the distance between the Forget! two vehicles after two hours? Define the variables used. b. What could be the average speed of each vehicle in kilometers perWWhhaatt ttoo UUnnddeerrssttaanndd hour? c. If the car travels at a speed of 70 kilometers per hour, what could be Have students take a closer look at some aspects of linear inequality in two variables the maximum speed of the bus? and their graphs. Provide them opportunities to think deeper and test further their d. If the bus travels at a speed of 70 kilometers per hour, is it possible understanding of the lesson by doing Activity 15. that the car’s speed is 60 kilometers per hour? Explain or justify your answer. Answer Key e. If the car’s speed is 65 kilometers per hour, is it possible that the Activity 15 bus’ speed is 75 kilometers per hour? Explain or justify your answer. 1. Linear inequalities in two variables are inequality that can be written in From the activity done, you were able to find out how linear inequalities in two one of the following forms: Ax + By < C, Ax + By ≤ C, Ax + By > C and variables are used in real-life situations and in solving problems. Can you give other Ax + By ≥ C while linear equations in two variables are mathematical real-life situations where linear inequalities in two variables are illustrated? Now, let’s go statements indicating that two expressions are equal and using the deeper by moving on to the next part of this module. symbol “=” 2. Infinite/many WWhhaatt ttoo UUnnddeerrssttaanndd 3. No 4. Maybe the amount of those canned goods she is buying is higher than In this part, you are going to think deeper and test further your understanding what she is expecting. of linear inequalities in two variables. After doing the following activities, you should x + y < 200 be able to answer the question: In what other real-life situations will you be able 5. a. Possible answers: 6m by 4m; 8m by 3m; 12m by 2m to find the applications of linear inequalities in two variables? b. Yes c. 2l + 2w = 20; 2l + 2w = 22; 2l + 2w = 28 Activity 15 THINK DEEPER…. Directions: Answer the following questions. Give your Complete solutions or explanations. 1. How do you differentiate linear inequalities in two variables from linear equations in two variables? 2. How many values of the variables would satisfy a given linear inequality in two variables? Give an example to support your answer. 3. Airen says any values of x and y satisfying the linear equation y = x + 5 also satisfy the inequality y < x + 5. Do you agree with Airen? Justify your answer. 4. Katherine bought some cans of sardines and corned beef. She gave the store owner Php 200 as payment. However, the owner told her that the amount is not enough. What could be the reasons? What mathematical statement would represent the given situation? 257

Before the students move to the next section of this lesson, give a short test 5. Jay is preparing a 24-m2 rectangular garden in a 64-m2 vacant square lot. (formative test) to find out how well they understood the lesson. a. What could be the dimensions of the garden? b. Is it possible for Jay to prepare a 2 m by 12 m garden? Why?WWhhaatt ttooTTrraannssffeerr c. What mathematical statement would represent the possible perimeter of the garden? Explain your answer. Give the students opportunities to demonstrate their understanding of linear inequalities in two variables by doing some practical tasks. Let them perform Activities QU ?E S T I ONS What new insights do you have about linear inequalities in two 16 and 17. You can ask the students to work individually or in group. Emphasize that variables? What new connections have you made for yourself? they must come up with some linear inequalities in two variables. Before giving the activity, present first how to make a budget proposal including its parts. Moreover, Now extend your understanding. This time, apply what you have students must be given the opportunity to solve the problems they have formulated. learned in real life by doing the tasks in the next section.Teacher’s Note and Reminders WWhhaatt ttooTTrraannssffeerr In this section, you will be applying your understanding of linear inequalities in two variables through the following culminating activities that reflect meaningful and relevant situations. You will be given practical tasks where in you will demonstrate your understanding. Activity 16 LET’S ROLE-PLAY! Directions: Cite and role-play at least two situations in real-life where linear inequalities in two variables are illustrated. Formulate problems out of these situations then solve them. Show the graphs of the linear inequalities drawn from these situations. RUBRIC: Real-life Situations on Linear Inequalities in Two Variables 4321 Don’t The situation is clear, The situation is The situation is not The situation is notForget! realistic and the use clear and the use of too clear and the use clear and the use of of linear inequalities linear inequalities in of linear inequalities linear inequalities in in two variables and two variables is not in two variables is not two variables is not other mathematical illustrated. The problem illustrated. The problem illustrated. The problem statements are formulated is related formulated is related formulated is not properly illustrated. The to the situation and the to the situation and the related to the situation problem formulated is answer is correct. answer is incorrect. and the answer is relevant to the given incorrect. situation and the answer is accurate. 258

Teacher’s Note and Reminders Activity 17 PLAN FIRST! Don’t Directions: Read the situation below then come up with the appropriate budget proposal. Forget! The budget proposal should be clear, realistic, and make use of linear inequalities in two variables and other mathematical statements. Due to the rising prices of food commodities, you decided to raise broiler chickens for your family’s consumption. You sought permission from your parents and asked them to give you some amount to start with. Your parents agreed to give you some money; however, they still need to see how you will use it. They asked you to prepare a budget proposal for the chicken house that you will be constructing, the number of chickens to be raised, the amount of chicken feeds, and other expenses. RUBRIC: Budget Proposal of Raising Broiler Chickens 4321 The budget The budget The budget The budget proposal is proposal is clear, proposal is not proposal is not clear, accurate, practical and too clear and clear and the practical, and the use of linear the use of linear use of linear the use of linear inequalities in inequalities in inequalities in two inequalities in two variables is two variables variables is not two variables illustrated. is not properly illustrated. and other illustrated. mathematical statements are properly illustrated. How did you find the different performance tasks? How did the tasks help you see the real world use of linear inequalities in two variables? You have completed this lesson. Before you go to the next lesson on system of linear equation and inequalities, you have to answer the following post-assessment. This module was about linear inequalities in two variables. In this module, you were able to differentiate between mathematical expressions and mathematical equations, differentiate between mathematical equations and inequalities, illustrate linear inequalities in two variables, and graph linear inequalities in two variables on the coordinate plane and solve real-life problems involving linear inequalities in two variables. More importantly, you were given the chance to formulate and solve real-life problems, and demonstrate your understanding of the lesson by doing some practical tasks. 259

Summary/Synthesis/Generalization: SUMMARYThis module was about linear inequalities in two variables. In this This module was about linear inequalities in two variables. In this module, you weremodule, you were able to differentiate between mathematical able to differentiate between mathematical expressions and mathematical equations,expressions and mathematical equations; differentiate between differentiate between mathematical equations and inequalities, illustrate linear inequalitiesmathematical equations and inequalities; illustrate linear in two variables, and graph linear inequalities in two variables on the coordinate plane andinequalities in two variables; graph linear inequalities in two solve real-life problems involving linear inequalities in two variables. More importantly, youvariables on the coordinate plane; and solve real-life problems were given the chance to formulate and solve real-life problems, and demonstrate yourinvolving linear inequalities in two variables. More importantly, understanding of the lesson by doing some practical tasks.you were given the chance to formulate and solve real-lifeproblems, and demonstrate your understanding of the lesson by GLOSSARY OF TERMS USED IN THIS LESSON:doing some practical tasks. 1. Cartesian coordinate plane – the plane that contains the x- and y-axesREFERENCES: 2. Coordinates of a point – any point on the plane that is identified by an ordered pairBennett, Jeannie M., David J. Chard, Audrey Jackson, Jim of numbers denoted as (x, y)Milgram, Janet K. Scheer, and Bert K. Waits. Holt Pre-Algebra, 3. Geogebra – a dynamic mathematics software that can be used to visualize andHolt, Rinehart and Winston, USA, 2005. understand concepts in algebra, geometry, calculus, and statistics.Bernabe, Julieta G. and Cecile M. De Leon. Alementary Agebra, 4. Half plane – the region that is divided when a line is graphed in the coordinate planeTextbook for First Year, JTW Corporation, Quezon City, 2002. 5. Linear equation in two variables/mathematical equation – a mathematical statementBrown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey and indicating that two expressions are equal and using the symbol “=”William L. Cole. Algebra, Structure and Method, Book I, Houghton 6. Linear inequality in two variables – a mathematical statement that makes use ofMifflin Company, Boston MA, 1990. inequality symbols such as >, <, ≥, ≤ and ≠Brown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey, 7. Mathematical expression – the left or the right member of any mathematicaland Robert B. Kane. Algebra, Structure and Method Book 2.Houghton Mifflin Company, Boston, 1990. statement 8. Plane divider – the line that separates the cartesian coordinate plane into two half planes 9. Slope of a line – the steepness of a non-vertical line 10. Solutions of linear equations – points in the coordinate plane whose ordered pairs satisfy the equality 11. Solutions of linear inequalities – points in the coordinate plane whose ordered pairs satisfy the inequality 12. Variables – any quantity represented by a letter of the alphabet 13. x-intercept – the x-coordinate of the point where a graph intersects the x-axis REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: REFERENCES: Bennett, Jeannie M., David J. Chard, Audrey Jackson, Jim Milgram, Janet K. Scheer, and Bert K. Waits. Holt Pre-Algebra, Holt, Rinehart and Winston, USA, 2005.Callanta, Melvin M. and Concepcion S. Ternida. Infinity Grade 8, Bernabe, Julieta G. and Cecile M. De Leon. Alementary Agebra, Textbook for First Year,Worktext in Mathematics. EUREKA Scholastic Publishing, Inc., JTW Corporation, Quezon City, 2002.Makati City, 2012. Brown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey and William L. Cole. Algebra, 260

Chapin, Illingworth, Landau, Masingila and McCracken. Prentice Structure and Method, Book I, Houghton Mifflin Company, Boston MA, 1990.Hall Middle Grades Math, Tools for Success, Prentice-Hall, Inc.,Upper Saddle River, New Jersey, 1997. Brown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey, and Robert B. Kane. Algebra, Structure and Method Book 2. Houghton Mifflin Company, Boston, 1990.Clements, Douglas H., Kenneth W. Jones, Lois Gordon Moseley Callanta, Melvin M. and Concepcion S. Ternida. Infinity Grade 8, Worktext in Mathematics.and Linda Schulman. Math in my World, McGraw-Hill Division, EUREKA Scholastic Publishing, Inc., Makati City, 2012.Farmington, New York, 1999. Chapin, Illingworth, Landau, Masingila and McCracken. Prentice Hall Middle GradesCoxford, Arthur F. and Joseph N. Payne. HBJ Algebra I, Second Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997.Edition, Harcourt Brace Jovanovich, Publishers, Orlando,Florida, 1990. Clements, Douglas H., Kenneth W. Jones, Lois Gordon Moseley and Linda Schulman. Math in my World, McGraw-Hill Division, Farmington, New York, 1999.Fair, Jan and Sadie C. Bragg. Prentice Hall Algebra I, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1991. Coxford, Arthur F. and Joseph N. Payne. HBJ Algebra I, Second Edition, Harcourt Brace Jovanovich, Publishers, Orlando, Florida, 1990. Fair, Jan and Sadie C. Bragg. Prentice Hall Algebra I, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1991.Gantert, Ann Xavier. Algebra 2 and Trigonometry. AMSCO Gantert, Ann Xavier. Algebra 2 and Trigonometry. AMSCO School Publications, Inc., 2009.School Publications, Inc., 2009. Gantert, Ann Xavier. AMSCO’s Integrated Algebra I, AMSCO School Publications, Inc.,Gantert, Ann Xavier. AMSCO’s Integrated Algebra I, AMSCO New York, 2007.School Publications, Inc., New York, 2007. Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 1, Applications, Equations, and Graphs. McDougal Littell, A Houghton Mifflin Company, Illinois, 2004.Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 2, Applications,Algebra 1, Applications, Equations, and Graphs. McDougal Equations, and Graphs. McDougal Littell, A Houghton Mifflin Company, Illinois, 2008.Littell, A Houghton Mifflin Company, Illinois, 2004. Smith, Charles, Dossey, Keedy and Bettinger. Addison-Wesley Algebra, Addison-Wesley Publishing Company, 1992.Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Wesner, Terry H. and Harry L. Nustad. Elementary Algebra with Applications. Wm. C.Algebra 2, Applications, Equations, and Graphs. McDougal Brown Publishers. IA, USA.Littell, A Houghton Mifflin Company, Illinois, 2008. Wilson, Patricia S., et. al. Mathematics, Applications and Connections, Course I, GlencoeSmith, Charles, Dossey, Keedy and Bettinger. Addison-Wesley Division of Macmillan/McGraw-Hill Publishing Company, Westerville, Ohio, 1993.Algebra, Addison-Wesley Publishing Company, 1992.Wesner, Terry H. and Harry L. Nustad. Elementary Algebra withApplications. Wm. C. Brown Publishers. IA, USA. 261

WEBSITE Links: WEBSITE Links as References and for Learning Activities: 1. http://algebralab.org/studyaids/studyaid.aspx?file=Algebra2_2-6.xml http://www.google.com.ph/imgres?q=budgeting&hl=fil&client=firefox- 2. http://edhelper.com/LinearEquations.htm a&tbo=d&rls=org.mozilla:en-US:official&channel=np&biw=1024&bih=497& 3. http://www.kgsepg.com/project-id/6565-inequalities-two-variables tbm=isch&tbnid=KVtCh7CW_sgkgM:&imgrefurl=http://www.lmnblog.com/ 4. http://library.thinkquest.org/20991/alg /systems.html lmn/2011/why-budgeting-is-the-answer-to-better-business-management-and- 5. http://math.tutorvista.com/algebra/linear-equations-in-two-variables.html better-productivity/&docid=mKdzgJNUMrdLxM&imgurl=http://www.lmnblog. 6. https://sites.google.com/site/savannaholive/mathed-308/algebra1 com/lmn/wp-content/uploads/2011/04/budget.jpg&w=424&h=281&ei=Knm9U 7. http://www.algebra-class.com/graphing-inequalities.html Ir4EqmOiAeptYDgCw&zoom=1&iact=hc&vpx=132&vpy=221&dur=7235&ho 8. http://www.beva.org/maen50980/Unit04/LI-2variables.htm vh=183&hovw=276&tx=155&ty=157&sig=103437241024968090138&page=- 9. http://www.classzone.com/books/algebra_1/page_build.cfm?id=lesson5&ch=6 1&tbnh=129&tbnw=224&start=0&ndsp=16&ved=1t:429,r:7,s:0,i:98 10. http://www.mathchamber.com/algebra7/unit_06/unit_6.htm 11. http://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php Answer Key 12. http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/ Summative Test U05_L2_T1_text_final.html 13. http://www.netplaces.com/algebra-guide/graphing-linear-relationships/graphing- Part I 5. B 9. D 13. B 17. B 1. A 6. B 10. C 14. D 18. D linear-inequalities-in-two-variables.htm 2. D 7. A 11. B 15. D 19. C 14. http://www.netplaces.com/search.htm?terms=linear+inequalities+in+two+variables 3. B 8. B 12. B 16. C 20. D 15. http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/MathAlgor/ 4. C linear.html Part II 16. http://www.purplemath.com/modules/ineqgrph.html 1. 4x – y ≥ 12 and 5x – 2y < 9 17. h t t p : / / w w w. s a d d l e b a c k . e d u / f a c u l t y / l p e r e z / a l g e b r a 2 g o / b e g a l g e b r a / i n d e x . 2. 3x + y = 10 and 3x – 5 ≤ 6 4. html#systems 18. http://www.tutorcircle.com/solving-systems-of-linear-equations-and-inequalities-to author pakidoble check po ito t71gp.html#close_iframe#close_iframe 19. http://www.wyzant.com/Help/Math/Algebra/Graphing_Linear_Inequalities.aspx Part III 1. Php4,800 WEBSITE Links for Videos: 1. http://www.phschool.com/atschool/academy123/english/academy123_content/wl- 2. Php35 book-demo/ph-237s.html 2. http://video.search.yahoo.com/search/video?p=linear+inequalities+in+two+variables 3. http://video.search.yahoo.com/search/video?p=systems+of+linear+equations+and+i nequalities WEBSITE Links for Images: 1. http://lazyblackcat.files.wordpress.com/2012/09/14-lex-chores-copy.png 2. http://www.google.com.ph/imgres?q=filipino+doing+household+chores&start= 166&hl=fil&client=firefox-a&hs=IHa&sa=X&tbo=d&rls=org.mozilla:en-US:offici al&biw=1024&bih=497&tbm=isch&tbnid=e6JZNmWnlFvSaM:&imgrefurl=http:// lazyblackcat.wordpress.com/2012/09/19/more-or-lex-striking-home-with-lexter- maravilla/&docid=UATH-VYeE9bTNM&imgurl=http://lazyblackcat.files.wordpress. com/2012/09/14-lex-chores-copy.png&w=1090&h=720&ei=4EC_ULqZJoG4iQfQro HACw&zoom=1&iact=hc&vpx=95&vpy=163&dur=294&hovh=143&hovw=227&tx=7 9&ty=96&sig=103437241024968090138&page=11&tbnh=143&tbnw=227&ndsp=1- 7&ved=1t:429,r:78,s:100,i:238 262

SUMMATIVE TESTPart I. Select the letter corresponding to your answer.1. Carl bought 10 big notebooks and 15 small notebooks. The total amount he paid was at most Php 550. If x represents the cost of big notebooks and y the cost of small notebooks, which of the following mathematical statements represent the given situation? a. 10x + 15y ≤ 550 c. 10x + 15y > 550 b. 10x + 15y ≥ 550 d. 10x + 15y < 5502. Which of the following is true about the number of solutions a linear inequality in two variables has? a. It has no solution b. It has one solution c. It has two solutions d. It has infinite number of solutions3. Which of the following ordered pairs is a solution of the inequality 2x – 3y > 1? a. (2, 3) b. (-3, -3) c. (5, 4) d. (-4, -1)4. Which of the following is a graph of a linear inequality in two variables? a. b. c. d. 263

5. The difference between Billy’s score and Alvin’s score in the test is not more than 4 points. Suppose Billy’s score is 26 points, what could be the score of Alvin? a. Between 22 and 30 b. 22 to 30 c. 30 and below d. 22 and above6. What linear inequality is represented by the graph at the right? a. x − y ≥ 2 b. x − y ≤ 2 c. -x + y ≥ 2 d. -x + y ≤ 27. Mrs. Abad gave the fish vendor Php 500-bill for 1.5 kg of bangus and three kg of tilapia that cost more than Php 350. Suppose a kilo of bangus costs Php 130. Which of the following could be the cost of a kilo of tilapia? a. Php 95 b. Php 105 c. Php 110 d. Php 1208. Which of the following is a linear inequality in two variables? a. 3a − 2 > 12 c. 2p ≥ 15 b. 15 + 8x < 14y d. 9m + 15 = 7n9. Grecia has some Php 50 and Php 20 bills. The total amount of these bills is less than Php 2,500. Suppose there are 35 Php 50-bills. Which of the following is true about the number of Php2 0-bills? IV. The number of Php 20-bills is less than the number of Php 50-bills. V. The number of Php 20-bills could be more than the number of Php 50-bills. VI. The number of Php 20-bills is equal to the number of Php 50-bills. a. I and II b. I and III c. II and III d. I, II, and III 264

10. A businessman would like to make a model which he can use as a guide in writing a linear inequality in two variables. He will use the inequality in determining the number of sacks of rice and corn that he needs to stock in his warehouse given the total cost (T), the cost (R) of each sack of rice and the cost (C) of each sack of corn. Which of the following models should he make and follow? I. Rx + Cy = T II. Rx + Cy ≤ T III. Rx + Cy ≥ T a. I and II b. I and III c. II and III d. I, II, and III11. In the inequality 6a + 4b ≥ 10, what could be the possible value of a if b = 2? 1 1 1 1 a. a ≤ 3 b. a ≥ 3 c. a < 3 d. a > 3 12. Which of the following shows the plane divider of the graph of y ≤ x + 2? a. c. b. d. 265

13. Ana and Marielle went to the grocery to buy cans of milk sachets of coffee. Ana paid Php 672 for 12 cans of milk and 24 sachets of coffee. Marielle bought the same cans of milk and sachets of coffee but only paid less than Php 450. Suppose each sachet of coffee costs Php 5.50. How many cans of milk and sachets of coffee could Marielle have bought? a. six cans of milk and 36 sachets of coffee b. eight cans of milk and 16 sachets of coffee c. 10 cans of milk and 12 sachets of coffee d. 12 cans of milk and 8 sachets of coffee14. A bus and a car left a place at the same time traveling in opposite directions. After 2 hours, the distance between them is less than 300 km. If the car travels at a speed of 70 kilometers per hour (kph), which of the following could be the speed of the bus? a. 100 kph b. 90 kph c. 80 kph d. 70 kph15. Darcy is making a design of window grill that is rectangular in shape. Suppose the perimeter of the window grill design is less than 30 cm. Which of the following could be the frame of the window grill design Darcy is making? a. c. b. d.16. There are at least 15 large and small tables that are placed inside a function room for at least 150 guests. Suppose only eight people can be seated around the large table and only six people for the small tables. Which of the following number of tables are placed inside the function room? a. ten large tables and 8 small tables c. eight large tables and 16 small tables b. nine large tables and 12 small tables d. six large tables and 15 small tables 266

17. Melanie is using two mobile networks to make phone calls. One network charges her Php 6.50 for every minute of call to other networks. The other network charges her Php 5 for every minute of call to other networks. In a month, she spends more Php 400 for these calls. Suppose she wants to model the total costs of her mobile calls to other networks using a mathematical statement. Which of the following mathematical statements could it be? a. 6.50x + 5y = 400 c. 6.50x + 5y ≥ 400 b. 6.50x + 5y > 400 d. 6.50x + 5y ≤ 40018. Mr. Miranda would like to increase his profit on hog and poultry raising to the maximum if possible. To do it, he has to prepare a business plan to determine the additional expenses and projected profit. Which of the following should Mr. Miranda prepare to come up with the business plan? I. Marketing Plan II. Operational Plan III. Financial Plan a. I only b. II only c. III only d. I, II, and III19. Mr. Tolentino would like to use one side of the concrete fence for the rectangular garage that he will be constructing. This is to minimize the construction materials to be used. To help him determine the amount of construction materials needed for the other three sides whose total length is more than 21 m, he drew a sketch of the garage. Which of the following could be the sketch of the garage that Mr. Tolentino had drawn? a. c. 6m 10 cm 7 cm 8m b. d.6 cm 5m 11 m 9 cm 267

20. A non-government organization is raising funds for the indigent families living in some remote areas by selling two kinds of concert tickets. They expect to raise at least Php 50,000 from the concert. After the concert, the officers of the organization need to account all sold tickets and their total cost then present it graphically to their members. Which of the following graphs could be prepared and presented by the officers considering the expected amount to be raised? a. c. b. d.Part II. Use the following mathematical statements to answer the questions that follow. 3x + y = 10 4x − y ≥ 12 3x − 5 ≤ 6 4x − 2y < 91. Which of the given mathematical statements are linear inequalities in two variables?2. Which of the given mathematical statements are not linear inequalities in two variables? Explain your answer. 268

3. Give three ordered pairs that satisfy each linear inequality in two variables. Show how you obtained these ordered pairs.4. Draw the graph of each linear inequality in two variables. Do the ordered pairs you have given in #3 are on the graph of the linear inequality? If NOT, explain why.5. Describe the solution set of each linear inequality in two variablesPart III. Solve the following problems.1. Mr. Villamayor rented a construction crane for five hr and a backhoe for seven hr. The total amount he paid is less than Php 9,000. Suppose the hourly rate for the crane is Php 800. What is the maximum amount he paid for the backhoe to the nearest hundreds?2. Wally paid at most Php 350 for the five notebooks and four pad papers that he bought. Suppose each notebook costs Php 42. What could be the greatest price of each pad of paper to the nearest peso?Part IV. Plan First! (GRASPS Assessment)Goal: Present simple budget proposal for raising broiler chickensRole: A son or daughter who wish to raise broiler chickens for family’s consumption 269

Audience: Your parentsSituation: Due to the rising prices of food commodities, you decided to raise broiler chickens for your family’s consumption. You sought permission from your parents and asked them to give you some amount to start with. Your parents agreed to give you some money, however, they still need to see how you will use it. They asked you to prepare a budget proposal for the chicken house that you will be constructing, the number of chickens to be raised, the amount of chicken feeds, and other expenses.Product: Simple Budget ProposalStandards: The budget proposal should be clear, realistic, and makes use of linear inequalities in two variables and other mathematical statements.Use the rubric below to check students’ work. RUBRIC: Budget Proposal of Raising Broiler Chickens4321The budget proposal is clear, The budget proposal is clear, The budget proposal is not The budget proposal is notaccurate, practical, and the practical and the use of linear too clear and the use of linear clear and the use of linearuse of linear inequalities inequalities in two variables is inequalities in two variables is inequalities in two variables isin two variables and other illustrated. not properly illustrated. not illustrated.mathematical statements areproperly illustrated. 270

TEACHING GUIDEModule 5: Systems of Linear Equations and Inequalities in Two VariablesA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of systems of linear equations and inequalities in two variables. Performance Standard: The learner is able to formulate real-life problems involving systems of linear equations and inequalities in two variables and solve these with utmost accuracy using a variety of strategies. UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics 1. Describe systems of linear equations and inequalities using practical situations andQUARTER: mathematical expressions.First Quarter 2. Identify which systems of linear equations have graphs that are parallel, intersect andSTRAND: coincide.Algebra 3. Graph systems of linear equations in two variables.TOPIC: 4. Solve systems of linear equations by (a) graphing; (b) elimination; (c) substitution.Systems of Linear Equations and Inequalities 5. Graph system of linear inequalities in two variables.in Two Variables 6. Solve a system of linear inequalities in two variables by graphing.LESSONS: 7. Solve problems involving systems of linear equations and inequalities in two variables.1. Systems of Linear Equations in Two Variables and Their Graphs ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:2. Solving Systems of Linear Equations in TwoVariables Students will understand that systems of linear How do systems of linear equations and3. Graphical Solutions of Systems of Linear equations and inequalities in two variables are inequalities in two variables facilitate findingInequalities in Two Variables useful tools in solving real-life problems and in solutions to real-life problems and making making decisions decisions? TRANSFER GOAL: Students will be able to apply the key concepts of systems of linear equations and inequalities in two variables in formulating and solving real-life problems and in making decisions. 271

B. Planning for Assessment Product/Performance The following are products and performances that students are expected to come up with in this module. a. Systems of linear equations drawn from real-life situation and the graph of each system b. Role-playing of real-life situations where systems of linear equations in two variables are applied c. Real-life problems involving systems of linear equations in two variables formulated and solved d. Design or sketch plan of an expanded school vegetable garden that demonstrates students’ understanding of systems of linear equations and inequalities in two variables. Assessment Map TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCEPre-Assessment/ Pre-Test: Part I Pre-Test: Part I Pre-Test: Part I Pre-Test: Part IDiagnostic Identifying systems of linear Graphing systems of linear Solving problems involving Products and performances equations and inequalities equations and inequalities systems of linear equations related to or involving in two variables and their in two variables and inequalities in two systems of linear equations graphs variables and inequalities in two Solving systems of linear variables equations and inequalities in two variables Pre-Test: Part II Pre-Test: Part II Pre-Test: Part II Identifying mathematics Illustrating mathematics Expressing understanding concepts previously learned concepts previously learned of mathematics concepts through the illustrations previously learned made 272

Pre-Test: Part III Situational Analysis Identifying the information Calculating unknown values Explaining how a Citing situations involving given in a problem mathematical statement linear equations in two Representing situations is derived from a given variables using mathematical situation expressions and statements Formulating and solving problems involving linear equations in two variablesFormative Quiz: Lesson 1 Quiz: Lesson 1 Quiz: Lesson 1 Identifying systems of linear Graphing systems of linear Representing situations equations in two variables equations in two variables using systems of linear equations in two variables Describing the solution sets of a systems of linear Explaining how to graph equations in two variables systems of linear equations using graphs in two variables Quiz: Lesson 2 Quiz: Lesson 2 Quiz: Lesson 2 Giving examples of systems Finding the solutions of Explaining how to obtain of linear equations in two systems of linear equations the solutions of systems variables in two variables graphically of linear equations in two and algebraically variables Identifying the information given in a problem involving Using the different methods Explaining why some systems of linear equations of solving systems of linear systems of linear equations in two variables equations in two variables in in two variables have one finding solutions to real-life solution, no solution, or problems infinite number of solutions Explaining how to check or verify results obtained Describing the advantages and disadvantages of using 273

the different methods of solving systems of linear equations in two variables Solving problems involving systems of linear equations in two variables Choosing and justifying the best option based on the solved problems involving systems of linear equations in two variablesQuiz: Lesson 3 Quiz: Lesson 3 Quiz: Lesson 3Giving examples of systems Determining whether an Explaining why someof linear inequalities in two ordered pair is a solution systems of linear inequalitiesvariables of a given system of linear in two variables have no inequalities in two variables solution or infinite number ofIdentifying the information solutionsgiven in a problem involving Solving systems of linearsystems of linear inequalities inequalities in two variables Explaining how the graphicalin two variables graphically solution of a system of linear inequalities in two variables is determined Describing the solution set of a system of linear inequalities in two variables Describing the advantages and disadvantages of finding the solution set of a system of linear inequalities in two variables graphically 274

Solving problems involving systems of linear inequalities in two variables Making and justifying the best decision based on the solved problems involving systems of linear inequalities in two variablesSummative Post-Test: Part I Post-Test: Part I Post-Test: Part I Post-Test: Part ISelf-Assessment Identifying systems of linear • Graphing systems of Solving problems involving Products and performances equations and inequalities linear equations and systems of linear equations related to or involving in two variables and their inequalities in two and inequalities in two systems of linear equations graphs variables variables and inequalities in two • Solving systems of variables linear equations and inequalities in two variables Part II Part II Part II Identifying systems of linear Solving systems of linear Describing the solution equations and inequalities in equations and inequalities set of systems of linear two variables in two variables graphically equations and inequalities and algebraically in two variables Part III Part IV: Solving problems involving GRASPS Assessment systems of linear equations and inequalities Journal Writing: Expressing understanding of systems of linear equations in two variables Expressing understanding of finding solutions of systems of linear equations in two variables graphically and algebraically Expressing understanding of systems of linear inequalities in two variables 275

Assessment Matrix (Summative Test)Levels of Assessment What will I assess? How will I assess? How Will I Score? The learner demonstrates understanding of key Paper and Pencil Test concepts of systems of linear equations and inequalities in two variables. Part I items 1, 2, and 8Knowledge 15% 1 point for every correct response Describes systems of linear equations Part II item 1 inequalities using practical situations and mathematical expressions. and Part IV item 1Process/Skills 25% Identifies which given systems of linear equations 1 point for every correct response have graphs that are parallel, intersect and coincide. Rubric on Problem Solving Part I items 3, 5, 10, 11, and 12 Rubric for drawing Criteria: Neat and Clear Graphs systems of linear equations in two variables. Part II item 3 Accurate Justified Solves systems of linear equations by Appropriate (a) graphing; Relevant (b) elimination; and (c) substitution. 1 point for every correct response Graph system of linear inequalities in two variables. Rubric for explanation Criteria: Clear Solve a system of linear inequalities in two variables Coherent by graphing. Part I items 4, 7, 9, 13, 15, and 17 JustifiedUnderstanding 30% Solve problems involving systems of linear Part II items 2 and 4 Rubric for drawing equations and inequalities in two variables. Criteria: Neat and Clear Part III Items 1 and 2 Accurate Appropriate Justified Relevant Part IV items 2, 3, and 5 Rubric on Problem Solving Rubric for explanation Criteria: Clear Justified Coherent 276

The learner is able to formulate real-life problems Part I Items 6, 14, 16, 18, 19, and 20 1 point for every correct response involving systems of linear equations and inequalities in two variables and solve these with Part IV item 4 utmost accuracy using a variety of strategies. Product/ GRASPS Assessment Rubric on Problem Posing/Performance 30% Make a design or a sketch plan of a Formulation and Problem Solving vegetable school garden with an area Criteria: Relevant Authentic of at least one hectare. Apply your understanding of the key concepts Creative Clear of systems of linear equations and Insightful inequalities in two variables. Then, use the design or sketch plan of the Rubric on Design/Sketch Plan garden in formulating and solving Criteria: problems involving systems of linear 1. Content equations and inequalities in two 2. Clarity of Presentation variables. 3. Accuracy of Measurements 4. Diversity of PlantsC. Planning for Teaching-LearningIntroduction: This module covers key concepts of systems of linear equations and inequalities in two variables. It is divided intothree lessons namely: Systems of Linear Equations and their Graphs, Solving Systems of Linear Equations, and GraphicalSolutions of Systems of Linear Inequalities in Two Variables. In Lesson 1, students will describe systems of linear equationsand their graphs and solution sets. The students will also draw the graphs of systems of linear equations using any graphingmaterials, tools, or computer software such as GeoGebra. In Lesson 2, the students will find the solution set of systems oflinear equations graphically and algebraically. The two algebraic methods of solving systems of linear equations that studentswill use are substitution method and elimination method. In Lesson 3, the students will determine the graphical solutions ofsystems of linear inequalities in two variables. Again, students will use any graphing materials, tools, or computer software.It would be more convenient for students to find the solution sets of system of linear inequalities if the use of GeoGebra isencouraged. 277

In all lessons, students are given the opportunity to use their prior knowledge and skills in learning systems of linearequations and inequalities. They are also given varied activities to process the knowledge and skills learned and deepen andtransfer their understanding of the different lessons. As an introduction to the main lesson, ask them the following questions: Have you ever asked yourself how businessmen make profits? How can farmers increase their yield or harvest? Howparents budget their income on food, education, clothing and other needs? How cellular phone users choose the best pay-ment plan? How students spend their daily allowances or travel from home to school?Entice the students to find out the answers to these questions and to determine the vast applications of systems of linearequations and inequalities in two variables through this module.Objectives:After the learners have gone through the lessons contained in this module, they are expected to: a. describe systems of linear equations using practical situations and mathematical expressions; b. identify which given systems of linear equations have graphs that are parallel, intersect, and coincide; c. draw the graph of systems of linear equations in two variables; d. find the solution set of systems of linear equations by (a) graphing; (b) elimination; (c) substitution; e. draw the graph of system of linear inequalities in two variables; f. determine the graphical solutions of a system of linear inequalities in two variables; and formulate and solve problems involving systems of linear equations and inequalities in two variables. 278

Pre-Assessment: III. PRE - ASSESSMENT Check students’ prior knowledge, skills, and understanding of mathematics Part I: Find out how much you already know about this module. Choose the letter concepts related to Systems of Linear Equations and Inequalities in that you think best answers the question. Please answer all items. Take Two Variables. Assessing these will facilitate teaching and students’ note of the items that you were not able to answer correctly and look for understanding of the lessons in this module. the right answer as you go through this module. Answer Key 1. Which of the following is a system of linear equations in two variables?Part I a. 2x – 7y = 8 c. x + 9y = 21. B 11. C b. 3x + 5y = -2 2x – 3y > 122. B 12. A3. D 13. C x – 4y = 9 d. 4x + 1 = 84. D 14. C 3y – 7 = 115. B 15. D6. C 16. A 2. What point is the intersection of the graphs of the lines x + y = 8 and 2x – y = 1?7. B 17. A8. C 18. A a. (1, 8) b. (3, 5) c. (5, 3) d. (2, 6) 9. B 19. C10. D 20. A 3. Which of the following is a graph of a system of linear inequalities in two variables? a. c. Part III:1. Php 20; Php 122.Number of Admission Fee Number of Admission Fee Adults Children 242 40 2 36 b. d. 3 483 60 4 60 5 724 80 65 1006 120 279

Teacher’s Note and Reminders 4. Which of the following shows the graph of the system 2x + y < 2 ? x – 4y > 9 a. c. b. d. Don’t 5. If 2x + y = 9 and 2x – y = 11 , what is the value of x?Forget! a. 4 b. 5 c. 10 d. 20 6. A car park charges Php 45 for the first three hours and Php 5 for every succeeding hour or a fraction thereof. Another car park charges Php 20 for the first three hours and Php 10 for every succeeding hour or a fraction thereof. In how many hours would a car owner pay the same parking fee in any of the two car parks? a. 2 hr b. 3 hr c. 5 hr d. 8 hr 7. How many solutions does a consistent and independent system of linear equations have? a. 0 b. 1 c. 2 d. Infinite 8. Which of the following ordered pairs satisfy both 2x + 7y > 5 and 3x – y ≤ 2? a. (0, 0) b. (10, -1) c. (-4, 6) d. (-2, -8) 280

Teacher’s Note and Reminders 9. Mr. Agpalo paid Php 260 for four adult’s tickets and six children’s tickets. Suppose the total cost of an adult’s ticket and a children’s Don’t ticket is Php 55. How much does an adult’s ticket cost? Forget! a. Php 20 b. Php 35 c. Php 80 d. Php 120 10. Which system of equations has a graph that shows intersecting lines? a. 2x + 4y = 14 c. 4x + 8y = 7 x + 2y = 7 x + 2y = 3 b. -3x + y = 5 d. 3x + y = 10 6x – 2y = 1 3x – y = 5 11. Mr. Bonifacio asked each of his agriculture students to prepare a rectangular garden such that its perimeter is at most 19 m and the difference between its length and its width is at least 5 m. Which of the following could be the sketch of a garden that a student may prepare? a. c. b. d. 12. Luisa says that the system 3x + y = 2 has no solution. Which of the 2y = 15 – 6x following reasons would support her statement? I. The graph of the system of equations shows parallel lines. II. The graph of the system of equations shows intersecting lines. III. The two lines as described by the equations in the system have the same slope. a. I and II b. I and III c. II and III d. I, II, and III 281

Teacher’s Note and Reminders 13. Jose paid at most Php 250 for the four markers and three pencils that he bought. Suppose the marker is more expensive than the pencil and their Don’t price’s difference is greater than Php 30. Which of the following could be the Forget! amount paid by Jose for each item? a. Marker: Php 56 c. Marker: Php 46 Pencil: Php 12 Pencil: Php 7 b. Marker: Php 35 Pencil: Php 15 d. Marker: Php 50 Pencil: Php 19 14. Bea wanted to compare the mobile network plans being offered by two telecommunication companies. Suppose Bea’s father would like to see the graph showing the comparison of the two mobile network plans. Which of the following graphs should Bea present to his father? a. b. c. d. 15. Edna and Grace had their meal at a pizza house. They ordered the same kind of pizza and drinks. Edna paid Php 140 for 2 slices of pizza and a drink. Grace paid for Php 225 for 3 slices of pizza and 2 drinks. How much did they pay for the total number of slices of pizza? a. Php 55 c. Php 165 b. Php 110 d. Php 275 16. The Senior Citizens Club of a certain municipality is raising funds by selling used clothes and shoes. Mrs. Labrador, a member of the club, was assigned to determine how many used clothes and shoes were sold after knowing the important information needed. She was asked further to present to the club how she came up with the result using graph. Which of the following graphs could Mrs. Labrador present? a. c. b. d. 282

Teacher’s Note and Reminders 17. The Math Club rented a sound system for their annual Mathematics Camp. They also rented a generator in case of power interruption. Don’t After the 3-day camp, the club paid a total amount of Php3,000, three Forget! days for the sound system and two days for the generator. If each is rented for one day, the club should have paid a total amount of Php1,100. What was the daily rental cost of the generator? a. Php 300 c. Php 800 b. Php 600 d. Php 2,400 18. Mrs. Soriano would like to keep track of her family’s expenses to have an idea of the maximum or minimum amount of money that she will allot for electric and water consumption, food, clothing, and other needs. Which of the following should Mrs. Soriano prepare? a. Budget Plan c. Pricelist of Commodities b. Compilation of Receipts d. Bar Graph of Family’s Expenses 19. A restaurant owner would like to make a model which he can use as a guide in writing a system of equations. He will use the system of equations in determining the number of kilograms of pork and beef that he needs to purchase daily given a certain amount of money (C), the cost (A) of a kilo of pork, the cost (B) of a kilo of beef, and the total weight of meat (D). Which of the following models should he make and follow? a. Ax – By = C c. Ax + By = C x+y=D x+y=D b. Ax + By = C d. Ax – By = C x–y=D x–y=D 20. Mrs. Jacinto would like to instill the value of saving and to develop decision-making among her children. Which of the following situations should Mrs. Jacinto present to her children? a. Buying and selling different items. b. A person putting coins in his piggy bank. c. Buying assorted goods in a department store. d. Making bank deposits in two banks that give different interests. 283

Teacher’s Note and Reminders Part II. Illustrate each mathematics concept in the given figure then describe it by completing the statement at the bottom. Lines Slope of a y - intercept of Line a Line Points on a Line Points Coordinates of Parallel Points Lines Intersecting Lines Linear Equations Linear Inequality Don’t My idea of (mathematics concept given) is _____________________Forget! ___________________________________________________________ ___________________________________________________________ _________________________________________________ 284

Teacher’s Note and Reminders Part III. Use the situation below to answer the questions that follow. Don’t One Sunday, a Butterfly Exhibit was held at the Quezon Memorial Circle Forget! in Quezon City. A number of people, children and adults, went to see the exhibit. Admission was Php 20 each for adults and Php 12 each for children. Questions: 1. How much did an adult pay for the exhibit? How about a child? 2. Complete the table below for the amount that must be paid by a certain number of adults and children who will watch the exhibit. Number of Adults Admission Fee Number of Admission Fee Children 2 2 3 3 4 4 5 5 6 6 3. How much would 10 adults pay if they watch the exhibit? How about 10 children? Show your solution. 4. If a certain number of adults watched the exhibit, what expression would represent the total admission fee? What mathematical statement would represent the total amount that will be collected from a number of children? Explain your answer. 5. Suppose six adults and 15 children watch the exhibit. What is the total amount they will pay for the admission? Show your solution. 6. If a number of adults and another number of children watch the exhibit, how will you represent the total amount they will pay as admission? Explain your answer. 7. Suppose the total amount collected was Php 3,000. How many adults and how many children could have watched the exhibit? 8. The given situation illustrates the use of linear equations in two variables. In what other real-life situations are linear equations in two variables applied? Formulate problems out of these situations then solve. 285

LEARNING GOALS AND TARGETS: Lesson 1 Systems of Linear Equations in TwoStudents are expected to demonstrate understanding of key concepts of Variables and theirsystems of linear equations and inequalities in two variables, formulate real- Graphslife problems involving these concepts, and solve these with utmost accuracyusing a variety of strategies.Lesson 1: SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES AND WWhhaatt ttoo KKnnoowwTHEIR GRAPHS Start Lesson 1 of this module by assessing your knowledge of theWWhhaatt ttoo KKnnooww different mathematics concepts previously studied and your skills in performing mathematical operations. These knowledge and skills may help you in Let the students draw and describe the graphs of some linear understanding Systems of Linear Equations in Two Variables and their Graphs. equations in two variables by doing Activity 1. This activity provides As you go through this lesson, think of the following important question: “How is the students the opportunity to recall graphing linear equations and to the system of linear equations in two variables used in solving real-life problems determine the characteristics of lines. and in making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or Answer Key peers or refer to the modules you have gone over earlier. To check your work, refer to the answer key provided at the end of this module. A1.c tivity 1 2. Activity 1 DESCRIBE ME!y y Directions: Show the graph of each of the following linear equations in a Cartesian coordinate plane. Answer the questions that follow. 1. y = 2x + 3 2. 3x – y = 2 xx 286

3. y 4. 3. y = 5x – 1 4. 2x – 3y = 6 x QU ?E S T I ONS a. How did you graph each linear equation in two variables? b. How do you describe the graphs of linear equations in twoLet the students find the slopes and the y-intercepts of the graphs of somepairs of linear equations. Then ask them to describe the solution set of each variables?pair of linear equations using their slopes and y-intercepts. Tell them toperform Activity 2. In this activity, the students will be able to see how the Were you able to draw and describe the graphs of linear equations in twoslopes and y-intercepts of two lines are related to the solution set of the variables? Suppose you draw the graphs of two linear equations in the samesystem of equations describing these lines. coordinate plane. How would the graphs of these equations look like? You’ll find that out when you do the next activity. Answer Key y Activity 2 MEET ME AT THIS POINT IF POSSIBLE…Activity 21. 2. Directions: Show the graph of each pair of linear equations below using the same Cartesian plane then answer the questions that follow. 1. 3x + y = 5 and 2x + y = 9 2. 3x – y = 4 and y = 3x + 2 x 287

3. 3. x + 3y = 6 and 2x + 6y = 12 QU NS?ES TIO a. How did you graph each pair of linear equations? b. How would you describe the graphs of 3x + y = 5 and 2x + y = 9?Activity 2(Graphs) How about 3x – y = 4 and y = 3x + 2? x + 3y = 6 and 2x + 6y = 12?a. Methods in graphing linear equations c. Which pair of equations has graphs that are intersecting?b. Intersecting lines; parallel lines; coinciding linesc. 3x + y = 5 and 2x + y = 9; one point of intersection; (-4, 17) How many points of intersection do the graphs have?d. 3x – y = 4 and y = 3x + 2 ; Its graph is parallel x + 3y = 6 and 2x + 6y = 12; Its graph is coinciding What are the coordinates of their point(s) of intersection?e. e.1) one e.2) none e.3) manyf. f.1) 3x + y = 5 slope = -3 y-intercept = 5 d. Which pair of equations has graphs that are not intersecting? Why? 2x + y = 9 slope = -2 y-intercept = 9 How do you describe these equations? f.2) 3x – y = 4 slope = 3 y-intercept = -4 e. Each pair of linear equations forms a system of equations. The y = 3x + 2 slope = 3 y-intercept = 2 f.3) x + 3y = 6 slope = y-intercept = 2 point of intersection of the graphs of two linear equations is the 2x + 6y = 12 slope = y-intercept = 2g. 3x + y = 5 slope = not equal y-intercept = not equal solution of the system. How many solutions does each pair of 2x + y = 9 3x – y = 4 slope = equal y-intercept = equal equations have? y = 3x + 2 x + 3y = 6 slope = equal y-intercept = equal e.1) 3x + y = 5 and 2x + y = 9 2x + 6y = 12 h. 3x + y = 5 There is one solution if the slopes and y-intercepts are not e.2) 3x – y = 4 and y = 3x + 2 2x + y = 9 equal. 3x – y = 4 There is no solution if the slopes are equal y = 3x + 2 and e.3) x + 3y = 6 and 2x + 6y = 12 the y-intercepts are not equal. x + 3y = 6 There are many solutions if the slopes and y-intercepts are equal. f. What is the slope and the y-intercept of each line in the given 2x + 6y = 12 i. Many possible answers pair of equations? f.1) 3x + y = 5; slope = y-intercept = 2x + y = 9; slope = y-intercept = f.2) 3x – y = 4; slope = y-intercept = y = 3x + 2; slope = y-intercept = f.3) x + 3y = 6; slope = y-intercept = 2x + 6y = 12; slope = y-intercept = g. How would you compare the slopes of the lines defined by the linear equations in each system? How about their y-intercepts? h. What statements can you make about the solution of the system in relation to the slopes of the lines? How about the y-intercepts of the lines? i. How is the system of linear equations in two variables used in solving real-life problems and in making decisions? 288

Let the students read and understand some important notes on systems Equations like x – y = 7 and 2x + y = 8 are called simultaneous linearof linear equations and their graphs before they perform the succeeding equations or a system of linear equations if we want them to be true for theactivities. Tell them to study carefully the examples given. same pair of numbers. The solution of such equations is an ordered pair of numbers that satisfies both equations. The solution set of a system of linear equations in two Teacher’s Note and Reminders variables is the set of all ordered pairs of real numbers that makes every equation in the system true. The solution of a system of linear equations can be determined algebraically or graphically. To find the solution graphically, graph both equations on a Cartesian plane then find the point of intersection of the graphs, if it exists. The solution to a system of linear equations corresponds to the coordinates of the points of intersection of the graphs of the equations. A system of linear equations has: a. only one solution if their graphs intersect. b. no solution if their graphs do not intersect. c. infinitely many solutions if their graphs coincide. Don’tForget! 289

Teacher’s Note and Reminders There are three kinds of systems of linear equations in two variables according to the number of solutions. These are: 1. System of consistent and dependent equations This is a system of linear equations having infinitely many solutions. The slopes of the lines defined by the equations are equal; their y-intercepts are also equal; and their graphs coincide. Example: The system of equations x – y = 5 is consistent and 2x – 2y = 10 dependent. The slopes of their lines are equal, their y-intercepts are also equal, and their graphs coincide. 2. System of consistent and independent equations This is a system of linear equations having exactly one solution. The slopes of the lines defined by the equations are not equal; their y-intercepts could be equal or unequal; and their graphs intersect. Example: The system of equations 2x + y = 5 is consistent and 3x – y = 9 independent. The slopes of their lines are not equal, their y-intercepts could be equal or unequal, and their graphs intersect. Don’t 3. System of inconsistent equationsForget! This is a system of linear equations having no solution. The slopes of the lines defined by the equations are equal or have no slopes; their y-intercepts are not equal; and their graphs are parallel. Example: The system of equations 2x + y = -6 is inconsistent. The 2x + y = 10 slopes of their lines are equal; their y-intercepts are not equal; and their graphs are parallel. 290

Pose the question: Systems of linear equations in two variables are illustrated in many real-life situations. A system of linear equations in two“How are the solutions to problems involving systems of linear equations variables can be used to represent problems that involve findingused in making decisions?” values of two quantities such as the number of objects, costs of goods or services, or amount of investments, solutions of whichWWhhaatt ttoo PPrroocceessss can also be described using graphs. But how are the solutions to problems involving systems of linear equations used in making Let students identify, describe, and give examples of systems of decisions? linear equations that are consistent and dependent, consistent and independent, or inconsistent. Ask them to perform Activity 3. WWhhaatt ttoo PPrroocceessss Answer Key Your goal in this section is to apply the key concepts of systems of linear equations in two variables and their graphs. Use the mathematical ideas and theActivity 3 examples presented in the preceding section to answer the activities provided.1. Consistent and independent Activity 3 CONSISTENT OR INCONSISTENT?2. Inconsistent3. Consistent and dependent Directions: Determine whether each system of linear equations is consistent4. Inconsistent and dependent, consistent and independent, or inconsistent.5. Consistent and independent Answer the questions that follow.6. Consistent and independent7. Consistent and dependent 1. 2x – y = 7 6. x – 2y = 98. Consistent and independent 3x – y = 5 x + 3y = 149. Inconsistent10. Consistent and independent 2. 2x + y = -3 7. 6x – 2y = 8 2x + y = 6 y = 3x – 4 Teacher’s Note and Reminders 3. x – 2y = 9 8. x + 3y = 8 Don’t 2x – 4y = 18 x – 3y = 8 Forget! 4. 8x + 2y = 7 9. 2y = 6x – 5 y = -4y + 1 3y = 9x + 1 5. -3x + y = 10 10. 3x + 5y = 15 4x + y = 7 4x – 7y = 10 291

Teacher’s Note and Reminders QU QU?E S T I O NS NS a. How were you able to identify system of equations that are consistent- dependent, consistent-independent and Don’t inconsistent? Forget! b. When do you say that a system of linear equations is consistent and dependent? consistent and independent? inconsistent? c. Give examples of systems of linear equations that are consistent and dependent, consistent and independent, and inconsistent. Were you able to determine which system of linear equations in two variables is consistent and dependent, consistent and independent, or inconsistent? In the next activity, you will describe the solution set of system of linear equations in two variables through its graph. Activity 4 HOW DO I LOOK? Directions: Describe the solution set of the system of linear equations as shown by the following graphs. Answer the questions that follow. 1. 3. In Activity 4, let the students describe the graphs of some systems of linear 2. 4. equations in two variables. Strengthen their understanding of consistent anddependent, consistent and independent, or inconsistent systems of linearequations by asking them to give examples. Let them draw and describe thegraphs of these systems of linear equations. Answer KeyActivity 4 ?E S T I O a. How many solution/s does each graph of system of linear equations have?1. one solution2. many solutions b. Which graph shows that the system of linear equations is3. no solution consistent and dependent? consistent and independent?4. one solution inconsistent? Explain your answer. c. When do you say that the system of linear equations as described by the graph is consistent and dependent? consistent and independent? inconsistent? d. Draw graphs of systems of linear equations that are consistent and dependent, consistent and independent, and inconsistent. Describe each graph. 292

Ask the students to draw the graphs of some systems of linear equations Activity 5 DESCRIBE MY SOLUTIONS!then describe the solution set of each. Let them perform Activity 5. If mathsoftware like GeoGebra is available, ask the students to make use of this. Directions: Graph each of the following systems of linear equations in twoGeoGebra is a dynamic mathematics software that can be used to visualize variables on the Cartesian coordinate plane. Describe the solutionand understand concepts in algebra, geometry, calculus, and statistics. set of each system based on the graph drawn. Answer the questions that follow. Answer Key 1. x + y = 8 4. x – 2y = 12 Activity 5 1. 4. x + y = -3 6x + 3y = -9 2. 3x – y = 7 5. 3x + y = -2 x + 3y = -4 x + 2y = -4 3. x + 6y = 9 2x + 6y = 18 2. 5. 3. 293

In some cases where students draw the graphs of some linear equations, QU ?E S T I ONS a. How did you graph each system of linear equations in twothe lines drawn may not appear to intersect because of the limited space variables?on the Cartesian coordinate plane used. In such cases, emphasize to thestudents that lines can be extended indefinitely and that the lines will meet b. How does the graph of each system look like?at a certain point. c. Which system of linear equations has only one solution? Why? How about the system of linear equations with no solution?Ask students to have a closer look at some aspects of the systems of linearequations and their graphs. Provide them opportunities to think deeper and infinite number of solutions? Explain your answer.test further their understanding of the lesson by doing Activity 6. In this section, the discussion was about system of linear equations in two variables and their graphs. Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision?Teacher’s Note and Reminders Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section. WWhhaatt ttoo UUnnddeerrssttaanndd Your goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of systems of linear equations in two variables and their graphs. After doing the following activities, you should be able to answer the following question: How is the system of linear equations in two variables used in solving real-life problems and in making decisions? Activity 6 HOW WELL I UNDERSTOOD… Directions: Answer the following. 1. How do you describe a system of linear equations in two Don’t variables?Forget! 2. Give at least two examples of systems of linear equations in two variables. 3. When is a system of linear equations in two variables used? 4. How do you graph systems of linear equations in two variables? 5. How do you describe the graphs of systems of linear equations in two variables? 6. How do you describe systems of linear equations that are consistent and dependent? consistent and independent? inconsistent? 294

Teacher’s Note and Reminders 7. Study the situation below: a. What system of linear equations represents the Don’t tJoocsoenwstarunctet da given situation? Forget! rectangular garden b. Suppose the system of such that its linear equations is graphed. perimeter is 28 m How would the graph look and its length is 6 like? times its width. c. Is the system consistent and dependent, consistent and independent, or inconsistent? Why? In this section, the discussion was about your understanding of systems of linear equations in two variables and their graph. What new realizations do you have about the systems of linear equations in two variables and their graphs? What new connections have you made for yourself? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. WWhhaatt ttooTTrraannssffeerrBefore the students move to the next section of this lesson, give a short test Your goal in this section is to apply your learning to real-life situations. You(formative test) to find out how well they understood the lesson. will be given a practical task which will demonstrate your understanding.WWhhaatt ttooTTrraannssffeerr Activity 7 HOW MUCH AND WHAT’S THE COST? Give the students opportunities to demonstrate their understanding of Directions: Complete the table below by writing all the school supplies that you systems of linear equations by doing a practical task. Let them perform use. Indicate the quantity and the cost of each. Activity 7. You can ask the students to work individually or in group. Emphasize to them that they must come up with some linear equations in two variables and that a pair of these equations must form a system. School Supply Quantity Cost 295

SUMMARY/SYNTHESIS/GENERALIZATION: Formulate linear equations in two variables based on the table. Then use some pairs of these equations to form different systems of equations. Draw the graph of This lesson was about systems of linear equations in two variables and each system of linear equations. Use the rubric provided to rate your work.their graphs. The lesson provided students opportunities to describe systems oflinear equations and their solution sets using practical situations, mathematical Rubric for Real-Life Situations Involving Systems of Linear Equations inexpressions, and their graphs. They identified and described systems of linear Two Variables and their Graphsequations whose graphs are parallel, intersecting, or coinciding. Moreover, thestudents were given the chance to draw and describe the graphs of systems 4 3 21of linear equations in two variables and to demonstrate their understanding ofthe lesson by doing a practical task. Students’ understanding of this lesson and Systematically listed Systematically listed Systematically listed Systematically listedother previously learned mathematics concepts and principles will facilitate their in the table the data, in the table the in the table the school in the table thelearning of the next lesson, Solving Systems of Linear Equations Graphically properly formulated school supplies, the supplies, the quantity, school supplies, theand Algebraically. linear equations quantity, and cost of and cost of each item quantity, and cost of in two variables each item, properly and formulated linear each item. Teacher’s Note and Reminders that form a system formulated linear equations in two of equations, and equations in two variables but unable accurately drawn the variables that form a to form systems of graph of each system system of equations equations. of linear equations. but unable to draw the graph accurately. In this section, your task was to cite three real-life situations where systems of linear equations in two variables are illustrated. How did you find the performance task? How did the task help you see the real world use of the topic? Don’tForget! 296

Lesson 2: SOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO 2 Solving Systems of VARIABLES Linear Equations in Lesson Two VariablesWWhhaatt ttoo KKnnooww Provide the students opportunities to represent a given situation using WWhhaatt ttoo KKnnoowwlinear equations in two variables, show the graphs of these equations, thenfind possible solutions. Ask them to perform Activity 1. This activity will lead Start the lesson by assessing your knowledge of the different mathematicsto students’ understanding of solving systems of linear equations. concepts previously studied and your skills in performing mathematical operations. These knowledge and skills may help you in understanding Solving Systems of Linear Answer Key Equations in Two Variables. As you go through this lesson, think of the following important question: How is the system of linear equations in two variables usedActivity 1 Amount Collected by Amount Collected by in solving real-life problems and in making decisions? To find out the answer, the Tricycle Driver in the Jeepney Driver in perform each activity. If you find any difficulty in answering the exercises, seek the Number of assistance of your teacher or peers or refer to the modules you have gone over earlier. Passengers Peso Peso 10 12 Activity 1 HOW MUCH IS THE FARE? 1 20 24 2 30 36 Directions: Use the situation below to answer the questions that follow. 3 40 48 4 50 60 Suppose for a given distance, a tricycle driver charges Php 10.00 every 5 100 120 passenger while a jeepney driver charges Php 12.00. 10 150 180 15 200 240 1. Complete the table below for the fare collected by the tricycle and jeepney 20 250 300 drivers from a certain number of passengers. 25 300 360 30 Number of Amount Collected Amount Collected Passengers by the Tricycle by the Jeepney Driver Driver 1 2 3 4 5 10 297

Teacher’s Note and Reminders 15 20 Don’t 25 Forget! 30 2. How did you determine the amount collected by the tricycle and jeepney drivers from their passengers? 3. Suppose in three round trips the tricycle and jeepney drivers had carried a total of 68 passengers. a. How would you find the number of passengers each had? b. What mathematical statement will you use to find the number of passengers each carried? What is the total amount of fare collected from the passengers by the two drivers? Explain how you arrived at your answer. c. How would you draw the graph of the mathematical statement obtained in 3b? Draw and describe the graph. 4. Suppose the total fare collected by the tricycle and jeepney drivers is Php 780. a. How would you find the number of passengers each had? b. What mathematical statement will you use to find the number of passengers each had? c. How would you draw the graph of the mathematical statement obtained in 4b? Draw the graph in the Cartesian coordinate plane where the graph of the mathematical statement in 3b was drawn. Describe the graph. 5. How do you describe the two graphs drawn? 6. What do the graphs tell you? 7. How did you determine the number of passengers each driver had? 298

Strengthen students’ skills in graphing systems of linear equations. At the How did you find the activity? Were you able to use linear equations in twosame time, provide them opportunities to examine different graphs drawn in variables to represent a real-life situation? Were you able to find some possiblea Cartesian coordinate plane. Tell them to perform Activity 2. Let them find solutions of a linear equation in two variables and draw its graph? In the next activity,out which graphs are intersecting, parallel, or coinciding. If intersecting, ask you will show the graphs of systems of linear equations in two variables. You needthem to determine their point of intersection and the meaning of this. this skill to learn about the graphical solutions of systems of linear equations in two variables. Answer Key Activity 2 LINES, LINES, LINES…Activity 2 Directions: Use the situation below to answer the questions that follow. y=x+7 3x + 8y = 12 1. y = -2x + 1 3. 8x – 5y = 12 x–y=6 2. y8x=+37xy–=215 4. 2x + 7y = -6 299