Quarter 1 Compiled Modules

(- 4)[������ + 1 = 6 (������ − 7)](- 4) (Multiply both side by – 4) and −4 (Apply the distributive property of multiplication) (Gather constant terms and the term with x-variable on one - 4������ − 4 = 6(������ − 7) equation) (for y = mx + b) - 4y – 4 = 6x – 42 (Divide the whole equation by the coefficient of y-variable and - 4y = 6x – 42 + 4 (Equation of a line in the form of y = mx + b) side of the (Apply the distributive property of multiplication) (Gather the terms with variables on one side of the - 4y = 6x – 38 equation, the constant terms on the other side and then (equation of a line in the form of ax + by = c) then simplify) y = 6x − 38 − 4 −4 y = − 3x + 19 22 or (for ax + by = c) - 4y – 4 = 6x – 42 - 4 + 42 = 6x + 4y or 6x + 4y = - 4 + 42 simplify) 6x + 4y = 38 Example 2. Find the equation of a line that passes through a point (7, - 5) and with slope m = 6. Solution: Substitute the coordinates of a point (7, - 5) for (������1, ������1) and the slope m = 6 respectively into the point-slope form of the equation of a line. ������ − ������1 = ������(������ − ������1) (Point-slope form) ������ − (−5) = 6(������ − 7) (Substitute) ������ + 5 = 6������ − 42 (Apply the distributive property of multiplication) (for y = mx + b) y = 6x – 42 – 5 (Gather constant terms and the term with x-variable on one side of the equation and then simplify) y = 6x – 47 (Equation of a line in the form of y = mx + b) or (for ax + by = c) 5 + 42 = 6x - y (Gather the terms with variables on one side of the or 6x – y = 5 + 42 equation, the constant terms on the other side and then simplify) 6x – y = 47 (equation of a line in the form of ax + by = c) Example 3. Write the equation of a line into the point – intercept form of the given below slope of the line is 3 and the y – intercept is (0, - 4). Solution for no. 1: Substituting the slope m = 3 and the value of y - intercept b = - 4 into y = mx + b. 101 | P a g e

y = mx + b (slope-intercept form) y = 3x - 4 (Substitute and simplify) (slope-intercept form) or (for ax + by = c) y = 3x + 4 (Gather the terms with variables on one side of the 4 = 3x – y equation and the constant terms on the other side) or 3x - y = 4 (equation of a line in the form of ax + by = c) PRACTICE EXERCISES 1 It’s your turn! Activity IV: Here are the pairs of points. What is the equation of a line? Find the equation of a line in the forms y = mx + b and ax + by = c that passes through the following pairs of points. Given points y = mx + b ax + by = c 1. (8, - 4) and (6, 12) 2. (-8, 3) and (- 7, 10) 3. (6, - 2) and (5, - 4) 4. (- 3, - 4) and (2, 2) 5. (9, 3) and (- 1, - 2) PRACTICE EXERCISES 2 Activity V: Who is this Famous Inventor and Mathematician? In 1596, the point-slope formula y – y1 = m (x – x1) was discovered by this French mathematician. Who is he? To find out, match the letter that corresponds to the answer to the numbered item on you left. The letter will spell out the name of this Famous Inventor and Mathematician. Given: Slope and a point of a line Equation of a line 1. m = 2; (4, 3) A) y = - 6x + 7 2. m = 2; (5, - 7) C) 3x – y = 8 3. m = − 3; (2, -2) D) y = - 7x + 2 E) 2x – y = 17 5 L) 2x – y = 5 N) 3x + 5y = - 4 4. m = 2; (10, 3) R) y = 2x – 5 5. m = - 7; (0, 2) S) 6x – 5y = 0 6. m = 2; (17, 0) T) y = 1x – 6 2 2 7. m = 3; (3, 1) _____ _____ _____ _____ _____ _____ _____ _____ 8. m = - 6; (-1, 13) 9. m = 2; (4, 3) 10. m = 1; (- 2, - 7) 2 11. m = 2; (4, - 9) 12. m = 6; (0, 0) 5 _____ _____ _____ _____ 12 3 4 56 78 9 10 11 1 2 102 | P a g e

PRACTICE EXERCISES 3 Activity VI. What is the equation of a line? Write the equation of a line into the point – intercept form of the following given below: 1. The slope of the line is 3 and the y – intercept is (0, - 4). 2. The slope of the line is 1 and the y – intercept is (0, 2). 2 3. The slope of the line is 1 and the y – intercept is (0, - 2). 2 4. The slope of the line is − 4 and the y – intercept is (0, - 1). 3 5. The slope of the line is 0 and the y – intercept is (0, 2). Did you find the equation of a line in the form of y = mx + b or ax + by = c using the Different form? PRACTICE EXERCISE 4 Activity VII: Solve my problem! Read, understand and analyze the situation below to answer the questions that follow. Situation 1: Cardo is a grade 8 student of Langaman National High School. He competed in the high jump competition and managed a personal best of 6 feet. When he was in grade 7 at the age of 13, he jumped 5 feet in the high jump competition. His coach decided that his progress in high jump could be modeled by an equation of a line. He let x represents Cardo’s age and y represents the height of his jump. Questions: 1. Write the equation of a line as described by the coach? 2. If the coach will use this equation to predict height of Cardo’s jump, what will be the height of his jump at age 16? Activity VIII Find, Fix and Justify. Situation 2: Felipe was asked to write the equation of a line that has a slope of 1 and passes through 3 the point (6, 4). Felipe made a common error and wrote the equation y = 1 x + 4 . Describe Felipe’s error 3 and write the correct equation. Situation 3: Determine a common mistake in finding the equation of a line. Describe the mistake and write the correct equation of the line in point-intercept form. Given: The slope is – 5 and the y – intercept is (0, 6). Incorrect equation: y = 6x + 5 REFLECTION! 1. How did you find the equation of a real-life problems involving linear equations in two variables? ________________________________________________________________________________ 2. What mathematical concepts or principles did you apply to derive the equation of a line? ________________________________________________________________________________ ________________________________________________________________________________ 3. What I have learned so far ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ 103 | P a g e

Integrated the development of the following learning skills: 1. Communication skills 3. Creativity A. Following instructions/directions a. Problem solving B. Reasoning C. Responding to ideas 2. Critical thinking 4. Character A. Observation a. Working independently B. Analysis b. Accuracy C. Problem solving c. Patience EVALUATION! ACTIVITY VI: CHECK YOUR KNOWLEDGE! A. Solve for the Equation of a Line of the following pair of points below. Write your answer in the form of y = mx + b and ax + by = c. 1. Find the equation of a line passing through the points (5, 7) and (2, –2). 2. Find the equation of a line passing through the points (3, –5) and (–2, 3). 3. Find the equation of a line passing through the points (–1, 3) and (–6, 5). 4. Find the equation of a line passing through the points (6, –3) and (1, –6). B. C. Prepared by: Karl Anthony M. Gonzales Pasay City North High School – MDC References for Further Enhancement: Book: Grade 8 Mathematics Learner’s Module, pp 192 -196 Online references: https://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation-of-line-given- two-points.php https://www.mathworksheets4kids.com/linear-equation/two-points/standard-level1-1.pdf 104 | P a g e

Module Code: Pasay-M8–Q1–W6-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 6/ Day 2 OBJECTIVE: Solves problems involving linear equations in two variables. YOUR LESSON FOR TODAY: This module will guide the learners to solves real-life problems involving linear equations in two variables. The application of linear equations in two variables can be seen in many different fields like business, industries, architecture, physics and in variety of mathematical problem. Also, the learners will apply their knowledge of the different mathematics concepts previously studied and their skills in performing mathematical operations. Those knowledge and skills may help them to understand the solutions to real-life problems involving linear equations in two variables. Unlocking of Difficulties Vocabulary Words • Variables - Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. • Independent variables are controlled inputs. • Dependent variables represent the output or outcome resulting from altering these inputs. • Rate – it is the ratio between two related quantities in different units. (Ex. slope = rise/run speed = distance/time) Let’s Recall! Activity I: Solve his problem! Consider the situation below and answer that questions that follow Manuel often rides a taxi from one place to another. The standard fare in riding a taxi is Php 40 as a flag-down rate plus Php 3.50 for every 200 meter or a fraction of it. Complete the table below Distance (in meters in x) Amount (in Php y) Questions: 1. What is the dependent variable in the given situation? Explain your answer. 2. What is the independent variable in the given situation? Explain your answer. 3. Based on the completed table, would the relation represent a line? 4. What is the y – intercept? Explain your answer. 5. What is the slope? Explain your answer. 6. Write the linear equation in two variable (y = mx + b) and answer the following questions. 105 | P a g e

a. If Manuel rides a taxi from his workplace to the mall with an approximate distance of 600 meters, how much will he pay? b. If he rides a taxi from his residence to an airport with an approximate distance of 6 kilometers, how much will he pay? c. If Manuel pays Php 75, how many kilometers did he travel? How about Php 90? Php 115? 7. Write the equation of a line in the form of ax + by = c using the answer in number 6. 8. Draw the graph of the equation you have formulated in item 7. How did you find the activity? Can you still recall the properties of a linear equations in two variables? Did you use them to solve the given problem? To better understand how the concepts of the linear equations in two variables can be applied to solve real-life problems, study the illustrative example presented below. Let’s Explore! Activity II. Study the problem. A regular pay phone services when using mobile cellular phone is Php 20 for the first three minutes and Php 12 for every minute additional or a fraction thereof. How much will a caller have to pay if his call lasts for 10 minutes? Solve the problem using any method. Solution: a. Let m be the rate of the price of the call per minute. Let x be the number of minutes of his call after the first three minutes call. Let y be the charge of his total calls. Let k be the charge of the call for the first three minutes. b. Expressing the problem in slope-intercept form of a line in two variables “y = mx + k”, and x = total minutes of call minus the first 3 minutes call. c. Substituting in the formula for the price of hos total call and solve for the charge. ������ = ������ℎ������12 (10 ������������������. − 3 ������������������. ) + ������ℎ������ 20 1 ������������������. ������ = ������ℎ������12 (7 ������������������. ) + ������ℎ������ 20 1 ������������������. y = Php 84 + Php 20 y = Php 104 d. The total charge of his call is Php 104. The previous activity allowed you to recall your understanding of the linear equations in two variables and gave you an opportunity to solve real-life problems that deal with linear equations in two variables. 106 | P a g e

PRACTICE EXERCISES 1 It’s your turn! Activity III: Word Problems. Solve the following problem involving linear equations in two variables. The difference between Nerbie’s age and Euvard’s age’s is 5 years. Eugene is older than Wyatt. Write an equation that represents all the possible different ages that Eugene and Wyatt can be. What will be the age of Nerbie if Euvard’s age is 17 years old? Solution a. Let N be the age of Nerbie, E be the age of Euvard. b. Express the problem in a linear equation with two variables. N–E=5 Difference between the age of Nerbie and Euvard c. Substituting in the formula to find the age of Nerbie if Euvard’s age is 17. N – 17 = 5 N = 5 + 17 N = 22 d. The age of Nerbie is 22 years old when Euvard’s age is 17. Try to solve this problem! 1. The length of a rectangular frame is trice as its width. What will be the length of the rectangular frame if the width is 21 inches long. Write a linear equation to represent the problem. PRACTICE EXERCISES 2 Activity IV: Predict the total enrollment! Read, analyze and solve the problem. The annual enrollment of Pasay National High School went from 500 in school year 2010 to 1250 in school year 2015 and it seem to follow a linear trend. Let 2010 be year so that 2005 is year 5. a. Graph the line. b. Find the slope of the line c. Write the equation of the line in slope-intercept form. d. If the enrollment follows the same trend, predict the total enrollment in 2015. Big Idea! Solving Real-Life Problems enhances the student’s mental skill, develop logical analysis and boost creative thinking. Possessing the ability to solve math word problem skills makes a huge difference in one’s career and life. Hence, it should be considered with seriousness and promoted to generate eagerness and interests to develop math word problem skills. 107 | P a g e

PRACTICE EXERCISES3 Activity V: Solve my problem! Read, understand and analyze the situation below to answer the questions that follow. Situation: Symbol is going to a trip from Manila to Tagaytay using his motor bike. He is already 15 km in his trip and the speed limit is 60 km per hour on the expressway. Let x be the number of hours from now and y be the total distance traveled. Questions: 1. Write the equation of a line as described by the situation? If you will use this equation to estimate the total distance he traveled, what will be the distance traveled if he already drives for 25 minutes? Show your solution and answer on the space provided on the next page. Solution: Activity VI: The more, the merrier! Read, analyze and solve the problem. Carlo Donuts charges Php 23 each for a special doughnut plus a fixed charge of Php 8 pesos for the box which can hold as many as 24 doughnuts. How many doughnuts would be in a box priced at Php 422? Questions: 1. Write the equation of a line as described by the situation? 2. What is the answer in the given problem? 3. If you will use the derived equation to estimate the number and the cost of doughnuts in a box, how much is the cost of 36 doughnuts? REFLECTION! • How did you find the equation of a real-life problems involving linear equations in two variables? ________________________________________________________________________________ • What mathematical concepts or principles did you apply to derive the equation of a line? ________________________________________________________________________________ • What I have learned so far ________________________________________________________________________________ 108 | P a g e

INTEGRATED THE DEVELOPMENT OF THE FOLLOWING LEARNING SKILLS: 1. COMMUNICATION SKILLS 3. CREATIVITY A. FOLLOWING INSTRUCTIONS/DIRECTIONS A. PROBLEM SOLVING B. REASONING C. RESPONDING TO IDEAS 2. CRITICAL THINKING 4. CHARACTER A. OBSERVATION A. WORKING INDEPENDENTLY B. ANALYSIS B. ACCURACY C. PROBLEM SOLVING C. PATIENCE EVALUATION! ACTIVITY VII: CHECK YOUR KNOWLEDGE Read, analyze and solve the problem. 1. Kale is tracking the progress of his plant’s growth. Today the plant is 5 cm high. The plant grows 1.5 cm per day. a. Write a linear equation in two variables that represents the height of the plant after d days. b. What will be the height of the plant after 20 days? 2. Jay opens a savings account with Php 3500. He saves Php1500 per month. Assume that he does not withdraw money or make any additional deposits. a. Write an equation that represents the total amount of money Jay deposits into his account after months. b. After how many months will Paul have more than Php 20,000? 3. A rental car company charges a daily price for renting a standard car, plus an additional Php 200 per km driven. The cost for renting a car for one day and driving it 5 km is Php 1500. a. Find the daily price for renting a car (not counting kilometrage). b. Write an equation that represents the relationship between the number of kilometers driven in one day and the cost for renting a car. Prepared by: Karl Anthony M. Gonzales Pasay City North High School – MDC References for Further Enhancement: Book: Grade 8 Mathematics Learner’s Module, pp 192 -193 Online references: https://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation- of-line-given-two-points.php https://www.mathworksheets4kids.com/linear-equation/two-points/standard-level1-1.pdf https://www.uen.org/core/math/downloads/8thGrade_Ch03_StudentWorkbook.pdf 109 | P a g e

Module Code: Pasay-M8–Q1–W6-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 6/ Day 3 OBJECTIVE: Illustrates a system of linear equations in two variables YOUR LESSON FOR TODAY: • Describe system of linear equation in two variables. TRY TO DISCOVER! In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. A pair of linear equation of the form {������������12 ������ + ������1 ������ = ������1 where ( a1 , b1 not both equal to 0 ) and ( a2 , b2 not both equal to ) ������ + ������2 ������ = ������2 is called a system of linear equation in two variables. Linear equation can be written in standard form or y-intercept form. Study the table below. Try to answer the questions that follow. For any solution of has a unique solution has no solution when and has an infinite linear equation in when number of solutions standard form when {������������12 ������ + ������1 ������ = ������1 ������ + ������ = 5 ������ + ������ = 4 ������ + ������ = 2 ������ + ������2 ������ = ������2 ������ − ������ = 1 ������ + ������ = 2 3������ + 3������ = 6 Questions: a. When can you say that the system of linear equation in to variables has: a.1) one solution? 110 | P a g e

a.2) no solution? a.3) infinitely many solutions? Algebraically, we can determine whether a system has a solution, no solution or infinitely many solutions if the following properties are satisfied. For any solution of has a unique solution has no solution when and has an infinite linear equation in when number of solutions standard form when {������������12 ������ + ������1 ������ = ������1 ������1 ≠ ������1 ������1 = ������1 ≠ ������1 ������1 = ������1 = ������1 ������ + ������2 ������ = ������2 ������2 ������2 ������2 ������2 ������2 ������2 ������2 ������2 For deeper understanding in determining the solution of system of linear equation in two variables, let’s proceed to the next activity. ARE YOU READY TO PRACTICE? Determine whether the statement is TRUE or FALSE. a. All system of linear equations in two variables has exactly one solution. b. The system of linear equation {27������������+−3������������==1192 has no solution. c. The solution of {4���2��� ���−��� +104���=��� =−58������ are infinitely many. Let us check your answers… Here’s the solution for the problem. a. False, because some system has no solution and others have many or infinite solution. b. False, because if we are going to find the value of a and b the result is ������1 ≠ ������1 ������2 ������2 , it means that it has a solution. c. True, because if we are going to find the value of a, b and c the result is ������1 = ������1 = ������1, it means that the solution is many or infinite. ������2 ������2 ������2 111 | P a g e

In this part, you are going to test further your understanding on the solution of a system of linear equation in two variables. Determine whether the ordered pair is a solution to the given system of linear equations. 1. x + y = 3 (5, 2 ) 2. x + y = 6 (1, 5) 3. x + 2y = 5 ( 1, 2 ) x+y=7 y = 3 + 2x y –x=1 Solution: 1. (5, 2) x + y = 3 ; x +y = 7 5+2=3 5+2=7 7≠3 7 = 7 Therefore, (5, 2) is not a solution. 2. (1,5) x + y = 6 ; y = 3 + 2x 1+5=6 5 = 3 + 2(1) 6=6 5=5 Therefore, (1, 5) is a solution. 3. (1,2) x + 2y = 5 ; y – x = 1 1 + 2(2) =5 2–1=1 1+4=5 1=1 5=5 Therefore, (1,2) is a solution. Let’s do it! PRACTICE EXERCISES 1: Determine whether the given linear systems has one solution, no solution, or infinite number of solutions. 1.{62���������+��� +124������������ = 8 _________________ = 24 _________________ _________________ 2. {124������������ + 2������ = 6 _________________ + 8������ = 18 __________________ 3. {123������������ − 6������ = 6 − 24������ = 9 4. {162������������++24������������==12 5. {3������������ − 2������ = 3 + 4������ = 18 PRACTICE EXERCISES 2: Determine whether the given ordered pair is a solution of the given system of linear equations. Write SOLUTION or NOT A SOLUTION. 1. {2���2��� ���+��� −3������������ = 4 ; (1, -2) _______________________ = −4 ; ( 4, -2) _______________________ ; (3, 2) _______________________ 2. {32������������++4������������==64 ; ( 2, 4) _______________________ ________________________ 3. {32������������ + 3������ = 15 ; (2, 1) + 6������ = 18 4. {62���������+��� +124������������ = 8 = 24 5. {2������������ − ������ = 3 − 2������ = 112 | P a g e

Remember: A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. A pair of linear equation of the form {������������12 ������ + ������1 ������ = ������1 where ( a1 , b1 not both equal to 0 ) and ( a2 , b2 not both equal to ) ������ + ������2 ������ = ������2 is called a system of linear equation in two variables. Linear equation can be written in standard form or y-intercept form. Let’s Try Harder! PRACTICE EXERCISES 3: Answer the following questions. 1. How can you check if an ordered pair is a solution of a system? 2. When does the system of equations have no solution? 3. Explain why (2,4) is a solution of a system of equations. 4. Write a system of equations for which: a. (3,2) is a solution. b. There is no solution. c. There is infinite number of solutions. Write your answer below. ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Character • Understanding of words/vocabulary • Consistency • Following instructions/directions. • Patience • Reasoning • Perseverance • Responding to ideas 3. Critical Thinking 4. Character • Observation • Values integration • Reflection • Honesty • Decision making • Working independently 113 | P a g e

EVALUATE YOURSELF! Directions: B. Determine whether the given ordered A. Determine whether the given linear pair is a solution to the given system systems has one solution, no solution, of equations or not a solution. or infinite number of solutions. 1.{������������+=������������=+−17 _______________________ 1.{32������������=− 3������ =5 ; (1, 2) 2. {23������������=+4������−=62������ ______________________ 10 + 2������ ( 2,4) _______________________ ____________________ ______________________ ______________________ 2. {������������ + ������ = 8 ; = ������ + 6 ____________________ 3.{������ ������ − ������ = 5 3.{������������ = −������ +7 ; ( -1, 8) + 5������ = −7 − ������ = −9 ( 1, 5 ) ( 6, 5 ) {2������������ + ������ = 4 ____________________ − 3������ = 3 4. 4. {2������������++6������������==111 ; {5������������=− 5������ − 5 ____________________ ������ = 15 5. 5. {5������������ + ������ = 11 ; + 4������ = 49 ____________________ HARD work ……pays OFF! -R. Bradbury Prepared by: GEMMA ORESCO-COTONER Kalayaan National High School References for further Enhancement: Grade 8 Learner’s Module, pp. 253-256 https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-systems-of-linear- equations-two-variables/ 114 | P a g e

Module Code: Pasay-M8–Q1–W6-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 6/ Day 4 OBJECTIVE: Graphs a system of linear equations in two variables. YOUR LESSON FOR TODAY: • Graphing system of linear equation in two variables. TRY TO DISCOVER! Solve for the slope (m) and y – intercepts. 1. y - x = 2 m = _____ ; y – intercept: ( ____, ___ ) 2. x + y = 4 m = _____ ; y – intercept: ( ____, ____) In previous lessons, you learned how to graph linear equation using slope and y-intercept. Now, your goal is to graph system of linear equation in two variables. Direction: Complete the table and graph. Then answer the questions that follow. System of Linear Equation Slope (m) y-intercept (b) b. In Two Variables ������. ������ = ������ − 42} ������ = ������ + ������. 2������ + 2������ = 6} ������ + ������ = 4 ������. 2������ + ���2���������==48} ������ + Graph: a. http://clipart-library.com/free/clipart-computer-black-and-white.html http://clipart- library.com/free/clipart-computer-black-and-white.html 115 | P a g e

c. http://clipart-library.com/free/clipart-computer-black-and- white.html a. How did you graph each pair of linear equations? b. How would you describe the graphs of ������ = ������ − 24}? ������ = ������ + How about 2������ + ���2���������==46}? 2������ + 2������ = 8}? ������ + ������ + ������ = 4 c. Which pair of equations has graphs that are intersecting? How many points of intersection do the graphs that are intersecting? What are the coordinates of their point(s) of intersection? d. Which pair of equations has graphs that are not intersecting? Why? How do you describe these equations? e. Each pair of linear equations forms a system of equations. The point of intersection of the graphs of two linear equations is the solution of the system. How many solutions does each pair of equations have? e.1) ������ = ������ − 42} ������ = ������ + e.2) 2������ + ���2���������==46} ������ + e.3) 2������ + 2������ = 8} ������ + ������ = 4 f. What is the slope and the y – intercept of each line in the given pair of equations? System of Linear Equation Slope (m) y-intercept (b) In Two Variables ������. 1 ������ = ������ − 24} ������ = ������ + ������. 2 2������ + 2������ = 6} ������ + ������ = 4 ������. 3 2������ + 2������ = 8} ������ + ������ = 4 How would you compare the slopes of the lines defined by the linear equations in each system? g. 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? How is the system of linear equations in two variables used in solving real-life problems and in making decisions? 116 | P a g e

ARE YOU READY TO PRACTICE? From the activities done, you were able to determine when two lines intersect and when they do not intersect. You were able to relate also the solution of the system of linear equations with the slope and y-intercepts of their graphs. If the lesson is clear for you, then you are ready for more practice… enjoy working and learning! PRACTICE EXERCISES 1: Direction: Complete the value in the table and graph the system of linear equation in two variables. System of Linear Equation in Slope (m) y-intercept Graph Two Variables (b) ������ = − 1 ������ + 2 { 1. 2 1 ������ = − 2 ������ + 2 13 2. {������ = − 2 ������ + 2 ������ = −������ + 6 ������ = ������ + 2 3. { ������ = ������ − 5 2 1. 2. 3. 117 | P a g e

PRACTICE EXERCISES 2: Direction: Complete the value in the table and graph the system of linear equation in two variables. System of Linear Slope (m) y-intercept (b) Graph Number of Equation in Two Solution Variables 1. {{3������ ������ + ������ = 5 − 15 = −3������ 2.{{123������������ − 6������ = 6 − 24������ = 1 3. {{2������������ = ������ − 1 = 3������ + 1 1. 2. 3. Remember: The graphing of system of linear equations in two variables has different ways. In this lesson the slope and y- intercept approach is used to graph to aid you in the next lesson. The graph of a system of linear equations in two variables can be intersected at a point where the intersection point is the system's only solution; parallel, meaning no solution; and coinciding which means infinitely many solution. 118 | P a g e

PRACTICE EXERCISES 3: Solve the problem. Use the graph to answer these questions. a. What was the company’s best year? b. When did the company break even? Was that good news? Explain. Answer: ______________________________________ ______________________________________ ______________________________________ _________________________________. e-math .Intermediate Algebra II. Oronce and Mendoza _________________________________________________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Character • Understanding of words/vocabulary: • Accuracy • Following instructions/directions. • Patience • Reasoning • Perseverance • Responding to ideas SELF EVALUATION: Direction: Graph each system of linear equations in two variables. Describe the graph and determine the number of solution. System of Linear Slope (m) y-intercept Graph Number of Equation in Two (b) Solution Variables 1. {������������ + ������ = 5 − ������ = 5 2. 2������ + 3������ = 4 2������ + 3������ = −4 1. 2. Prepared by: GEMMA ORESCO - COTONER Kalayaan National High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 254-256 E-math Intermediate Algebra II Worktext, pp.3—8 119 | P a g e

Module Code: Pasay-M8–Q1–W7-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 7/ Day 1 OBJECTIVE: Categorizes when a given system of linear equations in two variables has graphs that are parallel, intersecting, and coinciding. TRY TO DISCOVER! Study and describe each pictures. Identify the difference between them. https://i.imgur.com/BbDws2Q.jpg a. b. c. Observation: ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ____________ 120 | P a g e

YOUR LESSON FOR TODAY: • Categorizing when a given system of linear equations in two variables has graphs that are parallel, intersecting, and coinciding. In previous lesson, we learned how to graph system of linear equation in two variables using slope and y- intercept. This time, we are going to categorize a given system of linear equations in two variables whose graphs are parallel, intersecting and coinciding. Go back to the previous session, were you able to relate real life situation to our topic? If that so, let’s proceed to the next activity. Study the table below. System of Linear Properties of Linear Graph Equations Equations intersecting 1.{−2������������ + ������ = 7 21 parallel + ������ = 1 1≠1 coinciding 31 4 2. {33������������++������������==−45 3=1 ≠5 3. {2������������ − 2������ = −5 12 5 − 4������ = −10 2 = 4 = 10 A. When can you say that the graph of a system of linear equation in two variables is: 1) intersecting? 2) parallel? 3) coinciding? B. How does the graph look like? Let’s compare our answer… A. 1. The graph of a system of linear equation in two variables is intersecting if ������1 ≠ ������1 . ������2 ������2 2. The graph of a system of linear equation in two variables is parallel if ������1 = ������1 ≠ ������1. ������2 ������2 ������2 3. The graph of a system of linear equation in two variables is coinciding if ������1 = ������1 = ������1. ������2 ������2 ������2 B. The graph is: 2. 3. 1. Note: You can use GeoGebra to check if your graph is correct. Download GeoGebra Apps in your cellphone or computer. 121 | P a g e

ARE YOU READY TO PRACTICE? Now that you learned how to categorized system of linear equations in two variables, you may now try the activities in the next section. Let’s Do It! Categorize the given system of linear equations in two variables whether the graph is intersecting, parallel or coinciding. System Properties Graph 1. {2������������−−2������������==612 2. {42������������++2������������==13 3. {3������������ − ������ = 3 − 3������ = 9 4. {22������������−−������������==−16 5. {3������������−−������������==26 Have you encounter some problems in doing this activity? If yes, try to go back to the previous discussion for better understanding. Were you able to identify the property used in each system? Were you able to classify a given system of linear equation according to their graph? You will need this skill for deeper understanding on how to categorize system of equations. 122 | P a g e

Where Do I Belong? Categorize each system of equation according to graph. {������ + ������ = 12 {4������������=+136������������ = 7 {44������������−=������������=−174 ������ = 3������ − 5 {4���2��� ���−��� =105������������ − 4 {������ ������ = ������ − 5 = −8 − ������ = −10 {6������������=− 3������ − 2 {������������+=23������������=−−11 {������ ������ = −4������ 6������ = 4 + 4������ = 12 {42������������==−−2������������−−12 {6������������=+ 3������ + 2 6������ = 4 Intersecting Parallel Coinciding 123 | P a g e

Answer the problem. Let’s Try Harder! Which system of equation Write your answer. has the graph that shows intersecting lines? And why? ______________________________________________________ ______________________________________________________ a. {2������������++24������==174 ______________________________________________________ ______________________________________________________ b. {6−���3��� ���−��� +2������������ = 5 ______________________________________________________ = 1 ______________________________________________________ ______________________________________________________ c. {4������������++28������������==37 ______________________________________________________ ______________________________________________________ d. {33������������+−������������==150 _____ Remember: The graph of a system of linear equation in two variables is categorized into 3: intersecting, parallel, and coinciding. Integrated the Development of the Following Learning Skills: 2. Character 1. Communication Skills • Consistency • Understanding of words/vocabulary • Patience • Perseverance • Following instructions/directions. 4. Character • Reasoning • Values integration 3. Critical Thinking • Honesty • Observation • Working independently • Reflection • Decision making Evaluate Yourself! What I have learned? Write whether the graph of a system of equation is intersecting, parallel or coinciding. 1. {2������������ − ������ = 3 __________________________ − 2������ = 6 __________________________ ___________________________ 2. {������������ = 4������ + 1 ___________________________ = 4������ − 1 ___________________________ 3. {63==2������������−+������������ 4. {������������==25−−2������������ 5. {147������������ + ������ = 1 + 2������ = 2 Prepared by: GEMMA ORESCO - COTONER Kalayaan National High School References: Grade 8 Learner’s Module, pp. 254-256 E-math Intermediate Algebra II Worktext, pp.3—8 124 | P a g e

Module Code: Pasay-M8–Q1–W7-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 7/ Day 2 OBJECTIVE: Solves a system of linear equations in two variables by graphing. YOUR LESSON FOR TODAY: In this module, students will solve a system of linear equations in two variables by graphing. Unlocking of Difficulties Graphical solution to a system of equations- to find the solution to a system of equations graphically mean to find the coordinates of the point where the two linear equations crossed, or intersected e. g. x = 4 and x+ y = 5 intersects at point (4, 1). This implies that x =4 and y=1 or the ordered pair (4,1) is the solution to the system x = 4 & x + y =5 Point of Intersection – a distinct point where given lines crossed each other. TRY TO DISCOVER! Activity 1: Find the solutions of each of the following systems of linear equations graphically. a. x + y = 3 b. y – x = -4 x–y=1 2x+ y = 5 Just like what you were asked to do in previous lessons, you need to graph linear equations in two variables in the Cartesian plane. And as required before, you need to obtain some ordered pairs and write them in a table. The sample table of values were written below: In both equations, we may easily do so by setting first x to zero (to get the x-intercept) and y to zero (to get the y-intercept). This is because both coefficients of x and y is either 1 or -1. So, we have: a. x + y = 3 x 023 y 310 x–y=1 x 012 y -1 0 1 Q: How did you come up with the graphical solution in our first system of equations? Identify the point of intersection of the two limes. Is the solution to this system of linear equations distinct? ___________________________________________________________ 125 | P a g e

Precisely! In the previous example, x + y = 3 and x – y = 1 may be graphed by either the intercept method or by plotting points and since the given system Ax + By = C & ax + by = c is a consistent and independent system it has a distinct solution which is (2,1) It is almost too easy to notice why the solution is the coordinate pair x =2 and y =1 or (2, 1)if we are to examine the point series for the lines of x + y =3 and x – y =1. We can check that ( 2, 1 ) is the solution set by verifying that x = 2 and y = 1 makes both equations of the system, true at the same time. b. Solve the system y – x = -4 by graphing. 2x + y = 5 This time, it is more convenient to choose to graph each equation using the slope – intercept method. Activity 2: Graphing systems of equations that may be done ideally using slope and intercept. In letter b, we can see an example of a pair of linear equations in two variables that may be graphed conveniently using slope-intercept method. y= -2x+5 y=x - 4 As shown in the figure the lines intersect at (3, -1). The checking illustrated on the right of its graphical solution supports that clearly, x =3 and y =-1 satisfies both equations of the system. 126 | P a g e

What mathematical concepts is a prerequisite for you to do the graphing activity smoothly and accurately? Explain. ____________________________________________________________________________ ____________________________________________________________________________ Take Note! In solving systems of linear equations in two variables by using graphical methods, related mathematical skills and ideas are put into practice. a. Writing a linear equation from general to slope-intercept form; sometimes you may be required to prepare a table of values for the points you need to include in the graphing. b. The main idea here is you are applying previously learnt skill in graphing linear equations, just that you are graphing a pair of linear equations in two variables in one coordinate plane. c. You should know which system has no solution, one solution, and infinitely many solutions. An inconsistent system has no solution, while a dependent system has infinite solutions. The kind of system we are normally concerned with is an independent and consistent system which has a distinct solution (x, y). PRACTICE EXERCISES 1: Activity 2 Look closely at the solution set of each system of equations from its graph. Identify where the lines crossed in each of the coordinate plane. Verify your answer by plugging in the values of x and y into both equations. Solution: ________________ Solution: ________________ Solution: ________________ Were you able to find the solutions of each system of linear equations? Is the graphical method a convenient way to provide the exact solution to a system of equations? 127 | P a g e

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Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Collaboration 4. Critical thinking a. Following instructions/directions. 3. Character a. Analysis b. Reasoning a. Patience b. Observation b. Accuracy ____________________________________________________________________________ PRACTICE EXERCISES 3 Activity 6 : Who was he? This mathematician born in 600 BC was said to be the first mathematician to prove a theorem. https://greatestgreeks.wordpress.com/2016/03/11/ To find out, solve the following systems match the letter that corresponds to the answer. The letters will spell out the name of this famous mathematician. 1. x + 2y = 4 A (-3, -1) x = 6y - 4 B (2, 4) 2. x + 3y = 9 21 2x – y = 4 E (3 ,5) 3. 2x + y = 7 1 x – 2y = -1 F (2, - 3 ) 4. x + 2y = 8 H (3, 2) 2x – y = 6 L (4, 2) 5. 3x + 5y = 3 9x – 10y = 4 6. x + 3y = -1 3x – 6y = 12 Answer: ___ ___ ___ ___ ___ ___ 12 3 4 5 6 129 | P a g e

EVALUATION Solve the following by graphing. Please show the checking after your solution. 1. x- y = 3 2. x + y = 7 3. 2x + y = 4 y=-x+3 2x – y = -10 y – 2x = 0 Prepared by: Janet S. Dar-Juan Kalayaan National High School References for Further Enhancement: https://1.cdn.edl.io DLM 2 – Unit 1 Systems of Linear Equations and Inequalities Grade 8 Learner’s Module, p. 273 130 | P a g e

Module Code: Pasay-M8–Q1–W7-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 7/ Day 3 OBJECTIVE: Solves systems of linear equations in two variables by (a) substitution, (b) elimination. 131 | P a g e

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Module Code: Pasay-M8–Q1–W7-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 7/ Day 4 OBJECTIVE: Solves problems involving systems of linear equations in two variables. 137 | P a g e

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