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Quarter 1 Compiled Modules

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Remember: In simplifying rational algebraic expression is in simplest form when the numerator and the denominator of the expression have no common factors other than 1 and -1. To simplify a rational algebraic expression: 7. Factor both numerator and denominator simultaneously into their prime factors using Sum and Difference of Two Cubes, 8. Identify the Greatest Common Factor that is equal to 1. 9. Apply the Fundamental Property of Fractions If a represents a real number and b and c represents non-zero numbers, ������������ = ������ ������������ ������ PRACTICE EXERCISES 3: Activity 5. Read, think and Write 4. Describe the patterns: a. Sum of Two Cubes b. Difference of Two Cubes 5. Discuss how important the pattern in factoring Sum and Difference of Two cubes in simplifying rational algebraic expression. 6. Explain why ������3+������3 does not simplify to 1. (������+������)3 7. Give your own example of a Rational Number which can be simplified using Sum and Difference of Two Cubes. _________________________________________________________________________ 2. Character Integrated the Development of the Following Learning Skills: A. Accuracy 1. Communication Skills B. Patience A. Understanding of words/vocabulary: sum and difference of two cubes C. Perseverance B. Following instructions/directions. C. Reasoning D. Responding to ideas 51 | P a g e

EVALUATION: Simplify the following rational algebraic expressions using sum and difference of two cubes. 1. (������−3)(������2+2������+9) ������3−27 2. (4������−1)(16������2+4������+1) 64������3−1 3. ������3−1 18(������−1) 4. ������3−������3 ������2−������2 5. 27������3+125������3 3������+5������ Prepared by: Rosalyn C. Bantay Pasay City East High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 77-78 https://www.mathwarehouse.com/algebra/rational-expression/how-to-simplify-rational-expressions.php https://www.rappler.com/newsbreak/iq/241607-things-to-know-about-star-city 52 | P a g e

Module Code: Pasay-M8–Q1–W3-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 3/ Day 4 OBJECTIVE: Performs addition and subtraction on rational algebraic expressions. YOUR LESSON FOR TODAY: • Performing addition and subtraction on rational algebraic expressions. TRY TO DISCOVER! The idea on addition and subtraction of rational algebraic expression is the same with addition and subtraction of fractions. Finding the Least Common Denominator (LCD) is important in performing the operations. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.pixtastock.com Let us study the following illustrative examples: 1. 3 + 4 = 3+4 = 7 5������ 5������ 5������ 5������ 2. 12 − 7 = 12−7 = 5 ������2 ������2 ������2 ������2 3. 4������ − 3������ = 4������−3������ ������+������ ������+������ ������+������ As you can see the given rational algebraic expressions are similar. How will you know if it is similar? Rational Algebraic Expressions are similar if the denominators are the same. So, if you will add or subtract similar rational algebraic expressions add/subtract the numerators and copy the denominator. Take a look at the following examples. 4. 2������ + 2������ = 2������+2������ = 2(������+������) = 2 ������+������ ������+������ ������+������ ������+������ 5. ������2 + ������−12 = ������2+������−12 = (������−3)(������+4) = ������ − 3 ������+4 ������+4 ������+4 ������+4 When you add or subtract rational expressions it is important to write the final answer in simplest form. How will you know if the expression is in simplest form? A rational algebraic expression is in its simplest form if the numerator and denominator have no common factor except 1. Factoring is an essential knowledge in the process. Since, numerator and denominator has exactly the same terms we can be able to cancel. 53 | P a g e

ARE YOU READY TO PRACTICE? Let us try to enhance your mathematical skills, study the given examples: 1. 2 + 1 ������ 3������ Solution: Get first the least common denominator since rational algebraic expression are dissimilar. LCD: 3x = 2(3) + 1(3) 3������ 3������ = 6+7 To add/subtract dissimilar rational algebraic expressions do the following: 3������ 1. Get the complete factors of the =7 denominator and find the LCD. 3������ 2. Divide the LCD to each denominator then multiply the result to the 2. 2 − 5 + 4 numerator. 3������2 2������ ������ 3. Add or subtract the numerator and copy = 2(2)−5(3������)+4(6������) the ne denominator, LCD. 6������2 4. Factor the sum or difference if factorable = 4−15������+24������ and cancel the exact similar factors in 6������2 the numerator and denominator. = 4+9������ 5. Check if the final answer is already in 6������2 simplest form. 3. 6 − ������+������ https://www.google.com/url?sa=i&url=https%3 (������−������)2 ������−������ A%2F%2Fwww.colourbox.com https://www.google.com/url?sa=i&url=https%3 = 6−(������+������)(������−������) A%2F%2Fwww.freeimages.com (������−������)2 = 6−(������2−������2) (������−������)2 = 6−������2+������2 (������−������)2 4. 2 + 1 − 7 10. ������+1 − ������2+3������ + 5 ������2−5������+6 ������−3 ������−2 ������−4 ������2−16 2������−8 = 2+(������−2)−7(������−3) 2(������ + 1)(������ + 4) − 2(������2 + 3������) + 5(������ + 4) = 2(������ + 4)(������ − 4) (������−3)(������−2) = 2+������−2−7������+21 2(������2 + 5������ + 4) − 2������2 − 6������ + 5������ + 20 = 2(������ + 4)(������ − 4) (������−3)(������−2) = 21−6������ 2������2 + 10������ + 8 − 2������2 − 6������ + 5������ + 20 ������2−5������+6 = 2(������ + 4)(������ − 4) 9������ + 28 = 2������2 − 32 1. Does every step have a mathematical concept involved? 2. What makes that mathematical concept important to every step? 3. Can mathematical concept in every step be interchanged? How? 4. Can you make another method in adding or subtracting rational algebraic expressions? How? 54 | P a g e

PRACTICE EXERCISES 1: Perform the indicated operations and express your answer in simplest form. 1. 5 + ������+8 6. 9 + 4 ������2������ ������������2 ������ ������ 2. 10 − ������ 7. 3 + 1 − 8 3−������ 3−������ ������ 2������ 4������ 3. 4 + 15 − 3������ 8. 3 + 8 − 15 ������ ������2 2������ ������ ������−4 ������2 4. 5������ − −5������ 9. 3 − 2−������ ������−1 ������2−1 ������+������ ������+������ 1 2������+3 5. ������ + ������ 10. ������+4 + ������2−16 ������2−������2 ������2−������2 PRACTICE EXERCISES 2: Perform the indicated operations and simplify the final answer. 1. ������−������ + 7������−3������ 6. ������+6 + 2������+1 8������ 8������ 3������−6 3������−6 2. ������−3������ − ������+3������ 7. 6 − 5������ 6������3������ 6������3������ ������−1 4 3. ������−1 + ������+1 8. 4 + 3 10������2������4 15������6������2 ������−8 ������+7 4. 1 3 9. 7 − 9 ������ ������2 3 3������−2 ������ + − 2 1 10. ������2−4������+4 + ������2+������−6 5. 3 + 8 ������ ������−4 Reminder: Again, what have we learned today? • If the two rational algebraic expressions that you want to add or subtract have the same denominator you just add/subtract the numerators which each other and copy the denominator. When the denominators are not the same in all expressions that you want to add or subtract you have to find a least common denominator or PRACTICE EXERCISES 3: A. Perform the indicated operation. B. Perform as indicated, simplify your answer if Express your answer in simplest form. necessary. 1. 6 + 4 1. 3 − 4 ������−5 ������−5 ������+1 ������ 2. 7 − 5 2. 2������ − 3 ������2−9 ������−3 4������−1 4������−1 3. ������−2 + ������−2 3. ������+2 − ������+2 ������−1 ������−1 ������ 2 4. ������2+3������−2 + ������2−2������+4 4. 3 − 2 ������2−4 ������2−4 ������2−������−2 ������2−5������+6 5. ������2+3������+2 3������+3 5. ������+8 + 3������−2 ������2−2������+1 ������2−2������+1 ������2−4������+4 ������2−4 − 55 | P a g e

Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Character A. Following instructions/directions. A. Patience B. Understanding messages B. Perseverance D. Responding to ideas EVALUATION: Perform the indicated operations, then simply your final answer if necessary. 1. ������ + ������ 6. 4 − 6 ������ ������ 3������+9 4������+12 2. 1 − ������ 7. 6 + 2 2������2+2 ������2+������+1 ������ 3. 3 − 2−������ 8. 1 − 1−������ ������2−1 ������2+4������+3 ������−1 ������−1 4. ������ + 1 − 3 9. ������ − 3 + 1 ������ ������2 ������+2 ������4−16 2−������ 5. 4������−3 + 3 − 2������−1 10. 3������−2 + 2������−3 18������3 4������ 6������2 ������2−3������−4 ������2−������−12 Prepared by: Arlyn L. Esber Pasay City National Science High School References for Further Enhancement: Grade 8 Leaner’s Module Unit 1, pages 94, 97 Integrated Math for Grade 8, Leonor and Chavez, pages 139-142 https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.pixtastock.com https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.colourbox.com https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.freeimages.com https://www.google.com/url?sa=i&url=https%3A%2F%2Fclipartart.com 56 | P a g e

Module Code: Pasay-M8–Q1–W4-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 4/ Day 1 OBJECTIVE: Performs multiplication and division on rational algebraic expressions. YOUR LESSON FOR TODAY: In this module I will be leading you through how to multiply and divide rational algebraic expressions. Most of the times in math you apply past concepts and ideas to be able to work all the way through the new problems. In this module, you will learn how to factor, simplify rational algebraic expressions, multiply and divide polynomials to be able to perform multiplication or division of rational algebraic expressions. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.dreamstime.com TRY TO DISCOVER! Activity 1: Match It Down. Match each rational algebraic expression to its equivalent simplified expression form choices A to E. Write the rational expression in the appropriate column. If the equivalent is not among the choices, write it in column F. E. ������ A. -1 B. 1 C. a + 5 D. 3a 3 ������2 + 6������ + 5 ������3 + 2������2 + ������ 3������2 − 6������ ������ − 1 ������ + 1 3������2 + 6������ + 3 ������ − 2 1 − ������ (3������ + 2)(������ + 1) 3������3 − 27������ ������3 + 125 ������ − 8 3������2 + 5������ + 2 (������ + 3)(������ − 3) ������2 − 25 −������ + 8 18������2 − 3������ 3������ − 1 3������ + 1 ������2 + 10������ + 25 −1 + 6������ 1 − 3������ 1 + 3������ ������ + 5 ABCDE F How did you come up with your answer in the activity? ____________________________________________________________________________________ Correct, you only have to simplify each rational algebraic expressions and apply factoring polynomials. 57 | P a g e

Take Note! To multiply rational algebraic expressions: • Factor the numerators and denominators completely. • Cancel out exactly similar factors in the numerators and denominators. • Write the answer in simplest form. To divide rational algebraic expressions: • Get the reciprocal the rational algebraic expressions after the division symbol. • Proceed to multiplication. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.pngitem.com Let us study the following illustrative examples: 1. 3 ∙ 2 Express the numerators and denominators into prime factors if possible. 14 5 Cancel out common factors. = (3)(2) Simplify (7)(2)(5) =3 35 2. 12������3������ ∙ 75 25 8������������3 (3)(4)(25)(3)(������)(������)(������)(������) = (25)(4)(2)(������)(������)(������)(������) 9������2 Can you identify what law of = 2������2 exponent applied in these steps? 3. 4������−12 ∙ (������+1)2 ������2−9 ������2−1 4(������ − 3) (������ + 1)(������ + 1) What factoring methods were used in = (������ − 3)(������ − 3) ∙ (������ + 1)(������ − 1) this step? 4 ������ + 1 What are the rational algebraic = ������ − 3 ∙ ������ − 1 expressions equivalent to 1 in this step? 4(������ + 1) 4������ + 4 = (������ − 3)(������ − 1) ������������ ������2 − 4������ + 3 Have you followed the process on multiplying rational algebraic expressions? ________ How was it? Can you form the steps on how to get the product? _______________________________________________________________________ Multiplication of Rational Algebraic Expressions has the same process when you multiply fractions. Factoring and simplifying rational expressions is an essential tool to perform the operation. 58 | P a g e

Now let us try finding the quotient. Look at the illustrative examples below. 4. 6������������2 ÷ 9������2������2 Get the reciprocal of the divisor or an 4������������ 8������������2 expression after the division symbol. Proceed to multiplication. = 6������������2 ∙ 8������������2 4������������ 9������2������2 Express the numerators and denominators into prime factors if = (3)(2)(������)(������)(������) ∙ (2)(2)(2)(������)(������)(������) possible. Cancel out common factors. (2)(2)(������)(������) (3)(3)(������)(������)(������)(������) = 4������ 3������ Aside from cancellation you can also apply the Laws of Exponent most especially if the exponents has a large value. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.alamy.com 6. 2������2+������−6 ÷ ������2−2������−8 2������2+7������+5 2������2−3������−20 = 2������2+������−6 ∙ 2������2−3������−20 Why do we need to factor out the 2������2+7������+5 ������2−2������−8 numerators and denominators? = (2������−3)(������+2) ∙ (2������+5)(������−4) (2������+5)(������+2) (������−4)(������+2) = (2������−3)(������+2) ∙ (2������+5)(������−4) What happens to the common (2������+5)(������+2) (������−4)(������+2) = 2������+3 factors between numerator and denominator? ������+2 So, how was the division of rational algebraic expression performed? ________________________________________________________________________ It is much easier now because we already have an idea on multiplication, isn’t it? PRACTICE EXERCISES 1: Activity 2: Warm Up! Perform the indicated operations and express your answer in simplest form. 1. ������6 ∙ 16 6. 4������ ∙ ������+1 8 ������3 ������2+������ 8 2. 10������2 35������2 7. ������+3 ÷ 1 7������ 5������3 ������+2 ������+2 ∙ https://www.google.com/url?s 3. 12������2 ÷ 3������3 8. 3������−6 ÷ ������−2 a=i&url=https%3A%2F%2Fcli 25������ 5������2 partart.com 14 8 4. 36������4 ÷ 24������2 9. ������+3 ∙ 3������−18 25������3 9������5 4 3������+9 5. 27������3 ∙ 8������4 5������+15 ∙ 8 64������ 9������5 4������−4 ������+3 10. 59 | P a g e

PRACTICE EXERCISES 2: Activity 2. Enhance your Skills. Perform the indicated operations and simplify the final answer. 1. 32������4������ ∙ 5������6 6. 1 ÷ ������2+6������+9 45������3 8������8������2 ������2−4 ������2+������−6 2. 2������ ÷ (������ − 1) 7. ������2+7������−8 ∙ 2������+10 ������+1 2������+16 9������−9 3. ������2+3������ ∙ ������ 8. ������+3 ÷ ������2+2������−3 ������3 3������+9 ������+2 ������2−2������+1 4. 5������+20 ÷ ������2−16 9. ������4−81 ∙ 4−������ ������2−������−12 ������2−6������+9 15������ 3������−12 10.2������52���−���+32������0−2 ������2+2������−8 5. ������2−������2 ∙ 1 ÷ ������2−4 ������+������ ������+������ Remember: To multiply and divide rational algebraic expressions, don’t forget that: • If the two rational algebraic expressions that you want to add or subtract have the same denominator you just add/subtract the numerators which each other and copy the denominator. • When the denominators are not the same in all expressions that you want to add or subtract you have to find a least common denominator or LCD. Apply factoring polynomials. Simplify the expression if needed. PRACTICE EXERCISES 3: Activity 3: Think Critically. Perform as indicated. Express you answer in simplest form. 1. 4������2 − 1 ∙ ������2 + 2������ + 1 6. ������2 + 2������ − 3 ÷ 2������2 − ������ − 1 8������ + 4 2������2 + ������ − 1 2������2 + 7������ + 3 4������2 − 1 ������2 − 9 2. ������2 − ������ − 12 ∙ ������2 ������ −4 9 8������2 + 2������������ − 15������2 4������2 − 9������������ + 5������2 − 6������ + ������3 + ������3 ������2 − ������2 ÷ 3. ������2 − 1 ∙ ������3 − 1 7. ������2 − 2������ + ������2 − 2������ + ������2 − 9 1 1 4������ + 12 4. ∙ 9 − 16 ������2 8. (8������6 − 27������6) ÷ (4������4 + 6������2������2 + 9������4) 6������ + ������2 ������2 + 3������ − 18 5. − 6������ + 5 ∙ ������2 + ������ − 30 ������3 + 1 2������ − 1 ������ + 1 ������ −3 4������2 − 1 ������2 − ������ + 2������2 − 5������ 9. ∙ ÷ 1 −3 THOUGTHS TO PONDER! 10. ������2 + 2������������ − 15������2 ∙ ������2 + 5������������ − 6������2 ÷ (������ + 6������) ������2 + 4������������ − 5������2 ������2 − 2������������ − 3������2 • Does every step have a mathematical concept involved? What makes that mathematical concept important to every step? ________________________________________________________ ________________________________________________________ _________________________________________________________________________ 2. Character A. Accuracy Integrated the Development of the Following Learning Skills: B. Patience 1. Communication Skills C. Perseverance A. Understanding of words/vocabulary: common ratio, geometric sequence. B. Following instructions/directions. C. Reasoning D. Responding to ideas 60 | P a g e

EVALUATION: Activity 4: Test Yourself. Perform the indicated operations and simplify. 1. 10������������2 ∙ 6������2������2 6. 81������������3 ÷ 27������2������2 3������������2 5������2������2 36������ 12������������ 2. ������2 − ������2 ∙ ������2 7. 2������ + 2������ ÷ 4 2������������ ������ − ������ ������2 + ������������ ������ ������2 − 3������ ������2 3. ������2 + 3������ − 10 ∙ ������2 − − 4 6 ������ − ������2 ������2 4. ������2 + 2������ + 1 ∙ ������2 − 1 8. ������ − 1 ÷ ������2 1 − ������ 1 − 2������ + 1 − 1 ������ + 1 + 2������ + 5. ������2 − 2������������ + ������2 ∙ ������ − 1 ������2 ������2 − 1 ������2 − 1 ������ − ������ ������2 ������2 + 2������ + 9. + 2������ + 1 ÷ + 4������ + 3 1 10. 16������2 − 9 ÷ 16������2 + 24������ + 9 − 5������ − 4������2 4������2 + 11������ + 6 6 Prepared by: Arlyn L. Esber Pasay City National Science High School References for Further Enhancement: Grade 8 Leaner’s Module Unit 1, pages 79, 88-89, 91 Integrated Math for Grade 8, Leonor and Chavez, pages 119-120, 126-127 https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.dreamstime.com https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.vhv.rs https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.alamy.com https://www.google.com/url?sa=i&url=https%3A%2F%2Fclipartart.com 61 | P a g e

Module Code: Pasay-M8–Q1–W4-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 4/ Day 2 OBJECTIVE: solves problems involving rational algebraic expressions. YOUR LESSON FOR TODAY: In the previous lessons, you learned how to solve problems involving factoring polynomials. This time, you will apply and use rational algebraic expressions in solving problems. What you should know? https://www.google.com/url?sa=i&url=http%3A%2F%2Fwww.portageandmainpress.com Rational Equation is solved by: • Finding the Least Common Denominator (LCD). • Multiplying both sides by the LCD • Solving the equation using Properties of Equality. TRY TO DISCOVER! Activity 1: Do you know how? 1. To solve the equation, what should be multiplied on both sides? Write in the box. 7 −1= 5 +1 5������ 2 6������ 3 ������+4 − ������ = 3 ������+7 ������+3 8 2. Is x = -1, a solution for each equation below? Prove your answer. a. ������+4 = 3 b. 3−5������ = 2−4������ 77 43 Take Note! To solve rational algebraic expressions, do the following: • Represent the unknown. • Form a mathematical equation. • Find the Least Common Denominator (LCD). • Solve the equation using the different Properties of Equality. • Check your answer. Let us start with an illustrative examples. Example 1 Seven divided by the difference of a number and four is equal to one-fifth the difference of the number and two. Find the numbers. Let us begin by representing the unknown using any variable. Let x = the number Form the working equation. 62 | P a g e

7 = 1 (x − 2) Apply Distributive Property ������−4 5 7 = ������−2 ������−4 5 Before we solve, take note that there's a variable in the denominator which means we will have a restricted value. So, x ≠ 4 because that would make the denominator zero which would make the expression undefined. Now we can cross multiply and solve! ������2 − 6������ + 8 = 35 Nsoonwee(���o���k������f2n−t−oh9w6e)���(���sb������−oo+ltu2h37ti)to=h=ne0s0searweiloluwrorreks.tricted value ������2 − 6������ + 8 − 35 = 35 − 35 Theref���o��� r=e,9th���������e��� ���n��� =um−b3ers are 9 and -3. Example 2 If a and b are real numbers, such that a + b = 3 and a2 + ab = 7. Find the value of b. Solution: Can you identify what mathematical concept was applied? ������(������ + ������) = 7 ������ = 7 What Property of Equality was used? 3 Since a + b = 3 Then, 7 + ������ = 3 Substitution Property 3 ������ = 3 − 7 Subtraction Property of Equality 3 ������ = 2 3 Example 3 If Jarod can paint a fence in 6 hours and Jaden can do the same in 4 hours, how long will it take them to paint the fence if they work together? Problems in which two or more job, machine or people work together to complete a certain task are referred to as Work Problems. Part of task done Part of task done 1 by first person + =by second person (one whole task or machine. or machine. completed) https://www.google.com/u To determine the part of the task completed by each person or rl?sa=i&url=https%3A%2F machine, we use the relationship: %2Fwww.clipartlogo.com rate x time = part of the task completed 63 | P a g e

Representation: fractional part of job done Let x = the number of hours they work together. in 1 hour It is best to represent the data and unknown in table form. 1 total time in hours 6 1 Jarod 6 4 1 Jaden 4 ������ Together x Amount of work Amount of work Total amount of work done in 1 done by Jarod in 1 + done by Jaden in 1 = hour hour hour 1 1+1= ������ 64 Equation: 1+1 = 1 LCD: 12x Solution: 6 4 ������ 12������ (1 + 1 = 1) 12������ Answer: 6 4 ������ It will take 2 2 hours or 2 hours and 24 2������ + 3������ = 12 5 5������ = 12 ������ = 12 ������������ 2 2 minutes for Jarod and Jaden to finish painting of fence together. 55 Example 3 A car and truck travel at the same rate. The car travels for 3 hours and the truck for 5 hours. If the truck travels 104 more miles than the car, how far does the car travel? Representation: Let x = the distance the car travels; x + 104 = the distance the truck travels distance time rate https://www.google.com/url?sa=i&url=https%3A%2F%2F x3 ������ www.pngitem.com 3 x + 104 5 ������ + 104 5 https://www.google.com/url?sa=i&url=http%3A%2F%2Fclipart- library.com 64 | P a g e

Equation: ������ = ������+104 Since it both travel at the same rate. Can you identify what process was apply in this step? 35 What properties of equality were applied in these steps? Solution: 5x = 3(x + 104) Answer: The car travels 156 miles. 5x = 3x + 312 2x = 312 x = 156 PRACTICE EXERCISES 1 Activity 2. Warm Up Solve the following problems. 1. The sum of two numbers is 40. The ratio of the smaller number to the larger number is 3. Find the 5 numbers. 2. If ������ + ������ ÷ ������ = 12, ������ + ������ ÷ ������ = 9 ������������������ ������+������ = 7. ������������������������ ������ + ������ + ������. ������ 3. Cardo can repair a house in 4 hours and his brother can do the same job in 8 hours. Cardo began to work alone and worked for 2 hours. Then his brother joined him until the job is completed. How long did the entire repair job take? 4. A pipe can fill a pool in 3 hours. The drain can empty the pool in 10 hours. With the pipe on and the drain open, how long would it take the pool to be filled? 5. Nurse Judy Ann can cycle 16 miles going to her work, at the same time that she can walk 6 miles. If she rides 5 miles per hour faster than she walks, how fast does she walk. 6. A jogger ran 8 miles and then walked 6 miles. The jogger’s running speed was 5 miles per hour faster than her walking speed. The travel time for jogging and walking was 2 hours. What is the speed of the jogger’s walking? PRACTICE EXERCISES 2 Activity 3. Enhance you Skills Solve each of the following problems. 1. The sum of two numbers is 20. The ratio of the smaller number to larger number is 2. Find the 3 numbers. 2. Twice the reciprocal of a number added to 1 is equal to one. What is the number? 3 3. If ������ = ������ − ������ + ������ , ������������ = 5, ������ − ������ = 12, what is b? ������ ������ − ������ 4. DJ Loonyo can clean the house in 4 hours. His younger sister can clean it in 6 hours. How long will it take them to clean the house if they will work together? 5. A river has a current that moves at 2 miles per hour. A boat can travel 11 miles with the current in the same time that it can travel 9 miles against the current. Find the speed of the boat in still water. Remember To solve problems involving rational algebraic expressions, follow these steps: 1. Read and Analyze the problem carefully. 2. Represent the unknown. 3. Formulate the equation. 4. Identify the Least Common Denominator (LCD). 5. Solve and check the solution. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.clipart.e mail 65 | P a g e

PRACTICE EXERCISES 3: Activity 4. Think Critically Solve each of the following problems. 1. The numerator of 6 is multiplied by a number and the same number is added to the denominator, 5 the new fraction is 13 . Find the number. 3 2. The sum of two numbers is 90. When the larger number is divided by the smaller, the quotient is 3 and the remainder is 10. What are the numbers? 3. If a, b and c are real numbers, such that ������������ = 1 , ������������ = 1 ������������������ ������������ = 1 . ������ℎ������������ ������������ ������������������ ? ������+������ 2 ������+������ 5 ������+������ 8 ������������+������������+������������ 4. A tank has an inlet and an outlet pipe. An inlet pipe can fill the tank in 6 hours and an outlet pipe can drain the tank in 3 hours. How long will it take to fill the tank if both pipes are open? 5. A motorbike travels 40 kilometers and a bicycle travels 15 kilometers in the same time. The rate of the bicycle is 20 kph less than the rate of the motorbike. What is the speed of the motorbike and bicycle? THOUGHTS TO PONDER! How important is self discipline in answering problems involving rational algebraic expressions? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ _______________________________________________________________________ Integrated the Development of the Following Learning Skills: 2. Character 1. Communication Skills A. Accuracy B. Patience A. Understanding of words/vocabulary: common ratio, geometric sequence. C. Perseverance B. Following instructions/directions. C. Reasoning D. Responding to ideas EVALUATION: Solve the following problems. 1. One number is 3 times another. The sum of their reciprocals is 20. Find the numbers. 3 2. The sum of a number and 1 of its reciprocal is − 13. Find the number(s). 36 3. It takes 10 hours to fill a pool using the inlet pipe. It can be drained in 15 hours using the outlet pipe. If the pool is half full, how long will it take to fill it if both pipes are open? 4. If ������������������ = 1, ������������������������ ( ������ + ������ + ������ )2. ������������ + ������ + 1 ������������ + ������ + 1 ������������ + ������ + 1 5. Vhong from his home, jogs a certain distance at a rate of 8 kilometers per hour. He runs back home over the same track at the rate of 24 kilometers per hour. If it took him 2 hours and 30 minutes jogging and running, what is the total distance travelled? Prepared by: Arlyn L. Esber Pasay City National Science High School References: Grade 8 Learner’s Module, pp. 106 Integrated Math 8 (Leonor and Chavez), pp. 156-158, 169, 178, 183-184, 186-187 Exploring Mathematics 8 (Baccay, Esperanza and Reyes), pp. 148, 150, 152, 155-158 Elementary Algebra (Eferza) page 126 http://www.mathguide.com/lessons/SequenceGeometric.html https://www.expii.com/t/applications-of-rational-expressions-and-word-problems-4604 66 | P a g e

Module Code: Pasay-M8–Q1–W4-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 4/ Day 3 OBJECTIVE: Illustrates the Rectangular Coordinate System and its uses. YOUR LESSON FOR TODAY: • Rectangular Coordinate System / Cartesian Coordinate System TRY TO DISCOVER! I. What You’ll Learn: A. descried the rectangular coordinate plane (x-axis, y-axis, quadrant, origin) B. given a point on a coordinate plane, gives its coordinate C. given a pair of coordinates, plot the point D. determine the quadrant or axis location of a given coordinate II. Unlock Your Difficulties: A. Let’s Connect: • On a number line, you can determine the location of a point using single number. • A number line is a one-dimensional coordinate system since we can only move right or left on the line but not moving up or down. • From that concept, coordinate means location. NOW…. How about locating a point on plane? Determining the coordinate of point on a plane? …… Your lesson today will teach you HOW. B. Lets. Explore: Historical Note: The French Philosopher Rene Descartes ( ren-ay’ day-kart) marked the use of Rectangular Coordinate of rectangular coordinate system also known as Cartesian Coordinate System name after him. C. Definition of Terms: ➢ Cartesian coordinate plane or Rectangular Coordinate Plane = A two dimensional coordinate system ➢ X-axis = the horizontal number line on a plane. ➢ Y-axis = the vertical number line on a plane ➢ Origin = the intersection of the perpendicular lines with coordinate (0,0) ➢ The x and y axes = divide the set of points on the plane into four subsets or Quadrants 67 | P a g e

D. Discussion: ❖ The Cartesian plane is composed of two perpendicular number lines that meet at the point of origin (0,0) and divides the plane into four regions called Quadrants. ❖ Each pointin the coordinate system is defined by an ordered pair of the form (x, y), where x and y ∈ ℜ (elements of real number). ❖ The first coordinate of a point is called the x-coordinate or abscissa and the second coordinate is called the y-coordinate or ordinate. We call (x, y) an ordered pair because it is different from (y, x).Thus, the order matters. ❖ The signs of the first and second coordinates of a point vary in the four quadrants as indicated be Quadrant I x > 0 or x is positive y > 0 or y is positive or (+, +) Quadrant II x < 0 or x is negative y > 0 or y is positive or (-, +) Quadrant III x < 0 or x is negative y > 0 or y is negative or (-, -) Quadrant IV x > 0 or x is positive y > 0 or y is negative or (+, -) There are also points which lie in the x- and y-axes. The points which lie in the x-axis have coordinates (x, 0) and the points which lie in they-axis have coordinates (0, y), where x and y are real numbers. Illustrated below is a Cartesian plane ARE YOU READY TO PRACTICE? Determine the location of a point: Steps: 1. Look where the given point is situated in the x-axis. That is your x-coordinate or also known as abscissa. 2. Then look for the location of a given point in the y-axis and that is your y-coordinate or ordinate 68 | P a g e

F A B E D CB B Example • Point A falls on positive 5 in the x-axis and positive 4 on the y-axis. Therefore, the coordinates of Point A are ( 5, 4 ) and located in quadrant I.(QI) • Point B falls on the negative 3 in the x-axis and positive 1 in the y-axis. Therefore, the coordinates of Point B are ( -3 , 1 ) and located in quadrant II (QII) • Point C falls on the negative 4 in the x-axis and negative 4 in the y-axis Therefore, the coordinates of Point C are ( -4, -4 ) and located in quadrant III (QIII) • How about Point D ? The coordinates of Point D are (____, ___ ) and located in quadrant ____ ? Q ___ • Point E falls on positive 3 in the x-axis and zero in the y-axis Therefore, the coordinates of Point E are (__,__) and located in ___ -axis? • Point F falls on zero in the x-axis and negative 5 in the y-axis. Therefore, the coordinates of Point F are ( __, ___) and located in ___-axis. 69 | P a g e

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Module Code: Pasay-M8–Q1–W4-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 4/ Day 4 OBJECTIVE: Illustrates linear equations in two variables. YOUR LESSON FOR TODAY: Linear Equations in Two Variables. TRY TO DISCOVER! I. What you will learn: • Writing linear equation ������������ + ������������ = ������ in a form ������ = ������������ + ������ and vice versa • Determine the y-intercept and the slope of the given equation. • Find the equation of a line given the value of slope(m) and the intercept. II. Unlock your difficulties Definition of terms • Variables = a symbol for a number we don’t know yet, It is usually a letter like ������ ������������ ������. • Constant = a fixed value, one that value does not change. All real numbers are constant terms, for example in the equation -������������ + ������ = ������, the constant term is 6 • Numerical coefficient = a fixed number that is multiplied to a variable. For example, in the term −������������������, the numerical coefficient is -5 • m = means slope or it refers to a slope • ������������ + ������������ = ������= general form of linear equation • ������ = ������������ + ������= slope intercept formor sometimes called the y-form. • III. Let’s do this Linear Equation in two variables > the exponent is 1 ➢ the two variables are not part of the same term ➢ the variable must not appear in the denominator of the fraction Activity I. Determine whether or not each equation is a linear equation in two variables. If yes, rewrite in general form ������������ + ������������ = ������ then, identify ������, ������ ������������������ ������ Example: 2������ + 4 = 2������ans. YES, itis a linear equation. 2������ − 2������ = -4, is the general form, therefore ������ = 2, ������ = −2 ������������������ ������ = −4 • 5������������ = 7 ans. NO because the variable x and y appear in the same term • 2������2-4������= 12 ans. NO because the exponent is 2 • -− 3 − 6 = 18 ans. NO because the variable is in the denominator of the fraction. 4X 73 | P a g e

1. 6������ = 10 + 4 ������> _________ 6. 4������ = 1/ 4������>__________ 2. ������ = 4������ + 16 > ________ 7. 2( ������ + 4 ) = 6������>_________ 3. 4������2 – ������ = 10> ________ 8. 26������ + 4 = ������ 2 > _________ 4. 6 + ������ = 7 >_________ 9. 3������ = 8������������ + 4 > _________ ������ 10. 5������ = 2������ + 9 > __________ 5. ������������ = 4 > _________ ARE YOU READY TO PRACTICE? Activity 1. Write the linear equation in the form ������������ + ������������ = ������ into the form ������ = ������������ + ������ Steps: a. Make the term containing variable y on the left side of the equation. b. Make the term containing variable x and the constant term on the right side of the equation (NOTE: apply properties of linear equation) c. Divide the equation by the numerical coefficient of y. d. Identify the numerical coefficient of x as m, and the constant term as b, then, e. rewrite the equation in the form ������ = ������������ + ������. An example is done for you. 1. 2������ + 3������ = 5 ������. 4������ − ������ = 9 ������. ������ + 2������ = 4 3������ = 2������ + 5 3������ = −2������ + 5 33 3 ������ −2������ + 5 =3 3 25 ������ = − 3 ������ = 3 ������ = ������ = ������ = ������ = Activity 2. Write the linear equation ������ = ������������ + ������ into the form ������������ + ������������ = ������. Identify A,B,C. Steps: a. Make the term containing variables on the left side of the equation. b. The constant term must be on the right side of the equation. NOTE: (apply the properties of linear equation) c. Make the numerical coefficient of the variables x & y a whole number. (if it is in a fraction form) d. The numerical coefficient of x must be a positive number thus forming an equation ������������ + ������������ = ������. 74 | P a g e

An example is done for you. 1. ������ = ������ = +1 2. ������ = 2������ + 4 3. ������ = −������ = 4 2 ������ ������ = ������ = ������ = ������ − 2 = 1 (2) ������ + ������) = (2)(1) (− 2 (−1)(−������ + 2������) = (−1)(2) ������ − 2������ = 2 ������ = 1 ������ = −2 ������ = 2 ������ = ������ = ������ = Activity 3. Analyzed the results of your activity and complete the following statement. A ___________________is an equation in two variable and can be written in two forms first form is ___________________, Where ������, ������, ������ are _________________such that ������ ≠ 0 ������������������ ������ ≠ 0. Second form is ___________________, where _______ is the slope and ______ is the y-intercept, where m and b are real numbers. PRACTICE EXERCISES 1: Rewrite the following linear equations ������������ + ������������ = ������ in the form ������ = ������������ + ������ or vice versa. Identify the slope (������) and the ������ – ������������������������������������������������������ (������) in case of slope- intercept form. Identify the numerical coefficient of ������ ( ������ℎ������ ������ ), numerical coefficient of ������ ( ������ℎ������ ������ ) and the constant ( ������ℎ������ ������ ) in case of general form. 3������ + 7������ = 14 2 2������ + ������ = 6 ������ = 3 ������ = 2 PRACTICE EXERCISES 2: −3������ + 3������ − 1 = 0 5������ − 7������ = 2 ������ = 2������ = 1 75 | P a g e

Remember: A linear equation in two variables is an equation that can be written in the form Ax + By = C where A, B, C are real numbers, but A and B cannot both be zero. Rewriting the general form (Ax + By = C) into slope-intercept form (y = mx + b) will make it easier to identify the slope (m) and the y- intercept (b) PRACTICE EXERCISES 3: Given a value of slope (������) ������������������ ������ − ������������������������������������������������������ (������). Find the equation of a line. An example is done for you. 1. ������ = 2 ������������������ ������ = −2 ➢ ������ = ������������ + ������ > the slope – intercept form ➢ ������ = 2������ – 2 > replace ������ by the given value which is ������, then replace b by given value which is −������ Ans: ������ = 2������ – 2 > the equation of the line. 2. ������ = 3 ������������������ ������ = 6 3. ������ = 1 ������������������ ������ = −3 5 4. ������ = −4 ������������������ ������ = 2 4 5. ������ − 6 ������������������ ������ = 0 6. ������ = 2 ������������������ ������ = 1 7. ������ = 0 ������������������ ������ = −8 8. ������ = 5 ������������������ ������ = 6 9. ������ = 2 ������������������ ������ = − 4 33 10. ������ = − 4 ������������������ ������ = 3 55 ____________________________________________________________________________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Creativity 3. Character 4. Critical Thinking A. Developed the understanding of words/ key concept A. Writing A. Accuracy A. Decision Making B. Carefully following instructions B. Visual Art B. Patience B. Responding to Ideas C. Reasoning C.Independent C. Reflexion EVALUATION: I. Which of the following are linear equation in two variables? If not, state the reason. Yes NO Reason 1. 4������ + 2������ = 2������ – 5 __________ ___________ _______________ ___________ _______________ 2. 3������ = 6������������ – 4 __________ ___________ _______________ ___________ _______________ 3. 5( ������ + 2 ) = 10������ − 1 __________ 4. 27������2 + 4 = 2������ __________ 5. 4������ + 3 + 5 __________ ___________ _______________ 3������ 76 | P a g e

II. Rewrite the following linear equations ������������ + ������������ = ������ in the form ������ = ������������ + ������ or vice versa 1. ������ + 2������ = 2������ – 5 2. ������ = −4/3������ + 2/3 3. 2������ + ������ = −4 4. ������ = 3������ − 1 5. 2������ = 5������ III. Given a slope (������) and y-intercept (������). Find the equation of a line 1. ������ = −4 ; ������ = 6 2. ������ = 8 ; ������ = −3 3. ������ = 2/3 ; ������ = 9 4. ������ = −5 ; ������ = −2 IV. REFLECT: Answer in 2 to 3 paragraphs. Use separate paper if necessary. Q; Many things in life operate like a linear equation in two variables where an Input result into an output. For example, investing in a friendship you gain happiness. Working hard you will succeed. Now, it’s your turn to cite an example in your life as a grade 8 students that appears to be linearly related. ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________ Prepared By: Mary Grace R. Ocasion Pasay City West High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 181-182 Intermediate Algebra Simplified Concepts and Structures pp.17 – 19 Grade 8 Mathematics Patterns and Practicalities pp.144 – 146 77 | P a g e

Module Code: Pasay-M8–Q1–W5-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 5/ Day 1 OBJECTIVE: Illustrates and finds the slope of a line given two points, equation, and graph. YOUR LESSON FOR TODAY: In any given linear equation, the steepness and direction of the line can be determine by means of slope. TRY TO DISCOVER! I. What you will learn: • Identify the ������1 , ������2, ������1 , ������2 • By graphing the coordinates of a given line you can solve for the slope. • Given an equation, identify the slope II. Unlock your difficulties • if two point ( ������1, ������1 ) ������������������ ( ������2 , ������2 ) are on a line, then the slope of the line is defined as ������ = ������2− ������1 or ������ = ������1−������2 ������2−������1 ������1−������2 • In equation y = mx + b the slope is the numerical coefficient of x. III. Let’s do this B(2,5) Run rise rise = 4 y-intercept A(0,1) run=2 x-intercept • Study the graph of a line with point ������ ( 0 , 1 ) ������������������ ������������������������������ ������ ( 2 , 5 )? • As we move from A to B there is a change in the ������ – coordinates and change in the ������ – coordinates. Slope = ������������������������ = 4 = 2 ������������������ 2 • If we let point ������ ( ������1, ������1 ) be the first point, point ������ ( ������2 , ������2 ) be the second point and m be ������ = ������������������������ = ������ℎ������������������������ ������������ ������ = the slope of the line, then Slope ������������������ ������ℎ������������������������ ������������ ������ ������1−������2 ������������ ������2− ������1 ������1−������2 ������2−������1 The ratio between the change in ������over the change in������is called the slope of a line. Let us determine the slope of the given two point, ������ ( 0,1 ) and ( 2 , 5 ) . ������ = ������2− ������1 = 5−1 4 = 2������ = ������1−������2 = 1−5 −4 ������2 − ������1 2−0= 2 ������1−������2 0−2= −2 = 2 Clearly, it does not matter which point is called ( ������1, ������1 ) and which is called ( ������2 , ������2 ) Just make sure that you subtract the ������ – coordinates and the ������ – coordinates in the same order. 78 | P a g e

ARE YOU READY TO PRACTICE? Activity 1. Complete the table below. An example is done for you. Ordered Pair ������1 ������2 ������1 ������2 ������2 − ������1 ������2 − ������1 ������2−������1 ������ 1 526 5-1=4 6-2=4 1 1 (1,2) and (5,6) ������2−������1 6−2 4 2 (-3,3) and (2,-2) 5−1 = 4 3 (-1,5) and (2,5) 4 (-2,6) and (-2,2) 5 (9,7) and (3,4) Activity 2. Analyzed the results of your activity and complete the following statement. The slope of a line given two points is the ____________________ of the difference of the change in ___ and the change in ____. Where the difference in the change in ������ is the _____________________ and the difference in change in ������ is the ___________________. In symbol ������ means ___________. Numerator Denominator ������ ������ slope quotient PRACTICE EXERCISES 1: Find the slope of the given two points. Ordered Pair ������1 ������2 ������1 ������2 ������2 − ������1 ������2 − ������1 ������2−������1 ������ ������2−������1 1 (-10, 11) and (5, -6) 2 (-3, 12) and (9, 12) 3 (-7, 2) and (-7, -8) 4 (-1, 5) and (-6, -2) 5 (8, -3) and (-9, 4) PRACTICE EXERCISES 2: ������������ ������ = ������2− ������1 Determine the slope of a line passing through 2 points. Used ������ = ������1−������2 ������2−������1 ������1−������2 1. (1, 2) and (5, 5) 2. (-5, -5) and (-5, 5) 3. (-5, 4) and (5, 1) 4. (-2, 1) and (6, 1) 5. (-3, -4) and (5, 4) 79 | P a g e

Remember: To get the slope of a line that passes through two points, we use the formula ������ = ������������������������ = ������ℎ������������������������ ������������ ������ = ������1−������2 ������������ ������2− ������1 always remember that the slope of a line is ������������������ ������ℎ������������������������ ������������ ������ ������1−������2 ������2−������1 the same throughout any given two point in the line. Make sure to subtract the y – coordinate and the x – coordinate in the same order. IF the equation is given, transformed it into y = mx +b; thus, you can identify the slope and it is the numerical coefficient of x.. PRACTICE EXERCISES 3: Given the graph determine the coordinates of the point and find the slope of the line passing through these points. _________________________________________________________________________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Creativity 3. Character 4. Critical Thinking A.Writing A. Decision Making A.Developed the understanding of words/ key concept B. Visual Art A. Accuracy B. Responding to Ideas B.Carefully following instructions B. Patience C.ReasoningC.Independent C. Reflexion 80 | P a g e

EVALUATION: I. Find the slope of the line that passes through the given points. Plot the points in the given Cartesian Plane. Identify the trends (positive, negative, undefined, zero) of the graph a. (3, 4) and (1, 2) b. (-6, 0) and (5, -3) c. (-4, 3) and (-4, 1) d. (2, 6) and (5, 1) e. (7, 2) and (-1, 2) a. b. Trends:_______________________ Trends: ______________________ c. d. Trends: ______________________ Trends: _______________________ e. Trends : ___________ 81 | P a g e

II. Given an equation, Identify the slope of the line. An example is done for you. 1. ������ = 3 ������ – 1 ans. ������ = 3 6. 5������ – ������ = −10 ans. ������ = 5������ + 10 so. m= 5 44 7. 7������ – 7������ = 0 8. 2������ – 5������ = −8 2. ������ = ������ – 3 9. 3������ – ������ = 0 3. ������ = 2������ + 5 10. 3������ – 4������ = −6 4. ������ = 4������ 5. ������ = −9 Writing: Reflect on this: We loses our HEALTH in search for WEALTH We work hard, we Sweat, We Save. Then we lose our wealth in search for Health Only to find out, nothing is left. Therefore, Health is Wealth. 1.How can you connect this passage to your lesson slope of a line? Describe the slopes of life that is embedded in that given passage. Use extra sheet of paper if necessary ________________________________________________________________________________________________________ ____________________________________________________________________________________ 2. As a grade 8 student what are the slopes of life that you have encounter and how did you go over it? Explain in paragraph. Use extra sheet of paper if necessary. ________________________________________________________________________________________________________ ____________________________________________________________________________________ Prepared by: Mary Grace R. Ocasion Pasay City west High School References: Grade 8 Learner’s Module, pp. 181-182 Bookman Elementary Algebra pp 241 – 246 Grade 8 Mathematics Patterns and Practicalities pp 160 - 169 https://www.google.com/search?q=clever+owl+clip+art&sxsrf=ALeKk03ZSSVN0eEoE7Ctq5ZiByJDFQYfhw :1592043089032&source=lnms&tbm=isch&sa=X&ved=2ahUKEwi_rKfWxv7pAhVF- WEKHWMaBBwQ_AUoAXoECAwQAw&biw=2133&bih=1041#imgrc=QtlJhJpo2N1I6M 82 | P a g e

Module Code: Pasay-M8–Q1–W5-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 5/ Day 2 OBJECTIVE: Writes linear equation Ax + By = C in the form of y = mx + b and vice versa. 83 | P a g e

YOUR LESSON FOR TODAY: A.) WRITING LINEAR EQUATION Ax + By = C into Y = mx + b. Any equation of the form Ax + By = C can be transformed into its equivalent linear equation y = mx + b which is known as slope-intercept form. In this new form, y is the dependent variable, x is the independent variable, m and b are constant. ILLUSTRATIVE EXAMPLE #1 : SUGGESTED STEPS: 1) 3x + 5y – 15 = 0 Express Ax + By – C = 0 into Ax + By = C (applying 3x + 5y – 15 = 0 the addition property of equality). 3x + 5y = 15 Make sure that the linear equation is express in the form of Ax + By = C, before you identify the value of A= 3 A, B, and C. B= 5 C = 15 In Identifying the value of A, B and C. Don’t disregard the sign of the coefficient. Serve as formula: Using y = −������ ������ + ������ , SUBTITUTE the value of A, B, & SUBTITUTE: y = −������ ������ + ������ A = 3 B = 5 C = 15 ������ ������ ������ ������ C. y = −3 ������ + 15 Perform the indicated operation. ( 15 ÷ 5 = 3) since, −3 55 5 y = −3 ������ + 3 is already in lowest term then copy. 5 Express the final equation in simplified form. ILLUSTRATIVE EXAMPLE #2: X = 8y - 2 Express x =8y – 2 first as Ax + By = C. Make sure that all terms with variables are on one side of the equation (left) and constant on the other side (right). X – 8y = - 2 Apply addition property of equality to cancel 8y on the right A=1 and be placed on the left side of the equation. -2 will B=-8 remain on the left side since it is constant term. C=-2 Identify the values of A, B and C. REMEMBER: Do not SUBTITUTE: disregard the sign of the coefficient. y=−1 ������ + ( −2 ) SUBTITUTE TO y = −������ ������ + ������ −8 −8 ������ ������ y = ������ x + ������ Remember the rules of sign numbers ������ ������ ( −1 = 1 ) and express it to its lowest term ( −2 = + 1 ). −8 8 −8 4 Better express your equation in its simplified form. 84 | P a g e

Remember: Suggested steps in writing Ax + By = C into the form of Y = mx + b 1. Make sure 1st that all terms with variables are on the same side and the constant on the other side. Express linear equation is in the form of Ax + By = C. 2. Identify the value of A, B, and C. Do not forget to copy the sign of the coefficient. 3. Substitute the value of A, B, and C to y = −������ ������ + ������ . ������ ������ 4. Perform the indicated operations. 5. Express the final equation in its simplified form. 85 | P a g e

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Enrichment: Give one best example of linear equation. Write it in the box below. Write the Ax + By = C form write the y = mx + b form ▪ What made you choose to write that linear equation? ▪ What are the significance of those numbers in your life? Say something. _________________________________________________________________________________________________ _________________________________________________________________________________________________ ______________________________________________________________________________________ ___________ _________________________________________________________________________________________________ ______________________________. ___________________________________________________________________________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 3. Creativity * Understanding of words/vocabulary * writing * Following instructions/directions. * visual art * Reasoning * Communication skills * . Responding to ideas 2. Critical thinking 4. Character * Observation * values integration * reflection * decision making * honesty * working independently EVALUATE YOURSELF: Directions: A. Complete the trend by writing the step by step solution in transforming Ax + By = C into y = mx + b and vice versa. Given A,B,C Formula Substitution simpified y=mx+b 1. 9x + 3y = 6 2. 4x - 2y = 5 3. 5x + 2y = 3 Given A&B Multiply+ B Product Simplified Ax+By=C 4. y =5x + 7 5. y =−������ x + 5 ������ 87 | P a g e

A. Below are the suggested steps in writing linear equation Ax + By = C into the form of y=mx + b. Arrange in correct order by writing the numbers 1 to 5 in the Substitute the value of A, B, and C to y = −������ x + ������ . ������ ������ Express the given linear equation into the form of Ax + By = C. Identify the value of A, B, and C from the given linear equation. Always express your final equation in its simplified form. Perform the indicated operations. Prepared by: BONA BAYHON - BERNALES Pasay City South High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 182-183 Elementary Algebra I, pp.149 - 151 https://sciencing.com/how-to-convert-slope-intercept-form-to-standard-form-13712257.htm 88 | P a g e

Module Code: Pasay-M8–Q1–W5-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 5/ Day 3 OBJECTIVE: Graphs linear equation given (a) any two points; (b) the x- and y-intercepts. YOUR LESSON FOR TODAY: Graphing Linear Equations in two Variables using : a.) TWO POINTS b.) X- and Y-INTERCEPTS Before going further, do you know how this 4 quadrant plane started? The Cartesian Coordinate System was developed by the mathematician RENE’ DESCARTES during an illness. As he lay in bed sick, he saw a fly buzzing around on the ceiling which was made of square tiles. As he watched he realized that he could describe the position of the fly by the ceiling tile he was on. After this experience he developed the Coordinate plane to make it easier to describe the position of an objects. Your past lesson had Today, let us describe linear equations in two variables by taught you that one way of its graph, using any two points. As we all knew that, “two describing linear equations points determine a line.” in two variables is by writing it in Ax + By = C or y = mx + b. TRY TO DISCOVER: ILLUSTRATIVE EXAMPLE #1: Plotting of points: ������������( 1, 3 ) meaning from the origin, move 1 unit to Graph the equation y = 2x + 1 the left,then 3 units upward. * In every point, Assign value for x to find y: ������������( 2, 5 ) meaning from the origin, move 2 units X____1____2___ to the right and 5 units upward. corresponds an ordered Y35 pair (x, y). SUBSTITUTE the value of x to • ������������( 2, 5 ) *Assigning value of x y=2x+1 to get the vale of y. • ������������( 1, 3 ) depends on you, but better If x=1, then If x=2, then assigned a smaller numbers. y = 2(1) + 1 y = 2(2) + 1 *After getting the value of x y=2 +1 y = 4 +1 y=3 y=5 and y then you have now ������1( 1, 3 ) ������2( 2, 5 ) = an ordered pair which can be located on the graph. 89 | P a g e

Another way of x- and y- intercepts are points describing linear where the graph crosses the equation by graphing coordinate axes. [ x-intercept on is by using their the x-axis (x,0) ] and [ y- x- and y-intercepts. intercept on the y-axis (0,y)] ILLUSTRATIVE EXAMPLE #2: *assign 0 as value of y to Graph the equation 2x + 3y = 6 by using x-and y-intercepts. get the x-intercept and; Solution: 2x + 3y = 6 (0,2) (3,0) * 0 for x to get the y- x-intercept y-intercept y-intercept • • intercept. assign y=0 assign x=0 x-intercept *substitute to the original 2x + 3(0)=6 2(0) + 3y=6 equation. 2x =6 3y=6 *perform the indicated 2x= 6 3y= 6 operations 22 33 *the value of x indicates X=3 y=2 that the graph will pass x-intercept ( 3,0 ) y-intercept ( 0,2 ) through on the 3 of x-axis and on 2 of y-axis. *Plot the points and connect. ARE YOU READY TO PRACTICE? Round 1: Given the graph of each linear equations. Write the coordinates of the indicated two point and its x- and y-intercepts. 1.) 4x + y = 12 1.) 4x+y=12 A(___,___) B(___,___) 2.) x-y=-9 x-intercept (___,___) y-intercept (___,___) • 2.) x – y = - 9 • C(___,___) D(___,___) C• A• x-intercept (___,___) y-intercept (___,___) D• E• • • 3.) 3x + y = - 6 B• E(___,___) F• F(___,___) 3.) 3x + y = - 6 x-intercept (___,___) y-intercept (___,___) 90 | P a g e

4.) In your own words. What is x- and y-intercept? _______________________________ ______________________________________________________________________ 5.) How do you describe the graph of a linear equation in two variables? _____________________________________________________________________________________ ROUND 2 : 1. Graph the linear equation y = 3x – 4 by using two points on the line. X____1____2____... y SUBSTITUTE: (___,___) (___,___) 2. Graph y = 5 – 3x that passes through Point ( 1,____ ) and point ( 3,____ ). X_____1________3____ Y SUBSTITUTE: ( 1, ____ ) ( 3, ____ ) 3. Graph 2x – 5y = 10 by using their x – and y-intercepts. x-intercept y-intercept ( ___,___ ) ( ___,___ ) 91 | P a g e

4. Show the graph of 4x + 5y = 20 using its x- and y-intercepts. X – intercept y - interceptS Remember ❖ A line is composed of points. ❖ Two points determine a line. ❖ An ordered pair ( x,y ) is used to locate a point in a coordinate plane. ❖ The graph of a linear equation is a straight line. ❖ The x-intercept of a line is the abscissa of the point (x,0), where the graph crosses the x-axis. ❖ The y-intercept of a line is the ordinate of the point (0,y), where the graph crosses the y-axis. It’s more fun in Mathematics ! Graphing Linear Equations & Catching Catch the Pokemons by graphing the following linear equations. (You may use any method of graphing ) 1.)15x – 12y = -180 2.)18x + 14y = 252 3.)17x +13y = -221 4.)19x – 11y = 209 Integrated the Development of the Following Learning Skills: 92 | P a g e

1. Communication Skills 3. Creativity A. Understanding of words/terms Problem solving, visual art, B. Following instructions/directions. C. Reasoning Open-mindedness D. Understanding message 2. Critical Thinking 4. Character Patience, hard work, and Include observation, interpretation, problem solving. perseverance. TEST YOURSELF: A. Fill in the blanks to complete the sentence. Choose your answer from the box bellow. 1. One way of describing linear equation is by writing it in the form of _____________ or _______________, also known as slope-intercept form. 2. Another way of describing linear equation is by ___________, using any __________ on the line or by using the _____________________. 3. ____________ is the vertical axis of the Cartesian plane, while _____________ is the horizontal axis. 4. The ______________ of the line is the point where the graph crosses the x-axis and _________________ is the point where the graph crosses the y-axis. 5. The graph of a linear equation is a ________________. X – intercept y – intercept Ax + By = C y = mx + b Line straight line Two points x- axis graphing x- and y-intercept y-axis Cartesian Plane B. Describe the given linear equation by graphing using: 1. Two points on the line Given: y = 8x – 10 2. X- and y-intercept Given: 5x – 3y = 15 Prepared by: BONA BAYHON - BERNALES Pasay City South High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 187 - 188 Elementary Algebra I, pp. 147-148,152S 93 | P a g e

Module Code: Pasay-M8–Q1–W5-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 5/ Day 4 OBJECTIVE: Describes the graph of a linear equation in terms of its intercept and slope YOUR LESSON FOR TODAY: “Describing the graph of a linear equation in terms of its intercept and slope” An-nyeong-ha-se-yo! This can be done by identifying first Today, we will study another the slope and the y-intercept of the method of graphing linear linear equation. Remember the equation. This time is by using y=mx + b? The m indicates the the slope and its y-intercept. slope of the line and b is the y- intercept or the point on the y-axis. Let us discover how. Then, identify the slope Supposed, you are asked to graph which is the coefficient of x. y= 5x – 2. First, identify the y- so m = 5. Meaning Rise 5 intercept ,which is the b. Meaning units and run 1 unit (the b = -2. This simply means, that denominator of whole the graph will pass through on -2 number) to the right since of the y - axis. the slope is positive. b = -2 • y = 5x - 2 Great job, • Rapmonster ! m= 5 Do you want to try, Jhope? 1 from -2 of y- axis, rise 5 units and run 1 unit to the right (+slope) Sure, Jimin ! Let us have another example. Illustrative example: RUN Start here! 1.) Graph the equation y = - ������ x + 2 . ( 3, 4 ) •R ������ I •S y = - ������ x + 2 E ������ y = - ������ x + 2 y-intercept ������ Slope Step 1. Locate +2 on the y-axis. Step 2. Rise 2 units. Step 3. Run 3 units to the left (- slope) Step 4. Connect the starting point and The last point. 94 | P a g e

ARE YOU READY TO PRACTICE: Directions: Let’s do some twist on cross word puzzle. Find the group of linear equation, its slope and its y-intercept. Please see example. y = 3x + 7 m = - ������ b=-7 Y = ������ x - 1 m = ������ b=-1 ������ ������ ������ y = ������ x – 1 m = ������ y = sx + 7 b = - i m=7 m = ������ ������ ������ ������ y = 3x – 7 m = ������ b=7 y = 7x - 1 b = ������ y = ������ x - 1 ������ ������ ������ m=8 y = 5x - 2 m=5 b=-2 y = ������ x + 7 m=-1 b=5 y = 3x + 8 m=8 b=3 y = ������ x + 5 ������ ������ m = ������ m = - ������ ������ b=3 m = ������ y = ������ x + 3 ������ b=5 y = 5x + 1 ������ ������ Practice Exercise #2: Graph the following linear equation using Slope and y-intercept. 1.) y = ������ x + 1 ������ 2.) 3x + 4y = 15 3.) Y = 7x - 2 95 | P a g e

Remember Positive negative slope ❖ The slope indicates the trend slope undefined of the graph of a linear equation. slope ❖ y = mx + b, WHERE m = ������������������������ and b is the y-intercept ������������������ zero ❖ Ax + By = C, WHERE m = −������ and b = ������ slope ������ ������ Practice #3: A. Identify the slope and the y-intercept of the given linear equation below and indicate the movement in graphing. Number 1 is done for you. 1.) y = −������ x - 7 m = −9 −3 b = -7 From -7 of y-axis move 3 units UP and 4 units to the LEFT ������������ = 12 4 2.) y = ������ x + 5 ������ 3.) y = 5 x - 8 4.) 5x + 2y = 0 5.) x–y=9 B. Given the graph, write its slope and its y-intercept.. 1.) m= b= Integrated the Development of the Following Learning Skills: 3. Character 1. Communication Skills A. Accuracy A. Understanding of words/vocabulary B. Patience B. Following instructions/directions. C. Reasoning C. Perseverance 2. Critical Thinking Include observation and interpretation 96 | P a g e

It’s more fun in Mathematics! Find the slope, the y-intercept and the equation of each graph. Answers are just around the corner. Gather it by writing on the correct box. b=3 m = −������ 3x -4y= -12 b=7 7x + 5y = 35 m = ������ ������ b = - 10 m = ������ y = −������ x + 7 ������ 5x – 4y = 40 ������ ������ y = ������ x + 3 y = ������ x - 10 ������ ������ 97 | P a g e

EVALUATION: Directions: A. Read the statement/question carefully. Encircle the letter that corresponds to your answer. 1. It is a plane composed of two perpendicular axes that meet at the point of origin. a. Cartesian plane c. Coordinate System b. Rectangular Coordinate system d. All of the above 2. It refers to the steepness of the line and it tells also the trend of the graph of a linear equation. a. ordered pair c. x-axis b. slope d. y-axis 3. It is composed of the abscissa and ordinate and used to locate a point on the plane. a. Ordered pair c. x-axis b. Slope d. y-axis 4. Another way of writing linear equation is in the form of y = mx + b, What do you call this form of linear equation? a. Slope-intercept form c. general form b. Standard form d. point form 5. What is the slope of the line, given the equation y = 5x + 7? a. m = 1 c. m = 5 b. m = 7 d. m = 0 6. What is the “b” or the y-intercept of x + 8 y = 16? a. m = 4 c. m = 8 b. m = 2 d. m = 1 B. Finish the statement by describing the slope of a line. 7. The slope is positive if _______________________________________________ ___________________________________________________________________. 8. The slope is negative if ________________________________________________ ______________________________________________________________________. 9. If the slope is zero then, _____________________________________________ __ ______________________________________________________________________. 10. The slope is undefined if ______________________________________________ _______________________________________________________________________. C. Draw your own vesion of Mr. Slope man and write something about it. Secure your future, Prepared by: through EDUCATION! BONA AYHON - BERNALES Keep safe, buddy! Pasay City South High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 188 - 189 Elementary Algebra, pp. 154 – 155 Math-only-math.com 98 | P a g e

Module Code: Pasay-M8–Q1–W6-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 6/ Day 1 OBJECTIVE: The learner finds the equation of a line given the following: • two points. • the slope and a point. • slope and its intercept. YOUR LESSON FOR TODAY: This module will guide the learners to find the equation of a line given two points. The learners will use the Two-Point Form of the Equation of a Line, ������ − ������1 = ������2−������1 (������ − ������1), where (������1, ������1) and (������2, ������2) ������2−������1 are the coordinates of the two given points P1 and P2, respectively, the Point-Slope Form of the Equation of a Line, ������ − ������1 = ������(������ − ������1), where (������1, ������1) are the coordinates of a given point and “m” is the slope, and the Slope - Intercept Form, ������ = ������������ + ������, where “m” is the slope and “b” is the y-intercept. Also, the learners will apply mathematical knowledge and skills to derive the equations of a line given any two points in the forms of y = mx + b or ax + by = c. Unlocking of Difficulties ❖ The Two-point Form of a Line The equation of the line passing through (������1, ������1) and (������2, ������2) is given by ������ − ������1 = ������2−������1 (������ − ������1). ������2−������1 ❖ The Point-Slope Form of the Equation of a line The equation of the line passing through (������1, ������1) and with the slope m is given by ������ − ������1 = ������(������ − ������1). ❖ Slope m = ������������������������ = ������2 − ������1 ������������������ ������2 − ������1 ❖ The Slope – Intercept Form The equation of the line with a slope, m and y – intercept, b is given by ������ = ������������ + ������ Let’s Recall! Activity I: Transform me in the form of y = mx + b! Rewrite the following linear equations ax + by = c in the form of y = mx + b. 1. 4x + 2y = 5 answer: y = - 2x + 5 3. 5x – y + = 10 answer: 2 2. x + 3y = - 9 answer: 4. 2x - 5y = - 9 answer: 99 | P a g e

Activity II: Set me to your Standard! Write the following linear equations in the form of y = mx + b in the form of ax + by = c. 1. y = 3x - 6 answer: 3x – y = 6 3. y = 2x + 5 answer: 2. y = 2x + 4 answer 4. y = − 3x − 3 answer: 3 5 Question: How was the activity you have just done? Was it easy for you to write the equation of a line in the form of y = mx + b or in ax + by = c? Please Remember! A. In finding the equation of a line given two points using the Two-point form, ������2−������1 ������ − ������1 = ������2−������1 (������ − ������1), you need to do the following: 1. Substitute the coordinates of the given points (������1, ������1) and (������2, ������2) into the two-point form of the equation of a line. 2. Simplify by adding or subtracting the terms in the ������2−������1. ������2−������1 3. Multiply both sides by the value of the denominator in right side of the equation and apply the distributive property of multiplication. 4. If you want to find the equation of a line in the form of y = mx + b, gather constant terms and the term with x-variable on one side of the equation, divide the whole equation by the coefficient of y-variable and then simplify. Or, 5. If you want to find the equation of a line in the form of ax + by = c, gather the terms with variables on one side of the equation, the constant terms on the other side and then simplify. B. In finding the equation of a line given the slope and a point using the Point-Slope form ������ − ������1 = ������(������ − ������1), you need to do the following: 1. Substitute the coordinates of a given point (������1, ������1) and the value of the slope “m” into the point – slope form of the equation of a line. 2. Apply distributive property of multiplication by distributing the value of the slope to each term inside the parenthesis. 3. If you want to find the equation of a line in the form of y = mx + b, gather constant terms and the term with x-variable on one side of the equation, and then simplify. Or, 4. If you want to find the equation of the line in the form of ax + by = c, gather the terms with variables on one side of the equation, the constant terms on the other side and then simplify. C. In finding the equation of a line given the slope and y-intercept using the Point-intercept form, y = mx + b, you only need to do is to substitute the slope “m” and the value of y - intercept “b” into y = mx + b and then simplify. Example 1. Find the equation of the line that passes through the points (7, - 1) and (3, 5). Solution: Substituting the coordinates of the points (7, - 1) and (3, 5) for (������1, ������1) and (������2, ������2), respectively into the two-point form of the equation of a line. ������ − ������1 = ������2−������1 (������ − ������1) (Two-point form) ������2−������1 (Substitute) ������ − (−1) = 5 −(− 1) (������ − 7) (Simplify) 3−7 ������ + 1 = 6 (������ − 7) −4 100 | P a g e


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