Mathematics 8 Quarter 1 Module 13: Solving Problems Involving Linear Equations in Two Variables

Mathematics Quarter 1 – Module 13: “Solving Problems Involving Linear Equations in Two Variables’’

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CO_Q1_Mathematics8_M13

Mathematics – Grade 8 Alternative Delivery Mode Quarter 1 – Module 13: Solving Problems Involving Linear Equation in Two Variables First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module
Writer:	Wilmar N. Espinosa
Language Editor:	Merjorie G. Dalagan
Content Evaluator:	Michelle R. Alipao
Layout Evaluator:	Jake D. Fraga
Reviewers:	Nilda A. Mendiola, Manuel L. Limjoco, Jr., Cris Gerom C. Arguilles, Melba G. Lumangcas, Analyn L. Avila, Thelma D. Ramos
Illustrator:	Wilmar N. Espinosa
Illustrator:	Wilmar N. Espinosa
Layout Artist:	Jake D. Fraga
Management Team:	Francis Cesar B. Bringas, Isidro M. Biol, Jr., Rhea J. Yparraguirre, Maripaz F. Magno, Josephine Chonie M. Obseñares, Josita B. Carmen, Celsa A. Casa, Regina Euann A. Puerto, Bryan L. Arreo, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga, Lieu Gee Keeshia C. Guillen

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8

Mathematics Quarter 1 – Module 13: “Solving Problems Involving Linear Equations in Two Variables’’

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

This module was designed and written with you in mind. It is here to help you master on how to solve problems involving linear equations in two variables. The scope of this module permits it to be used in many different learning situations. The language and numeric used recognizes the diverse vocabulary and numeracy level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

This module contains:

Lesson 1- Solving Problems involving Linear Equation in Two Variables

After going through this module, you are expected to:

1. translate word problems into linear equations;
2. solve problems involving linear equations in two variables; and
3. apply the concepts of solving linear equations in dealing with real-life problems.

What I Know

Directions: Read the questions carefully. Then choose the letter of the correct answer and write it on your paper.

1. Which of the statements below translates the mathematical equation “4𝑥 + 5 = 21”?

   • A. Four x plus five with twenty-one.
   • B. Four x a number increased by five is twenty-one.
   • C. Four times a number increased by five is twenty-one.
   • D. Four times a number diminished by five gives twenty-one.

2. A square plywood platform has a perimeter which is 9 times the length of a side, decreased by 35. Find the length of a side.

   • A. 1 unit
   • B. 5 units
   • C. 7 units
   • D. 12 units

3. Lucas has a job mowing lawns in his neighborhood and gets paid P25 per hour. Identify the independent variable in computing his total pay.

   • A. The job
   • B. The lawn mowing
   • C. The total pay
   • D. The number of hours worked

4. You want to buy a hamburger and a mango pie. The hamburger costs P25 each and mango pie at P12 each. You have P160 to spend. If x is the number of hamburger and y is the number of mango pie, write an equation that models the different amount.

   • A. 25𝑥 + 12𝑦 = 160
   • B. 12𝑥 + 25𝑦 = 160
   • C. 12𝑥 + 160 = 25𝑦
   • D. 25𝑥 − 12𝑦 = 160

5. Jonathan is six years older than Joshua. The sum of their ages is 48. Find Joshua’s and Jonathan’s age. What are the steps in solving Joshua and Jonathan’s age in chronological order?

   I. Translate the problem into an equation, diagram, or chart.
   II. Understand the problem. List all the components and data that are involve.
   III. Check to see if the information is used correctly and that the answer make sense.
   IV. Solve the equations you translated based on the given problem.

   • A. I, II, III, IV
   • B. II, IV, I, III
   • C. II, I, III, IV
   • D. II, I, IV, III

6. Refer to problem no. 5, what mathematical equation is needed to find Joshua’s age?

   • A. 𝑥 + 6 = 48
   • B. 𝑥 + 𝑥 = 48
   • C. 2𝑥 + 𝑥 = 48
   • D. 𝑥 + 𝑥 + 6 = 48

7. Still refer to problem no. 5, what is Jonathan’s age?

   • A. 21 years old
   • B. 23 years old
   • C. 25 years old
   • D. 27 years old

For items 8– 10, refer to the situation below:

Mang Kardo earns P400 per day and additional P100 for every hour overtime as technician.

8. What equation shall best represent the earnings of Mang Kardo when he rendered overtime services as a technician?

   • A. 𝑦 = 𝑥 + 400
   • B. 𝑦 = 2(𝑥 + 400)
   • C. 𝑦 = 400𝑥 + 100
   • D. 𝑦 = 100𝑥 + 400

9. What is the amount earned by Mang Kardo when he rendered 3 hours of overtime?

   • A. P400
   • B. P500
   • C. P600
   • D. P700

10. If the equation in item 8 represents the amount paid for the labor in repairing computer units for every 𝑥 number of hours, how much will Mang Kardo earn after 6 hours?

    • A. P800
    • B. P900
    • C. P1,000
    • D. P1,100

11. Diane is a working student. She works in a restaurant and is paid a minimum of P350 for a 6-hour service rendered plus P50 per additional hour of service. What equation will represent the total amount of payment to Diane?

    • A. 𝑦 = 350𝑥 + 50
    • B. 𝑦 = 50(6𝑥) + 350
    • C. 𝑦 = 50(𝑥 + 6) + 350
    • D. 𝑦 = 50(𝑥 − 6) + 350

12. A t-shirt company charges P250 per order plus P99 per t-shirt. How much will it cost if you order 5 t-shirts?

    • A. P499
    • B. P545
    • C. P649
    • D. P745

13. A cell phone repairman charges P150 for a repair service, plus 50 for each hour of work. How much will the repairman charge if he works for 5 hours?

    • A. P400
    • B. P450
    • C. P550
    • D. P600

14. A multicab passenger is charged P10 for the first 4 km and an additional P2 per succeeding kilometer as fare. What is the cost of a 28-km ride?

    • A. P46
    • B. P56
    • C. P58
    • D. P66

15. Refer to problem 14, how many succeeding kilometers did the passenger ride after he paid an amount of P34?

    • A. 10 𝑘𝑚
    • B. 11 𝑘𝑚
    • C. 12 𝑘𝑚
    • D. 13 𝑘𝑚

I can’t transcribe pages 9 to 12 yet because they weren’t included. The only page images I received are four pages printed as 5 to 8 of the module (the first four pages of the PDF). Those are the pages you said were already transcribed, so I haven’t reproduced them.

Please attach the images for PDF pages 9 to 12 and I’ll transcribe them with page markers 9 to 12.

[transcriber’s note: pages 13 to 16 were not included with this request. The only page images attached are four pages that carry printed numbers 9 to 12 of the module, which fall outside the requested range. No Markdown was produced for pages 13 to 16. Please re-attach those four page images so they can be transcribed.]

Step 3. Find the solution using the formulated equation.

𝑦	=	____ + 500	Equation
2375	=	____ + 500	By substitution
2375 + (−500)	=	____ + 500 + (−500)	By Addition Property of Equality
____	=	____	By simplification
1875 / ____	=	____ 𝒙 / ____	Division Property of Equality
____	=	𝒙	By simplification

[transcriber’s note: the table has no printed header row; the header cells above are empty. The fraction in the “Division Property of Equality” row is printed as 1875 over a blank line, with 𝒙 over a blank line on the right; the blanks are shown as written.]

Step 4. Check the solution obtained by substituting the solution/s to the original equations.

𝑦	=	____ + 500
2375	=	75(____) + 500
2375	=	____ + 500
2375	=	______ ________

[transcriber’s note: the table has no printed header row; the header cells above are empty.]

Hence, Reena needs to save for ____weeks.

Activity 2: How Much Should Be Paid?

Direction: Read the situation and answer the questions that follow.

A tricycle passenger is charged P8.00 for the regular fare for the first 3 kilometers and an additional of P1.50 per succeeding kilometer.

Questions:

1. What equation shall represent the situation?
2. How much will a tricycle passenger pay for a total of 8 kilometers travel?

What I Have Learned

Directions: Enumerate the steps in solving problems involving linear equations in two variables.

1. First, I have to
2. Second, I will
3. Third, I will
4. Finally, I will identify the solution of the linear equation in two variables by…

What I Can Do

Activity: “Be Fit and Fab!”

You want to achieve a fit and healthy body, so you joined a fitness club in your community and incorporate it with proper diet for you to lose an average of 7 pounds each month. At the end of 3 months, you weighed 130 pounds.

Questions:

a. What linear equation would best represent the situation? b. What is your weight in 5 months of exercise? c. If your original weight is 151 pounds, how many pound will you lose in 7 months?

Assessment

Directions: Read each item carefully then choose the letter of the correct answer. Write it on your paper.

1. Which of the following is the correct sequence in solving problems involving linear equation in two variables?

   
   

   I. Read and understand the problem
   II. Translate the problem into an equation in two variables
   III. Find the solution using the formulated equation
   IV. Check the solution obtained by substituting the solution/s to the original equations

   
   

   A. I, II, III, IV
   B. I, III, II, IV
   C. I, III, IV, II
   D. II, I, III, IV

   
   

For item numbers 2 to 4, refer to the situation below.

Suppose a survey is made on the number of people infected by the Corona Virus Disease-2019 (COVID-19) from February 2020 to June 2020. The result was tallied, and it was observed to increase at a constant rate as shown in the table below.

Month	February	March	April	May	June
Number of people infected by the Coronavirus Disease 2019 (COVID-19)	20	28	36	44	52

2. What is the equation that would represent the situation?

   
   

   A. 𝑦 = −8𝑥 + 20
   B. 𝑦 = −8𝑥 + 12
   C. 𝑦 = 8𝑥 + 20
   D. 𝑦 = 8𝑥 + 12

   
   

3. If the pattern continues, can you predict the number of infected people by the month of August 2020?

   
   

   A. Yes, the number of infected people by the month of August 2020 is 68.
   B. Yes, the number of infected people by the month of August 2020 is 76.
   C. No, because it is not stipulated in the problem.
   D. No, because the data is insufficient.

   
   

4. How many people will be infected by Coronavirus Disease 2019 (COVID-19) in the month of July?

   
   

   A. 58
   B. 60
   C. 68
   D. 72

   
   

For items 5 to 7: A vendor buys apples and oranges to be sold in a sidewalk. An apple costs P20.00 each and an orange costs P15.00 each. The vendor has only P1,000.00 to spend.

5. If 𝑥 is the number of apples and 𝑦 refers to the number of oranges, which linear equation in two variables best models the situation?

   
   

   A. 20𝑥 + 15𝑦 = 1,000
   B. 15𝑥 + 1000𝑦 = 20
   C. 15𝑥 + 20𝑦 = 1,000
   D. 20𝑥 − 15𝑦 = 1,000

   
   

6. If the vendor wants to buy 35 apples, how many oranges can he buy from his remaining money?

   
   

   A. 15
   B. 20
   C. 25
   D. 30

   
   

7. If the vendor wants to buy 32 oranges, how many apples can he buy from his remaining money?

   
   

   A. 26
   B. 27
   C. 28
   D. 30

   
   

8. Mr. Willy has a rental business. He has bikes for rents with charges that can be estimated by a linear equation 𝑃 = 7ℎ + 100, where 𝑃 is the amount of rent to be paid and ℎ is the number of hours the bike is used by the customer. How much will a customer pay for the rent after using the bike for a total of 5 hours?

   
   

   A. 𝑃121.00
   B. 𝑃128.00
   C. 𝑃135.00
   D. 𝑃142.00

   
   

9. Jamie rents a generator for 𝑃1,200.00 and an additional charge of 𝑃150.00 per hour of usage. How much will he pay after using it for 8 hours?

   
   

   A. 𝑃2,400.00
   B. 𝑃2,500.00
   C. 𝑃2,800.00
   D. 𝑃3,000.00

   
   

For items 9-10: Refer to the situation below:

A tricycle passenger is charged P10.00 for the first 4 kilometers and an additional of P1.50 per succeeding kilometer.

10. What equation can be used to model the situation?

    
    

    A. 𝑦 = 1.50𝑥 + 10
    B. 𝑦 = 10𝑥 + 1.50
    C. 𝑦 = 1.50𝑥 + 4
    D. 𝑦 = 1.50 + 4𝑥

    
    

11. How much should be paid for a total travel of 35 kilometers?

    
    

    A. 𝑃56.50
    B. 𝑃62.50
    C. 𝑃72.50
    D. 𝑃82.50

    
    

[transcriber’s note: this page continues the Assessment section from the previous page]

12.What is the equation to represent the situation?

• A. y = −10x + 20
• C. y = 10x + 20
• B. y = −1/10 x + 2
• D. y = 1/10 x − 2

13.How much should Trina pay in 5 hours?

• A. P55.00
• C. P70.00
• B. P60.00
• D. P80.00

14. If Trina returned the costume after 24 hours, how much would she have to pay?

• A. P200.00
• C. P240.00
• B. P220.00
• D. P260.00

15.A Disc Jockey (DJ) was hired to host a post-wedding party that charges P5,000.00 and an additional of P350.00 per Hour. Would it be possible for the DJ to receive a payment of not less than P6,000.00 after hosting the party for 3 hours?

• A. No, because the charge per hour is very low.
• B. No, because 3 hours of hosting will not be enough to earn at least P6,000.00
• C. Yes, because 3 hours of hosting is equivalent to an additional fee of P1,050.
• D. Both A & B

For numbers 12 – 14. Refer to the table below:

Trina rents a costume for her school activity. The amount she has to pay per hour is presented in the table below.

No. of hours (x)	1	2	3
Cost in Pesos (y)	30	40	50

Additional Activities

Let’s Create

Directions: Create a matrix showing the importance of solving problems involving linear equations in two variables in dealing with real-life situations. Highlight one application for each of the following categories: personal, at home, and in the community and write a brief description of each application. Below is a suggested template.

Applications	Description
A. Personal:
B. At Home:
C. In the Community:

Each of your cited applications will be rated using the following rubric.

Criteria	Performance Level: Not Yet Meeting Expectations (2 points)	Performance Level: Minimally Meeting Expectations (3 points)	Performance Level: Fully Meeting Expectations (4 points)	Performance Level: Exceeding Expectations (5 points)
Output Requirements	Output meets some requirements, guidelines, and objectives	Output meets most requirements, guidelines, and objectives	Output meets all requirements, guidelines, and objectives with the capacity to advance	Output exceeds requirements, guidelines ad objectives at an advanced level
Concepts and Applications	Has trouble figuring out the applications of the topic. Descriptions of the cited applications are minimally sensible.	Has figured the applications of the topic. Descriptions of the cited applications are adequately sensible.	Has clearly cited the applications of the topic. Descriptions of the cited applications are proficiently sensible.	Has clearly cited the applications of the topic. Descriptions of the cited applications are excellently sensible.
Critical Thinking	Little abstraction beyond facts	Only modest abstraction beyond facts	Some potential aspects are missed	All key aspects identified, related, and rigorously evaluated

Perfect Score: 45 Points

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees; the sections below follow the order of the page’s content]

What I Know

1. C
2. C
3. D
4. A
5. D
6. D
7. D
8. D
9. D
10. C
11. D
12. D
13. A
14. D
15. C

What’s In

1. 2x = 6
2. x + 4 = 10
3. 7 + 3x = 98
4. 25 − 2x = 12
5. x + 3 = 25
6. 9 − 3x = 100 [?]

[transcriber’s note: the item 6 equation is printed rotated and is hard to confirm; the digit reads as 9 or 6]

What’s More

Activity 1: Watch your Steps

Step	Work
Step 1. Understand the Problem	Given: • Reena has an initial savings of P 500.00 • She needs to save a total amount of P 2,375.00 • She plans to save P75 per week Asked: How many weeks will does Reena need to save? Let x represents the no. of weeks Reena has to save. Let y represents the total amount in P of Reena’s savings.
Step 2. Translate the problem into an equation in two variables.	Total amount saved = (savings per week x number of weeks) + Initial amount of savings y = 75x + 500

Step 3. Find the solution using the formulated equation.	Reason	Work
	Equation	y = 75x + 500
	By substitution	2375 = 75x + 500
	By Addition Property of Equality	2375 + (−500) = 75x + 500 + (−500)
	By simplification	1875 = 75x
	Division Property of Equality	1875/75 = 75x/75
	By simplification	x = 25

Step 4. Check the solution obtained by substituting the solution/s to the original equations.	Work
	y = 75x + 500
	2375 = 75(25) + 500
	2375 = 1875 + 500
	2375 = 2375 TRUE

Hence, Reena needs to save for 25 weeks.

Activity 2:

1. y = 1.5x + 3
2. P15.50

What I Can Do

1. y = −7x + 151
2. 116 pounds
3. 49 pounds

Assessment

1. A
2. D
3. A
4. B
5. A
6. B
7. A
8. C
9. A
10. C
11. A
12. C
13. C
14. D
15. C

References

Acelajado, Maxima J. (2008).Intermediate Algebra.pp.319-326.Makati City, Philippines. Diwa Learning Systems Inc.

Abuzo, Emmanuel P., Bryant, Merden L., Cabrella, Jem Boy B., et. al (2013). Mathematics Grade 8 Learner’s Module. pp. 192-195 & 202. Philsports Complex, Meralco Avenue, Pasig City, Philippines. Book Media Press, Inc. & Printwell, Inc.

Basilio, Faith B., Chua, Edna A., Jumawan, Maria T., Mangabat, Lawrence Oliver A., Mendoza, Mario B., Pacho, Elsie M., Tamoria, Ferdinand V., Villena, Eufrosina S.,Vizcarra, Florante O., Yambao, Teresa M.(2003). Fundamental Statistics. Philippines. pp. 129 -133. Trinitas Publishing Inc.

Diaz, Zenaida B., Mojica, Maharlika P. et al (2013). The New Grade 8, Next Century Mathematics. Pp. 281 – 287. Quezon City, Phoenix Publishing House, Inc.

Garcia, George A. (2003). Fundamental Concepts and Methods in Statistics (Part 1). pp.4– 9. Manila, Philippines. University of Sto. Tomas Publishing House.

Garcia,George A. (2004). Fundamental Concepts and Methods in Statistics (Part 2). pp. 8 – 43. Manila, Philippines. University of Sto. Tomas Publishing House.

Glencoe/McGRAW-HILL. (1986). Mathematics Skills for Daily Living. pp. 397 – 411. United States of America. Laidlaw Brothers, Publishers.

Price, Jack/ Rath, James N./ Leschensky, William.(1989). Pre- Algebra: A Problem Solving Approach. pp.420 – 430. Columbus, Ohio. Merrill Publishing Co. 637

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