Lesson 6 - Algebra in Taglish: Solving Quadratic Equations by Factoring, Completing the Square, and Using the Quadratic Formula

 Lesson 6 – Quadratic Equations

        Matapos nating mapag-aralan ang tungkol sa linear equations at ang kaugnayan ng mga ito sa pagresolba ng mga problema sa pang-araw-araw na buhay, dadako naman tayo sa quadratic equations.

Ang quadratic equation na may isang variable/unknown ay isang equation kung saan ang pinakamataas na exponent ng variable ay 2.

        Ang karaniwang anyo ng isang quadratic equation na may isang variable na kinakatawan ng x ay: 

        ax² + bx + c = 0 

kung saan ang a, b, at c ay mga constant at ang a ≠ 0.

Ang pinakakaraniwang ginagamit na paraan sa paglutas ng mga quadratic equation ay ang mga sumusunod:

Solving Quadratic Equations by Factoring

1. Ang factoring o pag-factor ay ang pinakamadaling paraan ng paglutas ng mga quadratic equation. Gayunpaman, nangangailangan ito ng kasanayan sa iba't ibang mga pamamaraan ng factoring. Bukod dito, hindi lahat ng quadratic equation ay malulutas sa pamamagitan ng factoring.

        Upang malutas ang isang quadratic equation sa pamamagitan ng factoring, sundin ang mga hakbang na ito:

        a. Ilipat ang lahat ng mga termino sa kaliwang bahagi ng equation upang ang kanang bahagi ay maging katumbas ng zero.

        b. Pasimplehin at i-factor ang quadratic expression na karaniwan na isang trinomial.

        c. Ang bawat isa sa mga factor ay isa na ngayong linear equation. I-equate ang bawat factor sa 0 at lutasin ang bawat linear equation para sa hinahanap na variable.

HALIMBAWA 1

        Solve for x in the equation 12x² - 9  = -10x + 3 

Step 1: Ilipat ang lahat ng terms sa kaliwang panig.

          12x²  - 9 = -10x + 3

  12x² – 9 + 10x - 3 = -10x + 3 + 10x - 3

      12x² +10x – 12 = 0

Step 2: Simplihen at i-factor

12x² + 10x – 12 = 0

6x² + 5x - 6 = 0 

(2x + 3) (3x – 2) = 0

Step 3: I-solve nang magkahiwalay ang dalawang factors.

2x + 3 = 0 at 3x – 2 = 0

      2x = -3       3x = 2

  2𝑥/2 = −3/2    3𝑥/3 = 2/3

𝑥 = −3/2         𝑥 = 2/3

Step 4: State your final answer.

There are two roots, namely −𝟑/𝟐   and 𝟐/𝟑.

Solving Quadratic Equations by Completing the Square

2. Ang completing the square ay ginagawa sa pamamagitan ng pagsunod sa mga hakbang na ito:

        a. Ilipat ang lahat ng constant term sa kanang bahagi ng equation.

        b. Hatiin o i-multiply ang magkabilang panig ng equation upang ang maging coefficient ng x² ay 1.

        c. Tukuyin ang constant na idaragdag sa kaliwang bahagi ng equation upang ito ay maging isang perpektong trinomial. Idagdag din ang constant na ito sa kanang panig ng equation. Ang constant na idaragdag ay ang square ng kalahati ng numerical coefficient ng linear term (bx).

        d. I-factor ang kaliwang bahagi ng equation.

        e. I-extract ang square root ng bawat panig ng equation. Ang square root ng kaliwang bahagi ng equation ay linear na ngayon at ito ay katumbas sa 

± square root sa kanang bahagi ng equation.

        f. Lutasin ang bawat isa sa mga linear na equation para makuha ang mga root.

HALIMBAWA 2

        Solve for x in the equation 3x² – 36x + 42 = 0.

Step 1: Ilipat ang constant sa kanang panig ng equation.

  3x² – 36x + 42 = 0

            3x² -36x = -42

Step 2: Hatiin o i-multiply ang equation upang maging 1 ang coefficient ng x².

Sa ating problema, magagawa natin ito sa ating equation kung idi-divide natin ito ng 3.

            3x² -36x = -42

            x² – 12x = - 14

Step 3: Complete the square.

Paano ito gagawin?

Ang numerical coefficient ng linear term na -12x ay -12. Ang kalahati ng -12 ay -6 at ang square ng -6 ay 36.

Kaya ang constant na idaragdag sa magkabilang panig ng equation ay 36.

                        x² – 12x = - 14

x² -12x + 36 = -14 + 36

x² – 12x + 36 = 22

Step 4: I-factor ang kaliwang panig ng equation.

x² – 12x + 36 = 22

(x – 6) (x – 6) = 22

(x – 6)² = 22

Step 5: Kunin ang square root ng bawat panig ng equation.

     (x – 6)² = 22

     

Step 6: Kunin ang value ng x.

   

Step 7: State your final answer.

The roots of the equation are 

Solving Quadratic Equations Using the Quadratic Formula

3. Maaaring gamitin ang quadratic formula upang malutas ang anumang quadratic equation.

        Sundin ang mga hakbang na ito sa paggamit ng quadratic equation.

        a. Ilipat ang lahat ng mga termino sa kaliwang bahagi ng equation upang ang kanang bahagi ay maging katumbas ng zero.

        b. Pasimplehin ang equation.

        c. Gamitin ang quadratic formula kung saan:

a = ang coefficient ng quadratic term (ax²); 

b = ang coefficient ng linear term (bx) ; at 

c = ang constant.

HALIMBAWA 3

Solve for h in the equation 3h² – 15h = -18

Step 1: Ilipat ang lahat ng termino sa kaliwang panig ng equation upang maging zero ang kanang panig.

3h² – 15h = -18

3h² – 15h + 18 = -18 + 18

3h² – 15h + 18 = 0

Step 2: Gawing simple ang equation.

3h² – 15h + 18 = 0

h² – 5h + 6 = 0

Step 3: Tukuyin ang value ng a, b, at c sa equation.

h² – 5h + 6 = 0

a = 1 b = -5 c = 6

Step 4: Kunin ang value (roots) ng h gamit ang quadratic formula.

Thus, the roots of our equation are 3 and 2.

Step 5: Optional. Check the answer.

For h = 3 

              3h² – 15h = -18   

3(3)² – 15(3) ≟ -18          

      3(9) – 45 ≟ -18               

        27 – 45 ≟ -18        

               -18 = -18              

For h = 2

            3h² – 15h = -18

      3(2)² – 15(2) ≟ -18

            3(4) – 30 ≟ -18

               12 – 30 ≟ -18

                      -18 = -18

REMEMBER

1 - A quadratic equation is an equation in which the highest power (exponent) of any of its variables is 2.

2 - The standard form of a quadratic equation with a single variable, x, is 

ax² + bx + c = 0where a, b, and c are constants and a ≠ 0.

3 – There are three ways in solving quadratic equations:  factoring, completing the square, and using the quadratic formula.

4 – You can only use the completing the square method as long as the coefficient of x² is 1.

5 – The formula for the quadratic equation is:

6 - When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b² – 4ac.

     A quadratic equation with real numbers as coefficients can have the following:

        a. Two different real roots if the discriminant b² – 4ac is a positive number                 (greater than zero).

        b. One real root if the discriminant b² – 4ac is equal to 0.

        c. No real root if the discriminant b² – 4ac is a negative number (less than                         zero).

--o0o--

Sanggunian: ALS Module: Equations 2

ALS A&E REVIEWER 2021: LS 1 - Communication Arts - Filipino | Wastong Gamit (Based on Actual & Previous Tests)

ALS A&E Reviewer

LS 1 – Communication Skills:  FILIPINO  (Wastong Gamit)

(Image from https://www.pinoyexchange.com)

Panuto: Piliin ang titik ng tamang sagot at i-shade ito sa iyong answer sheet. Kung sa palagay mo ay walang tamang sagot, i-shade ang E.

1. Tuwing tag-ulan ________ ng palay ang Tatay.

A. tanim

B. nagtanim

C. nagtatanim

D. magtatanim

2. Palagiang maghugas ng kamay upang makaiwas sa sakit dulot ng mikrobyo.

Alin ang pangatnig sa pangungusap?

A. Palagiang

B. maghugas

C. upang

D. dulot

3. Si Abegail ay may dalawang aso at tatlong pusa. Lahat sila ay inahin subalit baog ang isa. Ang pangalan niya ay Beybz.

Alin sa mga sumusunod ang hindi totoo?

A. Lima ang bilang ng mga alaga ni Abegail.

B. Isa sa mga alaga ni Abegail ay lalaki.

C. Mas maraming pusa si Abegail kaysa aso.

D. Lahat ng alaga ni Abegail ay babae. 

4. Di mahapayang gatang si Aling Marites kaya kakaunti lang ang kanyang mga tapat na kaibigan.

Ang kahulugan ng mga salitang may salungguhit ay:

A. Hindi nagpapatalo

B. Ususera

C. Tsismosa

D. Masungit

5. Sa kambal, si Charlie ang __________.

A. may tangkad

B. matangkad

C. mas matangkad

D. pinakamatangkad

6. _________ pista lang ng baryo umuuwi ang Kuya Nilo.

A. Kapag

B. Kung

C. Habang

D. Gayunman

7.  Aling pangungusap ang may tamang bantas at/o baybay?

A. Paborito ni Lukas ang atis, mangga at tsiko.

B. Naku, po!

C. Dumede na ba ang Baby?

D. Dalhin mo ito bukas: ruler, lapis, at papel.

8. Malulusog ang tanim na palay ni Mang Pedro kaya pinangutang na niya ito. Kinabukasan, nasira ng bagyo ang mga pananim.

Anong salawikain ang nababagay sa pahayag?

A. Daig ng maagap ang masipag.

B. Huwag magbilang ng sisiw hanggang di pa napipisa ang mga itlog.

C. Kung may tiyaga, may nilaga.

D. Nasa Diyos ang awa, nasa tao ang gawa.

9. Alin sa mga pangungusap sa ibaba ang walang maling baybay?

A. Nakahuli si Jaime ng maraming apahap at dalagambukid. 

B. Masarap kumain ng halo-halo ngayong tag-araw.

C. Maganda ang kanyang imahe sa publiko.

D. Matibay ang inpraistrakturang ipinatayo ni Meyor Toto.

10. Pupunta raw ______ si Presidente Duterte upang pasinayaan ang bubuksang pampublikong pagamutan.

A. rito

B. dito

C. rin

D. din

11. Kagigising lamang ni Rosita ______ dumating ang kanyang mister.

A. ng

B. nang

C. kung

D. samantalang

12. Si Debbie at ako ay magtutungo sa batis. _______ ay maliligo roon.

A. Sila

B. Tayo

C. Kami

D. Ikaw

13. Nais pumasok ng paaralan si Tim subali’t siya ay nilalagnat.

Anong uri ng pangungusap ito?

A. payak

B. tambalan

C. hugnayan

D. tambalan-hugnayan

14. Pumanaog ka at _________ mo ang bakuran.

A. walisan

B. walisin

C. winalis

D. nawalis

15. Kahit sa siyudad lumaki, __________ ng adobong palaka si Rollie sa tuwing nauuwi ng probinsya.

A. nakain

B. kumakain

C. kinain

D. kakain

16. Malasutla ang kutis ng bagong hirang na Binibining Bacolod.

Anong bahagi ng pangungusap ang malasutla?

A. pangngalan

B. pang-abay

C. pang-angkop

D. pang-uri

17. Kinaon ng kuya ang nanay sa daungan.

Anong salita ang maaaring ipalit sa “kinaon”?

A. Sinundo

B. Inihatid

C. Sinamahan

D. Kinanlungan

18. ___________ mo ng mantekilya ang mga tinapay na ihahandog sa mga bisita.

A. Pahirin

B. Pahiran

C. Ipahid

D. Mapahid

19. Matapos makapagsaing, nanaog si Adelfa upang sunduin ang anak na si Biboy sa paaralan.

Alin ang simuno sa pangungusap?

A. makapagsaing

B. Adelfa

C. anak

D. Biboy

20. Ang katumbas ng “di maliparang uwak” ay ______.

A. malapad

B. malaki

C. malawak

D. mahaba

--o0o--

MGA SAGOT:

ALS A&E Reviewer 2021: LS 1 - Communication Arts - ENGLISH | Reading Comprehension (Based on Actual & Previous Tests)

Directions: Choose the letter of the correct answer and shade it on your answer sheet. If you think there is no correct answer, shade E. 

(Image from https://all-free-download.com)

1. The comedian pulled silly faces to make the audience laugh.
The word silly in this sentence means:

A. funny
B. stupid
C. scary
D. sensible

2. Franklin Delano Roosevelt (the only president who served three terms) instituted the New Deal reforms.

        What correction should be made to this sentence?

A. Place commas before and after the parentheses.

B. Capitalize the word “president.”

C. Make the initial letters of “New Deal” lowercase.

D. Capitalize the word “reforms.”

3. Josie has three ducks and four chickens. They are all hens, but one of the ducks seems infertile. Her name is Dalmy.

        Which of the following is true?

A: Josie has seven chickens in total.

B: Josie has less chickens than ducks.

C: One of Josie’s ducks is a male.

D: All of Josie’s animals are female.

4. Lota could no longer take her boyfriend’s incorrigible behavior.

The underlined word means:

A. reformed

B. incurable

C. repentant

D. frustrated

5. Once Karlo lifted his pen and made a start, writing the essay became easy.

        If we change the start of the sentence to:

    Writing the essay became easy........

        What will the ending be?

A. after Karlo starts writing.

B. after lifting his pen.

C.  once Karlo lifted his pen and made a start.

D. once he lifted his pen and made a start. 

6.    Read the paragraph below and answer the question that follows.

(Source: https://www.education.vic.gov.au)

        Genealogy is fun. Just as a piece of furniture or a picture takes on much more interest if you know its history, so does an individual become more real once the ancestral elements that shaped him are known. An in-depth family history is a tapestry of all those to whom we owe our existence.

        Which statement best conveys the theme of this paragraph?

A: Finding out about our ancestors is more interesting than researching the history of objects.

B: Genealogy is a study of people and their belongings in the past.

C: Genealogy is a study of family history.

D: Genealogical research can bring meaning and life to a family’s history.

7. Which statement has the correct punctuation mark and/or capitalization?

A. Bongbong likes pizza, spaghetti, and lasagna. 

B. All the dog’s dishes were full. 

C. Moana s listening to Hot and Cold by Katie Perry.

D. Please bring the following items to class:

pencil, paper, eraser and folder. 

8. What does this proverb suggest?

Beauty is in the eye of the beholder.

A. It takes good eyes to see beauty.

B. You are beautiful if you have blue eyes.

C. Beauty is subjective. 

D. Beauty is universal.

9. Which of the following sentences contains no spelling error?

A. Curiosity killed the cat. 

B. Pad and pencil are examples of stationaries.

C. After 20 years, the couple has decided to legally seperate.

D. To avoid bad breathe, brush your teeth at least twice a day.

10. We all love to win. However, we also have to know how to accept defeat.

        If we change the above into a single sentence and begin:

    We have to know how to accept defeat........

        What will the best ending be?

A: however, we all love to win.

B: but winning is better.

C: so we can also love to win.

D: even though we all love to win.

For Items 11 - 15: Read the following paragraphs to answer the next five questions.

(Source:  https://www.education.vic.gov.au) 

One of the modern world’s intriguing sources of mystery has been airplanes vanishing in mid-flight. One of the more famous of these was the disappearance in 1937 of a pioneer woman aviator, Amelia Earhart. On the second last stage of an attempted round the world flight, she had radioed her position as she and her navigator searched desperately for their destination, a tiny island in the Pacific.

        The plane never arrived at Howland Island. Did it crash and sink after running out of fuel? It had been a long haul from New Guinea, a twenty-hour flight covering some four thousand kilometers. Did Earhart have enough fuel to set down on some other island on her radioed course? Or did she end up somewhere else altogether? One fanciful theory had her being captured by the Japanese in the Marshall Islands and later executed as an American spy; another had her living out her days under an assumed name as a housewife in New Jersey.

        Seventy years after Earhart’s disappearance, ‘myth busters’ continue to search for her. She was the best-known American woman pilot in the world. People were tracking her flight with great interest when, suddenly, she vanished into thin air. Aircraft had developed rapidly in sophistication after World War One, with the 1920s and 1930s marked by an aeronautical record-setting frenzy. Conquest of the air had become a global obsession. While Earhart was making headlines with her solo flights, other aviators like high-altitude pioneer Wiley Post and industrialist Howard Hughes were grabbing some glory of their own. But only Earhart, the reserved tomboy from Kansas who disappeared three weeks shy of her 40th birthday, still grips the public imagination. Her disappearance has been the subject of at least fifty books, countless magazine and newspaper articles, and TV documentaries. It is seen by journalists as the last great American mystery.

There are currently two main theories about Amelia Earhart’s fate.

There were reports of distress calls from the Phoenix Islands made on Earhart’s radio frequency for days after she vanished. Some say the plane could have broadcast only if it were on land, not in the water. The Coast Guard and later the Navy, believing the distress calls were real, adjusted their searches, and newspapers at the time reported Earhart and her navigator were marooned on an island. No one was able to trace the calls at the time, so whether Earhart was on land in the Phoenix Islands or there was a hoaxer in the Phoenix Islands using her radio remains a mystery. 

Others dismiss the radio calls as bogus and insist Earhart and her navigator ditched in the water. An Earhart researcher, Elgen Long, claims that Earhart’s airplane ran out of gas within fifty-two miles of the island and is sitting somewhere in a 6,000-square-mile area, at a depth of 17,000 feet. At that depth, the fuselage would still be in shiny, pristine condition if ever anyone were able to locate it. It would not even be covered in a layer of silt.  Those who subscribe to this explanation claim that fuel calculations, radio calls, and other considerations all show that the plane plunged into the sea somewhere off Howland Island. 

Whatever the explanation, the prospect of finding the remains is unsettling to many. To recover skeletal remains or personal effects would be a grisly experience and an intrusion. They want to know where Amelia Earhart is, but that’s as far as they would like to go. As one investigator has put it, “I’m convinced that the mystery is part of what keeps us interested. In part, we remember her because she’s our favorite missing person.”

11. Amelia Earhart’s nationality was:

A. Australian

B. American

C. English

D. South African

12. All the following are theories about Amelia’s fate EXCEPT:

A. her plane ran out of fuel and crashed into the sea.

B. she crashed somewhere on Howland Island

C. she was captured by the Japanese and executed as a spy.

D. she escaped incognito and lived under an assumed name.

13. The most convincing evidence that Amelia crashed somewhere on land was:

A: the finding of aircraft remains.

B: sightings by islanders.

C: radio contact with the coastguard from the Phoenix Islands.

D: distress signals from the Phoenix Islands on her particular radio frequency

14. If the aircraft were ever recovered from its probable sea grave:

A. it would be hardly recognizable.

B. it would be in pristine condition and considered highly valuable.

C. it may reveal some grisly evidence.

D. B and C together.

15. The fate of Amelia Earhart still fascinates investigators for all the following reasons EXCEPT:

A. she was a famous female aviator and adventurer.

B. there are such conflicting theories about her disappearance.

C. she may have staged her own disappearance.

D: she presents one of the twentieth century’s great unsolved mysteries.

Read the passage below from © The Economist Newspaper Limited, London, 1999  about older people in the workforce and answer Items 16 – 19. 

(Source: https://www.ielts.org)

The general assumption is that older workers are paid more in spite of, rather than because of, their productivity. That might partly explain why, when employers are under pressure to cut costs, they persuade a 55-year old to take early retirement. Take away seniority-based pay scales, and older workers may become a much more attractive employment proposition. But most employers and many workers are uncomfortable with the idea of reducing someone’s pay in later life – although manual workers on piece-rates often earn less as they get older. So retaining the services of older workers may mean employing them in different ways.

        One innovation was devised by IBM Belgium. Faced with the need to cut staff costs, and having decided to concentrate cuts on 55 to 60-year olds, IBM set up a separate company called Skill Team, which re-employed any of the early retired who wanted to go on working up to the age of 60. An employee who joined Skill Team at the age of 55 on a five-year contract would work for 58% of his time, over the full period, for 88% of his last IBM salary. The company offered services to IBM, thus allowing it to retain access to some of the intellectual capital it would otherwise have lost.

        The best way to tempt the old to go on working may be to build on such ‘bridge’ jobs: part-time or temporary employment that creates a more gradual transition from full-time work to retirement. Studies have found that, in the United States, nearly half of all men and women who had been in full-time jobs in middle age moved into such ‘bridge’ jobs at the end of their working lives. In general, it is the best-paid and worst-paid who carry on working. There seem to be two very different types of bridge job-holder – those who continue working because they have to and those who continue working because they want to, even though they could afford to retire.

        If the job market grows more flexible, the old may find more jobs that suit them. Often, they will be self-employed. Sometimes, they may start their own businesses: a study by David Storey of Warwick University found that in Britain 70% of businesses started by people over 55 survived, compared with an overall national average of only 19%. But whatever pattern of employment they choose, in the coming years the skills of these ‘grey workers’ will have to be increasingly acknowledged and rewarded.

16. In paragraph one, the writer suggests that companies could consider

A. abolishing pay schemes that are based on age.

B. avoiding pay that is based on piece-rates.

C. increasing pay for older workers.

D. equipping older workers with new skills

17. Skill Team is an example of a company which

A. offers older workers increases in salary.

B. allows people to continue working for as long as they want.

C. allows the expertise of older workers to be put to use.

D. treats older and younger workers equally

18. According to the writer, ‘bridge’ jobs

A. tend to attract people in middle-salary ranges.

B. are better paid than some full-time jobs.

C. originated in the United States.

D. appeal to distinct groups of older workers.

19. David Storey’s study found that

A. people demand more from their work as they get older.

B. older people are good at running their own businesses.

C. an increasing number of old people are self-employed.

D. few young people have their own businesses.

20. Which of the following is a metaphor?

A. I'm drowning in paperwork.

B. You are as light as a feather.

C. Life is a broken-winged bird that cannot fly.

D. I came, I saw, I conquered.

ANSWERS:

ALS A&E Reviewer 2021: LS 5 - Understanding the Self & Society (Based on Previous Actual Test Questions)

Piliin ang titik ng tamang sagot.

1. “Huwag kang bibitiw! Subukan mo lang hanggang makamit ang tagumpay.” Anong mabuting katangian ang ipinahihiwatig ng mga pangungusap na ito?

A. makatarungan

B. matipid

C. matiyaga

D. maagap

2. Kalilipat mo lamang at ang iyong pamilya sa Barangay Loob. Nais mong makibahagi sa mga gawaing pampaunlad sa inyong barangay. Ano ang dapat mong gawin?

A. pumunta sa istasyon ng pulis upang magtanong

B. humingi ng tulong sa iyong kapitbahay

C. gumawa ng sarili mong proyekto

D. pumunta sa barangay hall at magtanong kung ano ang mga proyekto nito

3. Ano ang nabanggit sa ibaba ang itinuturing na karapatang sibil?

A. bumoto

B. makaroon ng ari-arian

C. maging malaya

D. pumili ng relihiyon

4. Ang pagluluwas ng ating mga produkto sa ibang bansa ay nagdudulot ng kabutihan para sa atin. Isa na rito ang ______.

A. pagtaas ng presyo ng mga bilihin

B. pagdami ng turista

C. paglabas ng dolyar sa bansa

D. pagpasok ng dolyar sa bansa

5. Makakamtan ang “Unity in Diversity” sa pagitan ng mga bansa kung ________.

A. Magsasama ang malalaki at mayayamang mga bansa

B. Kikilalanin at uunawain ng ang bawa’t bansa ay may iba’t ibang kultura

C. Sasapi ang maliliit na bansa sa mga samahang internasyunal

D. Itataguyod ng bawa’t bansa ang sarili nitong interes

6. Paano maisusulong ang proyektong Gender and Development (GAD) at gender equality?

A. Pagkilala na magaling sa Mathematics ang mga lalaki at sa English naman ang mga babae

B. Mga batang lalaki lamang ang maaaring maglaro ng baril-barilan

C. Maaaring maghugas ng plato ang mga anak na lalaki at babae

D. Kulay pink ang dapat isuot ng babae at blue naman sa lalaki

7. Bilang bahagi ng pamayanan, paano mo maipakikita ang iyong aktibong pakikilahok sa proyektong “Harap Mo, Linis Mo”?

A. Hindi ako makikinig sa mga pahayag ng mga nakilahok

B. Magmumungkahi ako ng mga makabuluhang paraan upang hindi kumalat ang basura

C. Makikipagkuwentuhan ako sa aking mga kaibigan habang nagsasalita ang iba

D. Uuwi ako kahit hindi pa tapos ang pulong

8. Ang pagiging matapat ay masasalamin sa pangungusap na ito:

A. Nagsasabi ako ng totoo anuman ang kahihinatnan

B. Iniipon ko sa alkansiya ang bahagi ng aking baon

C. Iniingatan ko ang aking mga gamit para di agad masira

D. Tinatapos ko agad ang aking takdang aralin bago manood ng TV

9. Napapabayaan ng iyong kaibigan ang kanyang pag-aaral dahil mas maraming oras niya ang iginugugol sa mga gawain ng kaniyang relihiyon. Ano ang maipapayo mo sa kanya?

A. Tumigil na lamang sa pag-aaral at magpakarelihiyoso

B. Ipagpatuloy ang ginagawa dahil pupunta siya sa langit kahit hindi nag-aaral

C. Pagtuunan ng pansin ang pag-aaral sa tuwing araw ng may pasok at ang paglilingkod naman sa araw ng pagsamba

D. Huwag siyang pansinin at gayahin

10. Alin sa mga sumusunod ang maituturing na pang-aabuso sa mga kababaihan?

A. Patigilin siya ng kanyang asawa sa pagtatrabaho

B. Pilitin siya ng kanyang asawa na ibigay ang buong sahod sa kanya

C. Iwanan siya ng kanyang asawa na may maraming utang na babayaran

D. Lahat nang nabanggit

11. Ano ang maaaring mangyari sa isang tao na hindi marunong magdala ng stress na kanyang nararanasan sa matagal na panahon?

A. Magiging desperado/desperada

B. Mabilis magalit

C. Magiging sakitin

D. Lahat nang nabanggit

12. Alin sa mga sumusunod na pagpapahalaga ang naglalarawan sa mga Pilipino ano man ang pangkat etnolingguistikong kinabibilangan?

A. matatag na paniniwala sa Diyos

B. pagsasabi ng po at opo

C. matatag na relasyon sa pamilya

D. pagiging mapagmuni

13. Ano ang makatarungang solusyon na dapat gawin ng mga bansang may hindi pagkakaunawaan sa kanilang teritoryo?

A. Paghahati ng teitoryo sa lahat ng umaangkin nito

B. Paggamit ng dahas

C. Dayalogo upang bumuo ng mutual agreement sa pagitan ng mga bansang sangkot

D. Pagbibigay ng teritoryo sa maliit na bansang umaangkin dito

14. Kung ang Sinulog Festival ay ipinagdiriwang sa Cebu, ano naman ang ipinagdiriwang sa Davao?

A. MassKara Festival

B. Ati-atihan Festival

C. Kadayawan Festival

D. Dinagyang Festival

15. Ano ang iyong mararamdaman kapag nabalitaan mong nasunugan ang isa mong kamag-anak?

A. malungkot

B. magagalak

C. masaya

D. magagalit

16. Kapag humingi ng tulong ang iyong kapatid sa paggawa ng kanyang proyekto sa susunod na araw, ano ang iyong gagawin?

A. sasabihin sa kanya na sa iba na humingi ng tulong

B. sisigawan siya at sasabihing huwag kang gambalain

C. hindi siya papansinin

D. tutulungan siya sa paggawa ng kanyang proyekto

17. Ang mga sumusunod ay matututunan sa pag-aaral ng Heograpiya maliban sa ___.

A. klima ng isang lugar

B. lokasyon ng lugar

C. kalagayang politikal ng isang lugar

D. populasyon ng lugar

Para sa bilang 18 – 20. Basahin ang talata sa ibaba at sagutin ang mga sumusunod na tanong.

        Nagtatrabaho sa isang pabrika si Upeng na may pitong anak. Malayo ang kanyang pinapasukan kung kayat maaga siyang umaalis sa umaga at gabi na kapag umuuwi. Bihira na niyang nakukumusta at nakakausap ang mga anak. Ang mga anak ay sa ama kumukunsulta hinggil sa mga gawain at asignatura sa paaralan.

18. Ano ang tungkulin ni Upeng sa kanyang mga anak?

A. tagapag-alaga ng asawa at mga anak

B. tagalaba ng mga damit ng pamilya

C. naghahanap-buhay para sa pamilya

D. tagaluto ng pagkain para sa mga anak at asawa

19. Bakit hindi na nagagampanan ni Upeng ang kanyang tungkulin sa asawa at mga anak?

A. Hindi niya mahal ang asawa at mga anak.

B. Mas nais niyang pagtuunan ng oras ang pagtatrabaho kaysa pamilya.

C. Nagtatrabaho siya, umaalis ng maaga at gabi na kapag nakauuwi ng bahay.

D. Hindi niya pinahahalagahan ang pamilya.

20. Sa iyong palagay, ano ang mabuting maidudulot nito sa kanyang mga anak habang sila ay lumalaki?

A. lalaki na may takot sa Diyos

B. lalaking maunawain at responsible

C. mawawalan ng pagmamahal sa magulang

D. hindi makapagtatapos ng pag-aaral

MGA SAGOT: 

2021 ALS & A&E Reviewer: LS 3 - Mathematical and Problem-Solving Skills

 Batch 2021 ALS A&E Test

Learning Strand 3 – Mathematical and Problem-Solving Skills

Direction: Choose the letter of the correct answer.

1.  If Set A = {All the primary colors} and Set B = {All the colors in the rainbow}, then A∩B =

A. {red, orange, yellow, green, blue, indigo, violet}

B. {red, yellow, blue}

C. {orange, green, violet}

D. (indigo}

 

2. A number divided by three, minus the same number multiplied by six, is one more than the number.

 

A. 3x – 6x = 1 - x

B. x/6 + 3x = x + 1

C. (6 – 3)/x = 1 + x

D. x + 1 = x/3 – 6x

 

3. The slope of a vertical line graph is _______.

 

A. 0

B. positive

C. negative

D. infinite

 

4. The y-intercept of the function: 3x + 5y – 6 = 0 is ______.

 

A. 3

B. 5/6

C. 6/5

D. 2

 

5. A square with a side of 6 cm and a rectangle with a width of 4 cm have the same area. What's the length of the rectangle?

 

A. 4 cm

B. 6 cm

C. 9 cm

D. 10 cm

 

6. Fifteen more than three times a number is the same as nine less than six times the number. What is the number?

 

A. 1/8

B. 2

C. 3

D. 8

 

7. The sum of the digits in a two-digit number is seven. If the tens digit is three less than the ones digit, find the number.

A. 25

B. 36

C. 47

D. 52

 

8. The area of a triangle is 30 units. If its base is 5 units, then its altitude is ________.

 

A. 6 units

B. 9 units

C. 12 units

D. 15units

 

9. Manolo reads three-fifth of 75 pages of his lesson. How many more pages he need to complete the lesson?

 

A. 15

B. 30

C. 45

D. 60

 

10. The school canteen posted its price list as follows:

Pancit                    ₱ 13.75
Spaghetti               ₱ 20.50
Hamburger            ₱ 24.75
Cheese sandwich   ₱ 12.50
Fruit juice              ₱ 10.00
Bottled water         ₱ 8.00

            Adele and her two friends ordered one pancit, two spaghetti, one hamburger, two fruit juices, and one bottled water. How much money did Adele give if she received a change of ₱42.50?

A. ₱107.50

B. ₱ 130.00

C. ₱ 150.00

D. ₱ 200.00

ANSWERS and SOLUTIONS

Lesson 5 - Applications of Linear Equations in Word Problems Using 2 Variables and Systems of Equations

 Lesson 5 – Applications of Linear Equations: Part 3

        Sa nakaraang aralin ay nai-apply natin ang ating natutunang konsepto at pamamaraan sa paglutas ng mga word problems na kinasasangkutan ng number, digit, uniform motion, money, age, investment, at mixture problems na may isang variable o unknown.

Sa bahaging ito ay pagtutunan naman natin ang pagsagot ng mga word problems gamit ang dalawang variables at/o dalawang equations.

Number Problems

        May mga word problems na nalulutas sa pamamagitan lamang ng isang variable o unknown at isang equation. Gayunpaman, may mga pagkakataon na mas madali ang pagsagot ng mga ito kung gagamit ng dalawang variables at dalawang sistema ng equations.

Tunghayan natin sa mga sumusunod na halimbawa:

HALIMBAWA 1

The sum of the two numbers is 17 and their difference is 7. Find the two numbers.

Let x = the  smaller number

y = the larger number

Given: x + y = 17 (Equation 1)

y – x  = 7         (Equation 2)

Add the two equations.

x + y = 17

+ y – x = 7

          2y = 24

      y = 12

Substitute the value of y into either Equation 1 or 2 to find x.

    y - x = 7 (Equation 2)

         12 – x  = 7

          12 – 12 – x = 7 – 12

        -x = -5 

                  -1( -x = -5)

         x = 5

Check our answer.

        For Equation 1:

x + y = 17

        5 + 12 ≟ 17

      17 = 17

        For Equation 2:

y - x = 7

        12 - 5 ≟ 7

        7 = 7

        Thus, our smaller number is 5 and the larger number is 7.

REMEMBER

We can solve many problems by translating them into systems of equations and using the following problem-solving guidelines:

1.    Understand the problem. Read it carefully and decide which quantities are unknown. (Unawain ang problema. Basahin itong mabuti at magpasya kung aling mga quantities ang hindi alam o unknown.)

2.    Develop a plan. Represent one of the unknown values by one variable and the second unknown by another variable. (Bumuo ng isang plano. Katawanin ang isa sa mga unknown ng isang variable at ang pangalawa ng isa pang variable.)

3.    Carry out your plan. Study the stated facts until you understand their meanings. Then translate the related facts into equations in two variables. Solve the system of equations. (Isagawa ang iyong plano. Pag-aralan ang mga nakasaad na katotohanan hanggang sa maunawaan mo ang mga kahulugan nito. Pagkatapos ay isalin ang mga kaugnay na katotohanan sa mga equation sa dalawang variable. Lutasin ang sistema ng mga equation.)

4.    If available, check your answers using the derivations you made and not the given equation itself. Write a statement to answer the question being asked in the problem. (Kung mayroon, suriin ang iyong mga sagot gamit ang mga derivasyon na iyong ginawa at hindi ang ibinigay na equation mismo. Sumulat ng isang pahayag upang masagot ang tanong sa problema.)

Age Problems

HALIMBAWA 2

        Olive is 4 years younger than Popeye. Twenty years ago, Popeye’s age was 13 years more than half the age of Olive. How old are they now?

        Let    x    =    Olive's age now

                 y    =    Popeye's age now

Labis na makatutulong kung ilalagay natin sa isang table ang mga ibinigay na impormasyon.

        From the table, we can get a system of two equations in x and y.

        x = y – 4 (Equation 1)

y – 20 = ½(x – 20) + 13 (Equation 2)

Simplifying Equation 2, we will have:

y – 20 = ½(x – 20) + 13 (Equation 2)

y – 20 = 1/2x – 10 + 13

y – 20 = 1/2x + 3

y = 1/2x + 23         (Equation 3

        Substitute the value of x in Equation 1 into Equation 3.

        y = ½(y – 4) + 23         (Equation 3)

        y = 1/2y – 2 + 23

1/2y = 21

        2(1/2y = 21)

y = 42

Substitute 42 as the value of y in Equation 1 to find the value of x.

        x = y – 4 (Equation 1)

        x = 42 – 4

        x = 38

Thus, Olive is 38 years old and Popeye is 42 years old now.

        Check our answer.

For Equation 3:

   y = ½(y – 4) + 23 (Equation 3)

42 ≟ ½(42 – 4) + 23

42 ≟ ½(38) + 23

42 ≟ 19 + 23

42 = 42

For Equation 2: 

y – 20 = ½(x – 20) + 13

        42 – 20 ≟ ½(38 – 20) + 13

        22 ≟ 19 – 10 + 13

        22 = 22

     Since we got equality, our answer is correct.

Digit Problems

HALIMBAWA 3

In a three-digit number, the hundreds digit is twice the units digit. If 396 be subtracted from the number, the order of the digits will be reversed. Find the number if the sum of the digits is 17.

Let h = the hundreds digit

t = the tens digit

u = the units digit

100h + 10t + u = the number

h = 2u (Equation 1)

        h + t + u = 17     (Equation 2)

                       (100h + 10t + u) – 396 = 100u + 10t + h

100h + 10t + u – 396 -100u – 10t – h = 396

                                   99h – 99u = 396

                          99h/99 - 99u/99 = 396/99

                                           h – u = 4          (Equation 3)

Substitute h = 2u into Equation 3.

h – u = 4

         2u – u = 4

         u = 4

Substitute  4 as u into Equation 1.

h = 2u

h = 2(4)

h = 8

        Substitute u = 8 and h = 4 into Equation 2 to find the value of t.

h + t + u = 17

4 + t + 8 = 17

  4 – 4 + t 8 – 8 = 17 – 4 – 8

                       t = 5

Substitute these 3 values into our number.

100h + 10t + u =

100(8) + 10(5) + 4 = 

800 + 50 + 4  = 854

Thus, our number is 854.

Let us check our answer by substituting our values into our three equations.

For Equation 1:

h = 2u

8 ≟2 (4)

8 = 8

For Equation 2:

h + t + u = 17

      8 + 5 + 4 ≟ 17

         17 = 17 

For Equation 3:

h – u = 4

8 – 4 ≟  4 

       4 = 4

Since our left and right terms for our three equations are balanced, our answer is correct.

Mixture Problems

HALIMBAWA 4

Tikyo wants to make a 1000 ml of 50% alcohol solution mixing a quantity of a 20% alcohol solution with a 70% alcohol solution. What are the quantities of each of the two solutions he has to use?

        Let     x    =    amount of 20% alcohol

                  y    =    amount of 70% alcohol

Analyzing the problem, we find that two conditions must be met.

Condition 1

liters of 20% solution + liters of 70% solution = liters of 50% solution

x + y = 1000

        Condition 2

        pure alcohol in 20% solution + pure alcohol in 70% solution = pure alcohol in 50% solution

                        0.20x + 0.70y = 0.50(1000)

The system we must solve is:

       x + y = 1000      (Equation 1) 

 0.20x + 0.70y = 0.50(1000)      (Equation 2)

        Simplify Equation 2.

0.20x + 0.70y = 0.50(1000)

0.20x + 0.70y = 500

10(0.20x + 0.70y = 500)

2x + 7y = 5000 (Equation 3)

Multiply Equation 1 by 2.

x + y = 1000

2( x + y = 1000)

2x + 2y = 2000         (Equation 4)

Subtract Equation 4 from  Equation 3.

2x + 7y = 5000         (Equation 3)

    -     (2x + 2y = 2000)         (Equation 4)

            5y = 3000

         5y/5 = 3000/5

              y = 600

Substitute 600 as the value of y into Equation 1.

x + y = 1000

          x + 600 = 1000

        x + 600 – 600 = 1000 – 600

            x = 400

Thus, Tikyo has to use 400 liters of 20% alcohol and 600 liters of 70% alcohol to obtain 1000 liters of 50% alcohol.

        Let us check to confirm our answer.

          x + y = 1000      (Equation 1) 

400 + 600 ≟ 1000

          1000 = 1000

0.20x + 0.70y = 0.50(1000)    (Equation 2)

      10(0.20x + 0.70y ≟ 500)

2x + 7y ≟ 5000

     2(400)x + 7(600) ≟ 5000

      800 + 4200 ≟ 5000

         5000 = 5000

Since we got equality in our original equations, our answer is correct.

Money Problems 

HALIMBAWA 5

The price of 3 chairs and 2 tables is ₱4500 and the price of 5 chairs and 3 tables is ₱7000. Find the price of 2 chairs and 2 tables.

Let c = the price of a chair

t = the price of a table

Given:

3c + 2t = 4500         (Equation 1)

5c + 3t = 7000         (Equation 2)

Subtract Equation 1 from Equation 2.

5c + 3t = 7000         (Equation 2)

  -   (3c + 2t = 4500) (Equation 1)

2c + t   =  2500

  2c – 2c + t = 2500 – 2c

          t = 2500 – 2c (Equation 3)

Substitute Equation 3 into either Eq’n 1 or 2.

If Equation 2:

5c + 3t = 7000

  5c + 3(2500 – 2c) = 7000

5c + 7500 – 6c = 7000

  -c + 7500 – 7500 = 7000 – 7500

-c = -500

-1(-c = -500) 

                                c = 500

        Substitute 500 for c into Equation 3 to find t.

t = 2500 – 2c (Equation 3)

t = 2500 – 2(500)

t = 2500 – 1000

t = 1500

Thus, the price of two chairs (2 x 500 = 1000) and two tables (2 x 1500 = 3000) is ₱ 4000 ( 1000 + 3000).

Let us check to confirm our answer.

                  3c + 2t = 4500 (Equation 1)

3(500) + 2(1500) ≟  4500

       1500 + 3000  ≟ 4500

                     4500 = 4500

                                 5c + 3t = 7000 (Equation 2)

5(500) + 3(1500) ≟ 7000

       2500 + 4500 ≟ 7000

                    7000 = 7000

        t = 2500 – 2c (Equation 3)

1500 ≟ 2500 – 2(500)

1500 ≟ 2500 – 1000

1500 = 1500

Since we got equality in our original equations, our answer is correct.

Investment Problems 

HALIMBAWA 6

Bongbong invested ₱11,000.  Part of his money is invested in bonds which yield 8% and the remainder is invested in the money market which yields 10%.  His total annual income from these investments is ₱1,020.  Find the amount he has invested in each kind of investment.

Let     b     = amount he invested in bonds at 8%

                    m         = amount he invested in money market at 10%

                0.08b = annual income from 8% bonds investment

        0.10m = annual income from 10% money market investment

b + m = 11000     (Equation 1)

       0.08b + 0.10m = 1020 (Equation 2)

        Multiply Equation 2 by 100.

0.08b + 0.10m = 1020         (Equation 2)

   100(0.08b + 0.10m = 1020)

          8b + 10m = 102000 (Equation 3)

        Multiply Equation 1 by 8.

b + m = 11000         (Equation 1)

8(b + m = 11000)

8b + 8m = 88000         (Equation 4)

Subtract Equation 4 from Equation 3.

         8b + 10m = 102000 (Equation 3)

     -   (8b + 8m = 88000) (Equation 4)

           2m = 14000

        2m/2 = 14000/2

             m = 7000

        Substitute 7000 as the value of m into Equation 1 to find b.

b + m = 11000         (Equation 1)

        b + 7000 = 11000

    b + 7000 – 7000 = 11000 – 7000

            b = 4000

Thus, Bongbong invested ₱4000 in bonds at 8% and ₱7000 in money market at 10%.

Let us check to confirm our answer.

b + m = 11000         (Equation 1)

          4000 + 7000 ≟ 11000

                 11000 = 11000

  0.08b + 0.10m = 1020         (Equation 2)

  100(0.08b + 0.10m ≟ 1020)

         8b + 10m ≟ 102000

 8(4000) + 10(7000) ≟ 102000

        32000 + 70000 ≟ 102000

             102000 = 102000

Since we got equality in our original equations, our answer is correct.

Motion Problems 

HALIMBAWA 7

A boat can travel 16 miles up a river in 2 hours.  The same boat can travel 36 miles downstream in 3 hours.  What is the speed of the boat in still water?  What is the speed of the current?

Let s = speed of the boat in still water

                v = speed of the current

        Kung hindi pa bihasa sa ganitong klaseng problema, makatutulong nang malaki ang paggawa ng table upang ilagay ang ating mga datos.

    

        Mapapansin na ang rate (s + v) ng boat downstream ay ang speed nito sa kalmang tubig (still water) PLUS ang speed ng current  (o daloy ng tubig)  dahil mas mabilis ang takbo ng bangka kung paibaba o paayon sa daloy ng tubig.  Kung salungat naman sa daloy ng tubig o upstream, magiging mabagal ang takbo ng bangka kaya ibabawas natin ang speed ng current sa kanyang speed sa still water upang makuha ang rate (s - v) nito.

        Mula sa talahanayan o table, narito ang ating nakuhang equation:

16 = 2(s -  v) (Equation 1)

36 = 3(s + v) (Equation 2)

Divide Equation 1 by 2.

16 = 2(s - v) (Equation 1)

       16/2 = 2(s - v)/2

    8 = s - v (Equation 3)

Divide Equation 2 by 3.

36 = 3(s + v) (Equation 2)

       36/3 = 3(s + v)/3

12 = s + v (Equation 4)

        Add Equation 3 and Equation 4.

8 = s - v (Equation 3)

   +    12 = s + v (Equation 4)

         20 = 2s

     20/2 = 2s/2

        10 = s

Substitute 10 for s in any of the equations to find v. Let’s take Equation 4.

12 = s + v (Equation 4)

12 = 10 + v

12 – 10 = 10 – 10 + v

2 = v

        Thus, the speed of the boat in still water is 10 miles per hour while the speed of the current is 2 miles per hour.

Let us check to confirm if our answer is correct.

16 = 2(s -  v) (Equation 1)

16 ≟ 2(10 – 2)

16 ≟ 2(8)

16 = 16

36 = 3(s + v) (Equation 2)

36 ≟ 3(10 + 2)

36 ≟ 3(12)

36 = 36

Since we got equality in our original equations, our answer is correct.

Pagsasanay 

Solve the following word problems and check your answers:

1.    If one number is three times as large as another number and the smaller number is increased by 19, the result is 6 less than twice the larger number. What is the larger number?

2.    A silver coin is 28 years older than a bronze coin. In 6 years, the silver coin will be twice as old as the bronze coin. Find the present age of each coin.

3.    The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

4.    You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?

5.    The cost of admission to a popular music concert was ₱162 for 12 children and 3 adults. The admission was ₱122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?

6.    Isko has invested in two savings accounts. One earns 10% and the other earns 15%. He invests ₱200 more in the account that earns 15%. The total interest earned for one year is ₱230. How much is invested in each account?

7.    A steamer goes downstream and covers the distance between two ports in 4 hrs., while it covers the same distance upstream in 5 hrs. If the speed of the stream is 2km/h, find the speed of the steamer in still water.

8.    The present ages of Bongbong and Sara are in the ratio 3:4. Five years from now, the ratio of their ages will be 4:5. Find their present ages.

Note: The above sample word problems were taken from different internet sites. 

ANSWERS: