Lesson 2 - Linear Equations in One Variable: ALS Module - Equations 1

 Lesson 2 – Linear Equations in One Variable

(Image from https://www.cmuse.org)

Simulan natin ang pag-aaral sa pamamagitan ng pag-alam sa mga konsepto ng mga sumusunod na termino:

1. Ang equality ay isang mathematical na pahayag kung saan ang dalawang expression ay pantay o may parehong value/halaga.

Kung gayon, ang 3x = 15 at 2y + 4y = 6y ay mga equation.

2. May dalawang uri ng equalities. Ang mga ito ay ang equations at identities.

3. Ang identity ay isang equality kung saan sa mga variable ay maaaring magkaroon ng anumang value at kung saan ang equality ay magiging tama sa lahat ng pagkakataon. Nangangahulugan ito na ang kaliwang bahagi ng isang equation ay palaging katumbas ng kanang bahagi kahit anong value/halaga ang ipinalit para sa variable.

Halimbawa, isaalang-alang natin ang ating equality sa itaas na:

2y + 4y = 6y

Kung ang value ng ating variable y ay 0 (zero), ang equation ay magiging ==>

2(0) + 4(0) = 6(0)

0 + 0 = 0

0 = 0

Kung ang y naman ay may value na 1, ang equation ay magiging ==>

2(1) + 4(1) = 6(1)

2 + 4 = 6

6 = 6

Mapapansin natin sa itaas na kahit ano ang value na ipalit natin sa ating variable, ang equation ay magiging tama o totoo.

4. Dahil walang mga paghihigpit o restrictions sa mga value na  maaaring mayroon ang y, ang 2y + 4y = 6y ay isang halimbawa ng unconditional equality.

5. Sa kabilang banda, ang isang equation ay isang conditional equality kung ang variable/s nito ay may partikular o iisang value lamang.

6. Ang solution o root ng isang equation ay anumang numero na, kapag pinalitan ang variable, ay gagawing pantay ang dalawang panig ng isang equation. Masasabi na ang root ang siyang nagbigay tama sa isang equation.

Halimbawa, sa equation na 5x – 13 = -3x + 19

Kung 4 ang ipapalit natin sa x sa equation, ito ay magiging ==>

5(4) – 13 = -3(4) + 19

   20 – 13 = -12 + 19

     7 = 7

Samakatuwid, ang 7 ay solution o root ng equation.

Samantala, kung 2 ang ipapalit natin sa x sa equation, ito ay magiging ==>

5(2) – 13 = -3(2) + 19

   10 – 13 = -6 + 19

   -3 ≠ 13

Dahil ang -3 ay hindi katumbas ng 13, masasabi natin na ang 2 ay hindi isang solution o root ng equation.

7. Ang set ng lahat ng mga solution ng isang equation  ay tinatawag na solution set. Upang malutas ang isang equation, nangangahulugang dapat nating  hanapin ang solution set nito.

8. Ang linear equation ay isang equation na ang lahat ng  term ay nasa first degree. Nangangahulugan ito na ang pinakamataas na exponent o kabuuan ng mga exponent ng alinman sa mga termino ng isang equation ay 1.

    Ang graph ng isang linear equation ay straight line. 

9. Ang linear equations ay maaaring uriin sa linear equations in one variable, linear equations in two variables, linear equations in three variables at iba pa.

10.   Ang linear equations in a variable, x, ay maaaring isulat sa pormang   

        ax + b = 0 kung saan ang a ay hindi maaaring zero (0).

11.   Ang equivalent equations ay algebraic equations na may magkaparehong solution sets o roots.  Ang pag-add o pag-subtract ng parehong numero o expression sa pagkabilang panig ng equation ay magreresulta ng equivalent equations.

Halimbawa, sa equation na x – 2 = 8, ang ating makukuhang sagot ay:

x -  2 = 8 

                      x = 8 + 2

      x = 10

Sa parehong equation na x – 2 = 10, 

kapag nagdagdag tayo ng 5 sa kaliwa at sa kanang panig ng equation, hindi pa rin magbabago ang equation. Subukin natin.

x – 2 + 5 = 8 + 5

      x + 3 = 13

            x = 10

Samakatuwid, ang x – 2 = 8 at x – 2 + 5 = 8 + 5 ay equivalent equations.

Tandaan lamang na kung ano ang ginawa sa kaliwang panig (left side) ay ganoon din ang gagawin sa kanang panig (right side) ng equation.

Hindi lamang addition at subtraction ang maaaring gawin sa magkabilang panig ng equation. Maaari ring gawin ang multiplication at division subalit ang numerong gagamitin ay dapat ay non-zero.

12.    Ang proseso ng paglipat ng isang termino sa kabilang panig at pagbabago ng sign nito ay tinatawag na transposition.

Halimbawa:

- 4m    =   - 5m – 3

-4m + 5m   =   - 5m – 3 + 5m

       m = 0 – 3

               m = -3

Sa pagso-solve ng linear equation in one variable, pagsamahin o ilipat natin ang variable sa kaliwang panig ng equation at ang constant naman ay nasa kanang panig.

13.    Ang literal equation ay isang equation kung saan ang ilan sa mga constant ay hindi tinukoy ngunit kinakatawan ng mga titik tulad ng a, b, c o d. Ang mga titik na ito ay tinatawag na  literal constants o arbitrary constants.

Samakatuwid,  ang 5x = Q ay isang halimbawa ng literal equation kung saan ang x ay isang variable at ang Q naman ay isang literal or arbitrary constant. 

Narito ang ilan pang halimbawa ng literal equations:

A = πr2         (area of a circle)

P = 2L + 2W         (perimeter of a rectangle)

C=5/9  (F −32)       (Fahrenheit to Centigrade)

P = 4s         (perimeter of square)

Mapapansin na ang mga equations sa itaas ay mga formulas. Ito ay dahil ang lahat ng formulas ay literal equations.

How to Solve Linear Equations in One Variable

Matapos nating mapag-aralan ang konsepto ng iba’t ibang termino, handa na tayong mag-solve ng linear equation na may isang variable. Paano natin ito gagawin?

Example A: Solve 5x – 7 + 4x = 2x – 28 + 4x.

Step 1 Gawing simple ang equation sa pamamagitan ng pagsasama-sama ng magkaparehong terms.

5x – 7 + 4x = 2x – 28 + 4x ==>

        9x – 7 = 6x - 28

Step 2 Ihiwalay ang variable at ilagay ito sa kaliwang panig at ang constant naman ay ilagay sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

         9x – 7 = 6x – 28

  9x – 7 + 7 = 6x – 28 + 7

      9x = 6x – 21

      9x – 6x = 6x – 6x – 21

      3x = -21

Step 3 Gamitin ang multiplication property of equality o i-divide ang dalawang panig ng  numerical coefficient ng variable

Step 3 Ihalili ang nakuhang value ng variable sa orihinal na equation upang matiyak na tama ang nakuhang solution, root, o sagot.

                                              x = - 7

                          5x – 7 + 4x = 2x – 28 + 4x 

                      5(-7) – 7 + 4(-7) = 2(-7) – 28 + 4(-7)

                     -35 – 7 – 28 = -14 – 28 -28

                           -70 = - 70

Samakatuwid, x = - 7 ay ang solution o root ng  5x – 7 + 4x = 2x – 28 + 4x dahil natugunan nito ang equation.

Example B: Solve 3(x –2) + 6 = 5x + 2.

Step 1 Gamitin ang multiplication property over addition upang maalis ang mga  parentheses.

3(x –2) + 6 = 5x + 2 ==>

       3x – 3(2) + 6 = 5x + 2

    3x – 6 + 6 = 5x + 2

          3x + 0 = 5x + 2

                3x = 5x + 2

Step 2 Ihiwalay ang variable at ilagay ito sa kaliwang panig at ang constant naman ay ilagay sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

   3x = 5x + 2

   3x – 5x = 5x – 5x + 2

          -2x = 2

Step 3 I-multiply ang magkabilang panig ng  multiplicative inverse ng –2, iyan ay −1/2.

-2x = 2

    −1/2 (−2𝑥) = −1/2 (2)

   x = -1

Step 4 Ihalili ang nakuhang value ng variable sa orihinal na equation upang matiyak na tama ang nakuhang solution, root, o sagot.

       x = - 1

                    3(x –2) + 6 = 5x + 2

          3(-1 – 2) + 6 = 5(-1) + 2

        3(-3) + 6 = - 5 + 2

            - 9 + 6 = -3

                   -3 = - 3

Samakatuwid, x = - 1 ay ang solution o root ng  3(x –2) + 6 = 5x + 2  dahil natugunan nito ang equation.

Example C: Solve 5(y + 3) - 4 = 5y - 8.

Step 1 Gamitin ang multiplication property over addition o distributive property upang maalis ang mga  parentheses.

       5(y + 3) - 4 = 5y - 8 ==>

 5(y) + 5(3) – 4 = 5y – 8

       5y + 15 – 4 = 5y – 8

Step 2 Gawing simple ang equation sa pamamagitan ng pagsasama-sama ng magkaparehong terms.

5y + 15 – 4 = 5y – 8

      5y + 11 = 5y - 8

Step 3 Ihiwalay ang variable at ilagay ito sa kaliwang panig at ang constant naman ay ilagay sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

  5y + 11 = 5y - 8

  5y – 5y + 11 = 5y – 5y – 8

          11 = 8 ?

Samakatuwid, dahil ang 11 ay hindi maaaring equal sa 8, ang equation na  

 5(y + 3) - 4 = 5y - 8 ay no solution  (walang solution).

Matuto Tayo

Upang malutas ang mga equation na kinasasangkutan ng mga fraction na walang mga variable sa denominator, maaari tayong magsimula sa pamamagitan ng paggamit ng multiplication property of equality upang mawala ang mga fractions sa equation. Maaari nating gamitin ang anumang common multiple tulad ng least common multiple (LCM) ng lahat ng denominator sa equation. (Ang LCM ng 2 o higit pang bilang ay ang pinakamaliit na numero na multiple o factor ng mga bilang na ito). Ang LCM ay tatawagin na natin ngayong least common denominator (LCD).

Upang matutunan ito, unawain natin ang mga sumusunod na halimbawa.

Example 1

Step 1 I-multiply ang magkabilang panig ng LCD, 8, upang mawala ang mga fractions sa equation.

Step 2 Ihiwalay ang variable at ilagay ito sa kaliwang panig at ang constant naman ay ilagay sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

6x + 16 = 3x – 32

6x – 3x + 16 – 16 = 3x  – 3x – 32 – 16

  3x + 0 = 0 - 48

        3x = - 48

Step 3 I-multiply ang magkabilang panig ng  multiplicative inverse ng 3, iyan ay 1/3.

       3x  =  - 48

         1/3(3𝑥  = −48)

             3𝑥/3 = −48/3

          x = - 16

Step 4 Ihalili ang nakuhang value ng variable sa orihinal na equation upang matiyak na tama ang nakuhang solution, root, o sagot.

       

                     

Samakatuwid, x = - 16 ay ang solution o root ng 3/4 𝑥 + 2 = 3/8𝑥 −4  dahil natugunan nito ang equation.

Example 2

        

Step 1 I-multiply ang magkabilang panig ng LCD, 3x, upang mawala ang mga fractions sa equation.

             

Step 2 Ihiwalay ang variable at ilagay ito sa kaliwang panig at ang constant naman ay ilagay sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

2x - 6 = 12

  2x - 6 + 6 = 12 + 6

      2x = 18

Step 3 I-multiply ang magkabilang panig ng  multiplicative inverse ng 2, iyan ay 1/2.

2x = 18

 1/2(2𝑥 = 18)

    2𝑥/2 = 18/2

 x = 9

Step 4 Ihalili ang nakuhang value ng variable sa orihinal na equation upang matiyak na tama ang nakuhang solution, root, o sagot.

                x = 9

          2/3  − 2/𝑥  =  4/𝑥  

           2/3  − 2/9 = 4/9   

 

Step 5 I-multiply ang magkabilang panig ng LCD, 9, upang mawala ang fractions.  

 

                    9(2/3 − 2/9 = 4/9)

                    18/3 −18/9 = 36/9

     6 – 2 = 4

           4 = 4

                     

Samakatuwid, x = 9 ay ang solution o root ng  2/3  −2/𝑥=4/𝑥   dahil natugunan nito ang equation.  

Example 3

          

Step 1 Ihiwalay ang constant sa kaliwang panig  at ilagay ito sa kanang bahagi gamit ang addition property of equality o sa pamamagitan ng transposition.

Step 2 Gamitin ang cross multiplication. I-multiply ang numerator ng kanang panig sa denominator ng kaliwang panig at numerator ng kaliwang panig sa denominator ng kanang panig.

Step 3 Ilipat sa kanang panig ang constant sa kaliwang panig sa pamamagitan ng transposition.

2y + 5 = 9

2y + 5 – 5 = 9 – 5

2y = 4

Step 4 I-multiply ang magkabilang panig ng  multiplicative inverse ng 2, iyan ay 1/2.

        2y  = 4

1/2(2𝑦)  = 1/2(4)

      2𝑦/2 = 4/2  

           y = 2

Step 5 Ihalili ang nakuhang value ng variable sa orihinal na equation upang matiyak na tama ang nakuhang solution, root, o sagot. 

                

               Samakatuwid, y = 2 ay ang solution o root ng   3/(2𝑦+5 )+2/3=1   dahil natugunan nito ang equation.

Summary

• A solution or root of an equation is any number that, when substituted for the variable, makes both sides of the equation equal.

• The set of all solutions is called the solution set.

• We say that two equations are equivalent if they both have the same solution set—any solution of one is also a solution of the other.

• A linear equation is an equation of the first degree in any number of variables.

• Linear equations in a variable, x, can be written in the form ax + b = 0 where a ≠ 0.

• The process of moving a term to the other side of an equation and changing its sign is called transposition.

• A literal equation is an equation in which some of the constants are not specified but are represented by letters such as a, b, c or d.

• Formulas are literal equations.

ANSWERS:

Lesson 1 - Mathematical Expressions: ALS Module - Equations 1

 ALS Module: Equations 1

(Image from https://happynumbers.com)

What Is This Module About?

Ang modyul na ito ay ang una sa dalawang modyul tungkol sa equations. Ipakikilala nito sa iyo ang pangunahing konsepto ng balance o equality na mathematically na kinakatawan ng mga equation. Ituturo nito sa iyo ang lahat tungkol sa mga konsepto ng mga equation at kung paano ito makatutulong sa iyo na malutas ang mga problemang nangyayari sa ating pang-araw-araw na buhay.

Ang modyul na ito ay binubuo ng tatlong aralin o lessons: 

Lesson 1—Mathematical Expressions 

Lesson 2—Linear Equations in One Variable

Lesson 3—Linear Equations in Two Variables

What Will You Learn From This Module?

Pagkatapos mong pag-aralan ang modyul na ito, dapat ay kaya mo nang:

• tukuyin kung ano ang mga linear na equation;

• gawing simple ang mga linear equation;

• tukuyin kung ano ang mga literal na equation;

• lutasin ang mga literal na equation;

• gawing mathematical expression ang mga parirala/pangungusap sa Ingles;

• lutasin ang mga word problems na kinasasangkutan ng mga linear equation;

• graph linear equation; at

• lutasin ang slope at y-intercept gamit ang mga linear equation.

Lesson 1 – Mathematical  Expressions

Bago natin talakayin ang mga equation at kung paano lutasin ang mga ito, suriin muna natin kung paano ipahayag nang maayos ang mga parirala at pangungusap sa matematika.

Suriin muna natin ang ilang mahahalagang termino bago kita hayaang gumawa ng ilang hands-on na pagsasanay.

Ang mga operational symbols ay mga simbolo na ginagamit para sa mga operasyon. Kabilang dito ang mga sumusunod:

1. Plus sign (+) Addition, such as m + n, means the sum of m and n.

2. Minus sign (–) Subtraction, such as p – q, means the difference of 

                                p and q or p is decreased by q

3. Multiplication sign (×) Multiplication,  such as r × s means the product of                                         r and s or  r multiplied by s

4. Division sign (÷) Division, such as d ÷ t means d divided by t or the                                                     quotient of d divided by t

5. Involution                 The operation of raising to a power, such as 

                                x² means the square of x, 

                                y³ means the cube of y, 

                                z⁴ means the fourth power of z

6. Radical sign (√) Evolution, the operation of taking a root, such as

                        √𝑥³ means the square root of x³ 

                        ∛𝑦⁴ means the cube root of y⁴ 

                        ∜𝑧  means the fourth root of z

Terms to Remember

1. Ang constant ay isang numero na ang value ay fixed o hindi nababago. 

Mga halimbawa ng constant:

                0, 3, -4, 17

2. Ang variable ay isang  numerical quantity na ang value ay nababago o naiiba. Kalimitan, gumagamit tayo ng mga titik para kumatawan sa mga variables.

Mga halimbawa ng variable:

            a,b, c,  m, n, x, y, z

3. Ang algebraic expression ay maaaring isang constant, isang  variable o  kumbinasyon ng  constants at variables na kinabibilangan ng operations ng addition, subtraction, multiplication at/o division.

Narito kung paano ginagamit ang mga algebraic expressions:

Algebraic Expression             Meaning

    10 + x             Add 10 and the variable x

   y – 3     Subtract 3 from the variable y

𝒂÷𝟐  or  𝒂/𝟐          Divide the variable a by 2

    9x     Multiply 9 by the variable x

  –3y     Multiply –3 by the variable y

4. Ang mga simbolo na ginamit upang ipakita na ang mga quantities na  nakapaloob sa kanila ay itinuturing bilang isang quantity ay tinatawag na grouping symbols or signs of aggregation.

Mga halimbawa ng grouping symbols

( ) parentheses 

[ ] brackets 

{ } braces 

Mga Halimbawa 1

Translate the following English phrases into algebraic expressions:

    a. The difference of two numbers represented by a and b

a – b

    b. Three more than the product of x and y 

xy + 3

    c. The age of Mary five years hence if she is now m years old

m + 5

    d. The quotient of thrice a number y and two more than z

3y ÷ (z + 2)

    e. What is the difference between 5 less than a number n and 5 less a number n?

(5 – n) – (5 – n)

5. Ang mga simbolo na ginagamit upang ipakita ang ugnayan ng dalawang quantities ay tinatawag na  relation symbols. Kabilang sa mga ito ang nasa talahanayan sa ibaba:

6. Ang  mathematical statement ay nagpapakita ng ugnayan ng dalawang quantities o expressions gamit ang mga relation symbols.

Mga Halimbawa 2

Translate the following English sentences into mathematical statements:

    a. 5 less than twice a number is equal to 3 more than the same number. 

Let x             =     the original number

2x – 5          =     5 less than twice a number

x + 3            =     3 more than the same number

2x – 5 = x + 3      is the mathematical statement 

2x – 5 and x + 3  are both algebraic expressions

    b. The sum of a and b is greater than 5.

Let a + b     =     sum of a and b

a + b > 5            is the mathematical statement

    c. A number increased by 9 equals 28.

Let n             =     the number

         n + 9 = 28     is the mathematical statement

    d. A number decreased by 7 equals 48.

Let n             =     the number

n – 7 = 48     is the mathematical statement

    e. 3000 is 75 times a number.

Let n             =     the number

3000 = 75n     is the mathematical statement

Unawain at Tandaan Natin

• Operational symbols are symbols used for operations in mathematics.

• A constant is a number whose value is fixed or never changes.

• A variable is a numerical quantity whose value changes. It is represented by             a  letter.

• An algebraic expression is a constant, a variable, or a combination of constants and variables involving any of the operations of addition, subtraction, multiplication, and/or division.

• Grouping symbols or signs of aggregation are symbols used to show that the quantities they enclose are treated as one quantity.

• Relation symbols are symbols used to show the relationship between two quantities.

• A mathematical statement shows the relation between two quantities or expressions by using the relation symbols.

Pagsasanay

A. Express the following in mathematical expressions.

1. One-half of the sum of a and b. 

2. The money of Mary after spending x pesos if she had y pesos originally

3. The money of Jane after buying x books at y pesos each if she had z pesos originally. 

4. The money of James after his father gave him 5 times his money if he had m money originally.

5. The age of Anita 5 years ago if her age now is a.

B. Use mathematical symbols to interpret the following and tell whether each is a mathematical statement or an algebraic expression.

1. The cost of n apples at p pesos each.

2. The difference of  two numbers a and b is less than c.

3. The age of Betty 2 years from now if she is already x years old.

4. The product of 9 and a is greater than one-half the quotient of b and 8.

5.  Five more than thrice a number n

C. Change the following word expressions into mathematical sentences and simplify them.

1. Each month Mark’s mother earns twice as much as he does and his father earns P5000 more than he does. Write an algebraic expression for the total monthly earnings of  Mark’s family.

2. Tony is saving money for a vacation during summer break. If he deposits P500 and then saves P120 a week for x weeks, write an algebraic expression for the amount that he will have at the end of x weeks.

3. Bonnie and Dan are buying apples. Bonnie buys 5 kilos of apples for $8. Dan buys 8 kilos of apples that cost $0.40 less per kilo than the apples Bonnie buys. Write an algebraic expression for the total amount Bonnie and Dan spent. 

ANSWERS:

The Correct Usage of Correlative Conjunctions

Correlative conjunctions are pairs of words that correlate two equally important clauses or phrases in one complete thought, that is, they link equivalent elements together (a verb to a verb, a noun to a noun, an adjective to an adjective).

Ang mga correlative conjunctions o mga pangatnig na nag-uugnay ay mga pares ng mga salita na nag-uugnay ng dalawang magkatulad na mahahalagang sugnay o parirala sa isang kumpletong kaisipan, ibig sabihin, pinag-uugnay nila ang mga katumbas na elemento (isang pandiwa sa isang pandiwa, isang pangngalan sa isang pangngalan, isang pang-uri sa pang-uri).

Correlative conjunctions are one of the three main types of conjunctions used in the English language to create smooth flow and reduce sentence fragments, along with coordinating conjunctions and subordinating conjunctions. Correlative conjunctions work in pairs to correlate two parts of a sentence of equal importance. Correlative conjunctions often connect two singular subjects with a singular verb, or two plural subjects with a plural verb. They apply a relation between two subjects or two verbs that act in tandem with each other.

 (https://www.masterclass.com/articles/correlative-conjunctions-explained#what-is-a-correlative-conjunction)

Examples of Correlative Conjunctions

A. Either ... or - used in a sentence in the affirmative sense when referring to a choice between two possibilities. It connects two positive statements of equal weight and things of the same types, phrases, clauses, or words. It indicates that there is a choice between the two choices, and only one can be selected. The verb agrees with the noun that is closer to it.

Ang either/or (alinman/o)ay ginagamit sa isang pangungusap sa affirmative sense kapag tumutukoy sa isang pagpipilian sa pagitan ng dalawang posibilidad. Ito ay nag-uugnay ng dalawang positibong pahayag na may pantay na bigat at mga bagay na magkapareho ang uri, parirala, sugnay, o salita. Ito ay nagpapahiwatig na mayroong pipiliin sa pagitan ng dalawang pagpipilian, at isa lamang ang maaaring mapili. Ang pandiwa ay sumasang-ayon sa pangngalan na mas malapit dito.

Examples/Mga Halimbawa

1. Jenny likes to eat cake and ice cream but her money is just enough for one. What will she order?

        She will order either cake or ice cream.

2. Edward wants to follow one of his parents' vocations. His mother is a doctor and his father is a lawyer. Next year, he will be entering university. What will Edward study?

        Edward will study either medicine or law.

3. Half of the class likes to play basketball. The other half likes to play volleyball.

        The class is to play either basketball or volleyball.

4. The excursionists don’t have a lot of time today, so they can either visit the zoo or watch an exhibition.

B. Neither ... nor - is used in a sentence in the negative sense when you want to say that two or more things are not true. It is used to connect the same kind of word or phrase in a sentence. It connects two negative statements of equal weight about two people or things. The verb agrees with the noun that is closer to it. 

The combination neither/nor indicates that neither of the two choices can be selected. In other words, neither choice is available.

Ang neither / nor (wala/ ni  o hindi / ni) ay ginagamit sa isang pangungusap sa negatibong kahulugan kapag gusto mong sabihin na ang dalawa o higit pang mga bagay ay hindi totoo. Ito ay ginagamit upang ikonekta ang parehong uri ng salita o parirala sa isang pangungusap. Nag-uugnay ito ng dalawang negatibong pahayag na may pantay na timbang tungkol sa dalawang tao o bagay. Ang pandiwa ay sumasang-ayon sa simuno (subject) pangngalan man o panghalip (either noun or pronoun) na mas malapit dito.

Ang kumbinasyong wala/ni o  hindi/ni ay nagpapahiwatig na wala sa dalawang pagpipilian ang maaaring piliin. Sa madaling salita, walang available na pagpipilian.

Examples/Mga Halimbawa

1. Jenny does not want cake. She does not want ice cream, too.

        Jenny wants neither cake nor ice cream.

2. Edward does not want to follow any of his parents' vocations. His mother is a doctor and his father is a lawyer. Next year, he will be entering university. What will Edward study?

        Edward will study neither medicine nor law.

3. Half of the class does not like to play basketball. The other half does not like to play volleyball.

        The class plays neither basketball nor volleyball.

4. The excursionists have no free time today, so they can neither visit the zoo nor watch an exhibition.

C. Both ... and  = implies a correlation between two subjects that are performing the same action. It refers to two things or people together. The combination both/and indicates that the two items are equally presented and included. The grammar is both A and B, that is A and B represent both nouns, verbs, or adjectives. The verbs always take plural forms.

Ang both ... and (Pareho ...at) ay nagpapahiwatig ng ugnayan sa pagitan ng dalawang paksa na nagsasagawa ng parehong aksyon. Ito ay tumutukoy sa dalawang bagay o taong magkasama. Ang kumbinasyong pareho/at ay nagpapahiwatig na ang dalawang aytem ay pantay na ipinakita at kasama. Ang gramatika ay parehong A at B, iyon ay, ang A at B ay kumakatawan sa parehong mga pangngalan, pandiwa, o pang-uri. Ang mga pandiwa ay laging may maramihang anyo.

Examples/Mga Halimbawa

1. Thomas likes chocolate ice cream. He also likes strawberry ice cream.

    Thomas likes both chocolate and strawberry ice cream.

2. The excursionists have a lot of time today. They can visit the zoo. They can also watch the exhibition.

    The excursionists can both visit the zoo and watch the exhibition.

3. Anne plays the piano. Jessica plays the piano, too.

    Both Anne and Jessica play the piano.

4. After shopping, the twins are tired. They are also hungry.

    After shopping, the twins are both tired and hungry.

4. After shopping, the twins are tired. Their mother is also tired.

    After shopping, both the twins and their mother are tired. 

D. Not only...but also -  used to connect and emphasize 2 words or 2 phrases at the same position. Both two phrases are being presented by the writer as surprising or unexpected, with the second one being even more surprising than the first. We use “not only but also” to give more information.

Ang not only ... but also   (hindi lamang...kundi ... rin) ay ginagamit upang ikonekta at bigyang-diin ang 2 salita o 2 parirala sa parehong posisyon. Ang dalawang parehong parirala ay ipinakita ng manunulat bilang nakakagulat o hindi inaasahan, na ang pangalawa ay mas nakakagulat kaysa sa una. Ginagamit natin ang "hindi lamang... kundi ...rin" upang magbigay ng higit pang impormasyon.

It can be used to list adjective qualities, nouns or verbs, to show complementary qualities, quantities or actions, events, and states. We use it when we have two things and we want to give a little extra emphasis to the second thing because it’s even better, or even worse, or more surprising, or more impressive, or more shocking than the first thing.

The combination not only/but also is similar to both/and because it shows that both items presented are included. However, the item after not only is normally something we expect the speaker to say, whereas the item after but also is often something unexpected:

Ang kumbinasyong hindi lamang/kundi ay katulad din sa pareho/at dahil ipinapakita nito na ang parehong mga item na ipinakita ay kasama. Gayunpaman, ang item pagkatapos ng hindi lamang ay karaniwang isang bagay na inaasahan nating sasabihin ng tagapagsalita, samantalang ang item pagkatapos ng kundi ay kadalasang isang bagay na hindi inaasahan.

When using not only . . . but also in a sentence, parallelism should be the goal. It means that the words following both parts of this correlative conjunction (i.e., not only and but also) should belong to the same parts of speech. For example, if a verb follows not only, then a verb should also follow but also. The verb agrees with the noun that is closer to it.

Tandaan na gamitin ang pagkakasunod ng mga bahagi ng pangungusap kung gagamitin ang "not only ... but also" sa unahan o gitna ng pangungusap.

Examples/Mga Halimbawa

1. Beginning/Unahan ng Pangungusap

    a. Not only + verb + subject…but also + subject + verb.

     Not only are Donnalyn's children inquisitive, but also they are clever.

    b. Not only + verb + subject…but + subject + also + verb…

     Not only did Bong Bong cook dinuguan but he also served them.

     Not only did Guillermo Tolentino sculpt the UP Oblation but he also did the Bonifacio Monument.

    c. Not only + verb + subject…but also…

    Not only will they paint the inside of the villa but also the outside.

    d. Not only + subject + but also + subject + verb ...

    Not only George but also Terry has come to the seminar.

    Not only bananas but also avocado is rich in potassium.

2. Middle/Gitna ng Pangungusap

    a. Subject + Verb + not only + Adjective + but also + Adjective
        (Simuno + Pandiwa + not only + Pang-uri + but also + Pang-uri)

         Madelyn is not only beautiful but also intelligent.

    b. Subject + Verb + not only + Adverb + but also + Adverb
         (Simuno + Pandiwa + not only + Pang-abay + but also + Pang-abay)

        Donald dances not only gracefully but also effortlessly.

    c. Subject + Verb + not only +  Noun + but also Noun
         (Simuno + Pandiwa + not only + Pangngalan + but also + Pangngalan)

        Sheila likes not only pancit but also spaghetti.

        Jenny likes to eat cake. She also likes to eat ice cream.
        Jenny likes to eat not only cake but also ice cream.

    d. Subject + not only + Verb + but also + Verb
         (Simuno +  not only + Pandiwa + but also + Pandiwa)

        Gabby not only writes music but also sings it.

Other Correlative Conjuctions

E. Whether/or = connects two possible actions of a subject. 

    Ang whether/or (kung ... o) ay nag-uugnay ng dalawang posibleng gawin ng simuno.

Examples/Mga Halimbawa

1. I was not sure whether Father would visit us or not.

2. I do not care whether Tommy comes to my birthday or not. 

3. Lucy may or may not come with us. We will have to go.

    We will have to go whether Lucy may come with us or not.

4. Danny does not like Faith. He will have to marry him.

    Donny will have to marry Faith whether he likes her or not.

    Whether or not he likes her, Donny will have to marry Faith.

F. Rather/than = presents a subject’s preference for one thing over another. 

    Ang rather/than (sa halip na) ay nagpapahayag ng pagpili ng simuno sa isang aksyon, tao, bagay, atbp. sa halip na ibang aksyon, tao, bagay, atbp.

Examples/ Mga Halimbawa

1. Cely would rather have tea than coffee.

2. Rene would rather call than text his mother.

3. My sister would rather cook than wash the dishes.

G. Such/that - connects two independent clauses in a way that applies a reason for an action. 

    Ang such.than [ganyan (ganito/ganoon)/kaya] ay nag-uugnay sa dalawang independiyenteng sugnay sa paraang naglalapat ng dahilan para sa isang aksyon.

Examples/Mga Halimbawa

1. It was a very hot afternoon. We stopped playing soccer.

    It was such a hot afternoon that we stopped playing soccer.

2. The lecture was very boring. The students felt asleep.

    The lecture was such boring that the students felt asleep.

3. Marie has very fine manners. Her classmates like her.

    Maris has such fine manners that her classmates like her.

H. As/as compares nouns using an adjective or an adverb.

    Ang as/as (kasing-) ay nagkukumpara ng panggalan gamit ang pang-uri o pang-abay.

Examples/Mga Halimbawa

1. The rose is fragrant. The sampaguita is fragrant.

    The rose is as fragrant as the sampaguita.

2. A lion can run 80 km/h. A cheetah can run 130 km/h.

    A lion cannot run as fast as a cheetah.

3. Norman weighs 60 kg. Ronnie weighs 60 kg.

    Norman is as heavy as Ronnie.

4. Sydney recorded 10oC yesterday. Melbourne recorded 8ooC yesterday.

    Sydney was not as cold as Melbourne yesterday.

I. Scarcely (Hardly)/when - used to combine or rewrite sentences denoting two simultaneous past actions.

    Ang scarcely (hardly)/when (Bahagya/nang) ay ginagamit upang pagsamahin o muling isulat ang mga pangungusap na nagsasaad ng dalawang magkasabay na nakaraang aksyon.

Examples/Mga Halimbawa

1. I reached the bus station. At once, the bus left.

    Scarcely/Hardly had I reached the bus station when the bus left.

2. As soon as Martha left the building, it collapsed.

    Scarcely/Hardly had Martha left the building when it collapsed.

3. No sooner did Tommy close the door than it rained heavily.

    Scarcely/Hardly had Tommy closed the door when it rained heavily.

J. Not/but is used when the subject has both the first and the second quality or the first quality is wrong, and the second is right.

Examples/Mga Halimbawa

1. Sheila does not have one mansion. She has two.

    Sheila owns not one but two mansions.

 2. Joemari cried loudly. It was not for sadness. It's for joy.

    Joemari cried loudly not for sadness but for joy

3.  I saw the dead bodies of a mother and her son.

    I saw not two dead bodies, but love and innocence.

5 Tips for Using Correlative Conjunctions 
(Source: https://www.masterclass.com)

There are a variety of helpful tips and rules for properly using these parts of speech in your sentences. Here are a few rules to follow when using correlative conjunctions.

1. Mind your subject-verb agreement. Subjects and verbs need to match when using correlative conjunctions. Singular subjects must match singular verbs, and plural subjects must match plural verbs. If you have multiple subjects, match the verb to the subject that is closest to the verb. For instance, youwould say: “Both the owner and his dogs run through the park,” instead of “Both the owner and his dogs runs through the park.”

Treat a subject that features either/or or neither/nor as singular if the elements after the conjunctions are singular. If one is plural, put it nearest to the verb and use a plural verb.

2. Ensure your pronoun agreement. Similar to subject-verb agreements, pronouns must also agree with their verbs when using correlative conjunctions. For example, “She plays tennis” is the correct agreement between pronoun and verb rather than “She play tennis.” This can be confusing when the pronoun’s antecedent is part of a correlative conjunction pair, however, if there is more than one subject, use the agreement for the closest noun or noun phrase. Let’s use the example sentence: “Not just my sister but my friends were all there as well.” In this case, “were” matches the plurality of “friends” rather than matching it to the singular “sister.”

3. Make sure your sentence has a parallel structure. Parallelism is important in grammatical structure for tracking the subjects of your sentences. Parallel structure deals with the grammatical form of your sentences, such as when discussing multiple items or making a list. For example, let’s look at the two sentences: “My mom not only likes to hike, but also is a fan of camping” and “My mom not only likes hiking but also camping.” In the second phrase, “hiking” and “camping” are parallel, while “to hike” and “camping” are not parallel with each other in the first sentence.

Position your correlative conjunctions in your sentence so the same type of word follows each one. In other words, use a parallel structure.

4. Use a comma with independent clauses. Only use a comma when your correlative conjunction separates two independent clauses, and avoid using it to separate the correlative conjunctions themselves. For example, let’s look at the two sentences: “Neither you, nor I should wear pastels,” and “Neither you nor I should wear pastels.” In this example, the latter is correct because both subjects depend on the same verb, which is “wear.”

Don't use a comma with a correlative conjunction unless the words after it could be a standalone sentence (i.e., contain a subject, a verb and convey a complete idea).

5. Watch out for double negatives. Neither/nor indicates a negative connotation, so be sure your main clause does not also contain a negative verb phrase on top of that. “I can’t neither see it nor hear it,” is not correct because “can’t” already provides a negative. The correct version would be: “I can neither see it nor hear it.”

Don't use a negative verb with neither/nor otherwise you'll create a double negative.

EXERCISES

Complete each sentence using the correct correlative conjunction pair from the parenthesis:

1. Shelly plans to take her annual leave  _________ in September _________ in December. (rather/than, whether / or, either / or, as / if)

2. _________ Mother is feeling happy _________ sad, she tries to keep a positive attitude. (Either / or, Whether / or, When / and, Neither/nor)

3. _________ had I taken my shoes off _________ I found out we had to leave again. (As / as, Rather / than, Scarcely/when, Whether / or)

4. _________ only is coconut water delicious, _________ it can be healthy. (Such/that, Whether / or, Not / but, Just as / so)

5. _________ I have roast beef for dinner, _____________________I cannot have ice cream for dessert. (Either/or, If /then, When / than, Whether / or)

6. _________ flowers _________ trees grow during warm weather. (Not only / or, Both / and, Not / but, Neither/nor)

7. _________ do we enjoy summer vacation, _________ we enjoy winter break. (Whether / or, Not only / but also, Either / or, As/as)

8. Terry is 5 feet tall. Her brother is also 5 feet tall. Terry is ______ tall _____ her brother. (either/or, neither/nor, both/an, as/as)

9. The clouds are very dark. It’s _________ going to snow _________ rain tonight. (neither/nor, as / if, either / or, as / as)

10. Savory dishes are _________ sweet _________ sour. (often / and, neither / nor, both / and, either/or)

Answers: 

Lesson 11 - Trigonometric Functions in Everyday Life: Introduction to Trigonometry in Taglish | ALS Module

LESSON 11 – TRIGONOMETRIC FUNCTIONS IN EVERYDAY LIFE

Sa mga nakaraang aralin , natutunan natin ang tungkol sa special at reference angles. Gayunman, dahil hindi lahat ng mga anggulo ay mga espesyal na anggulo o lahat sila ay may mga reference angles o anggulong sanggunian na 30 , 45 , o 60 °, natutunan naman natin ang paggamit ng isang scientific calculator  upang hanapin ang numeric values ng anim na trigonometric functions – sine, cosine, tangent, cosecant, secant, at cotangent. Sa ngayon, handa na tayong gamitin ang lahat ng ating natutunan upang tugunan ang mga problema  sa pang-araw-araw na buhay na may kinalaman sa trigonometric functions.

PAG-ARALAN AT SURIIN NATIN

Interesado si Kevin na sukatin ang taas ng puno sa larawan sa itaas. Paano niya ito nalutas? Tingnan natin kung paano niya ito ginawa.

Una, sinukat niya ang taas mula sa lupa hanggang sa antas ng kanyang mata. Ito ay 5 ft..

Pagkatapos ay gumamit siya ng isang protractor upang sukatin ang anggulo sa pagitan ng pahalang sa kanyang linya ng paningin at sa tuktok ng puno.  Ito ay may sukat nga 40o.  

Pagkatapos  noon ay sinukat niya ang kanyang distansya mula sa puno. Ito ay 12 ft.

Matapos ito, ginawa niya ang mga sumusunod na pagkalkula:

        tan 40o    =  opposite side / adjacent side  

                =  opposite side  / distance from the tree

                  =  opposite side  /  12 ft.

        opposite side = tan 40o (12 ft.)

             = 0.8391 (12 ft.)

             = 10.0692 or 10.07 ft.

Samakatuwid, ang taas ng puno ay 10.07 ft. + 5 ft. = 15.07 ft.  o 15 feet.

Muli nating subukin na sumagot ng isa pang problema upang matasa ang ating nalalaman.

Si Jack ay nasa tuktok ng isang tore na may taas na 100 talampakan. Tumingin siya paibaba sa  kalsada at nakita niya ang kanyang kotse na naka-parada di kalayuan sa tore. Gaano kalayo ang kanyang kotse sa pinakaibaba ng tore?

Lutasin natin ang problema ng step by step:

STEP 1 Determine the relationship of the given side with the given angle.                             (Tukuyin ang kaugnayan ng ibinigay na panig sa ibinigay na anggulo.)

In the right triangle in the picture, the height of the tower (100 ft.) is the side opposite the 60° angle. (Sa nabuong right triangle sa larawan, ang taas ng tore [100 ft] ay ang gilid katapat (opposite side) ng anggulong 60 degrees.)

STEP 2 Determine the relationship of the unknown side with the given angle.                     (Tukuyin ang kaugnayan ng hinahanap na gilid sa ibinigay na anggulo.)

The unknown side is the side adjacent to the 60° angle.

(Ang hinahanap na gilid ay ang gilid kalapit (adjacent) ng anggulong 60 degrees.)

STEP 3 Determine the appropriate trigonometric function to be used. Here we                     are going to use the cotangent function.

        (Tukuyin ang naaangkop na trigonometric function na gagamitin. Dito                     ay gagamitin natin ang cotangent function.)

        cot 60° = adjacent side / opposite side

STEP 4 Substitute the given values and solve for the unknown. (Ihalili ang                             ibinigay na mga value at hanapin ang nawawalang gilid.)

        cot 60°  =  adjacent side /  100 ft

        adjacent side = cot 60° (100 ft)

(Gamit ang iyong scientific calculator, ating makukuha na ang numeric value ng cot 60o = 0.5774 [rounded-off to 4 decimal places]).

        adjacent side = 0.5774 (100 ft)

        adjacent side = 57.74 ft

Samakatuwid, ang kotse ni Jack ay 57.74 talampakan ang layo mula sa base ng tore.

Paano kung sa halip na cotangent ay tangent ang ating ginamit  na trigonometric function? Tingnan natin ang pagkalkula.

        tan 60° = opposite  side / adjacent side

        tan 60° =  100 ft /  adjacent side

        adjacent side (tan 60°) = 100 ft

        adjacent side = 100 ft / tan 60°

(Gamit ang iyong scientific calculator, ating makukuha na ang numeric value ng tan 60o = 1.73205080757.)

        adjacent side = 100 ft / 1.73205080757

        adjacent side = 57.74 ft (rounded-off to 2 decimal places)

Samakatuwid, kahit tangent ang ginamit sa pagkalkula, ang kotse ni Jack ay 57.74 talampakan pa rin ang layo mula sa base ng tore.

SUBUKIN NATIN ITO

Upang matasa pa ang ating natutunan, subukin nating sumagot ng isa pang problema. 

Suppose a man is standing on top of a 35 ft. building. He looks down to an open manhole and estimates that the angle from the horizontal down to the manhole is 63°. If the man is 5 ft. tall, how far is the manhole from the base of the building?

STEP 1 Determine the relationship of the given side with the given angle.

        In the right triangle represented in the picture above, the height of the                 building, 35 ft., is the side opposite the 63°angle.

STEP 2 Determine the relationship of the unknown side with the given angle.

        The unknown side is the side adjacent to the 63° angle.

STEP 3 Determine the appropriate trigonometric function to be used.

        tan 63° = opposite side / adjacent side

STEP 4 Substitute the given values and solve for the unknown.

        tan 63° = opposite side / adjacent side

        tan 63° = (height of building + height of man) / adjacent side

        tan 63° = (35 ft + 5 ft) / adjacent  side

        tan 63° = 40 ft / adjacent side

(Using a scientific calculator, the value of tan 63° = 1.9626)

        1.9626 =  40 ft / adjacent side

        adjacent  side ( 1.9626)  =  40 ft 

        adjacent side = 40 ft / 1.9626

        adjacent = 20.38 ft (rounded-off to 2 decimal places)

      Thus, the manhole is 20.38 feet from the base of the building.

SUBUKIN NATIN IYO

You can also solve problems involving trigonometric functions by using a shortcut method. (Masasagot din natin ang mga problemang kinapapalooban ng trigonometric functions sa pamamagitan ng shortcut method.)

A scuba diver makes an angle of 50° with the vertical when diving into an ocean. How far must he swim to be 100 meters below the water surface?

A right triangle is formed with the acute angle 50°. The distance from the water surface to the desired position of the diver (100 meters) is the side adjacent to 50°. We are asked for the length of the hypotenuse.

We will use the secant function. Recall that secant is the reciprocal/inverse of cosine. Cosine involves adjacent side and hypotenuse. Remember SohCahToa?

        sec 50° = hypotenuse / adjacent side

        sec 50° = distance the diver must swim / 100 m

        distance the diver must swim = sec 50o (100 m)

(Using a scientific calculator, sec 50° = 1.5557)

                distance the diver must swim = 1.5557 (100 m)

     Thus, the distance the diver must swim is 155.57 m  or 155.6 meters.

SUMMARY

A. We do the following steps in solving problems involving right triangles:

STEP 1 Determine the relationship of the given side with the given angle.

STEP 2 Determine the relationship of the unknown side with the given angle. 

STEP 3 Determine the appropriate trigonometric function to be used.

STEP 4 Substitute the given values and solve for the unknown.

B. Upang madaling matandaan ang kaugnayan ng isang ibinigay na angle sa hinahanap o ibinigay na gilid o side, tandaan ito: SohCahToa, kung saan ==>

S = sine o = opposite side

C = cosine a = adjacent side

T = tangent h = hypotenuse

sine         = opposite side / hypotenuse

cosine     = adjacent side  / hypotenuse

tangent   = opposite side / adjacent side

C. Tandaan ang reciprocal/inverse  o kabaliktaran ng sine, cosine, at tangent.

1. Ang kabaliktaran ng sine ay cosecant. Kung kaya,

    sine         =     opposite side / hypotenuse

    cosecant =     hypotenuse / opposite side

2. Ang kabaliktaran ng cosine ay secant, Kung kaya,

    cosine     =     adjacent side  / hypotenuse

    secant     =     hypotenuse / adjacent side

3. Ang kabaliktaran ng tangent ay cotangent. Kung kaya,  

    tangent     =     opposite side / adjacent side

    cotangent =     adjacent side / opposite side

D. Pareho rin ang makukuhang sagot kung cosecant sa halip na sine, secant sa halip na cosine, o cotangent sa halip na tangent ang ginamit na trigonometric function hangga’t tama ang pagkaka-set up ng equation. Sa parteng ito, kakailanganin ninyo ang natutunan sa Algebra. Dapat ay alam mag-cross multiplication at pagta-transfer ng mga variables from left to right and vice versa.

E. Upang maiwasan ang pagkakamali sa pagpili ng trigonometric function base sa given at unknown side, gamitin ang nakasulat sa table:

Nais ipahiwatig ng nasa itaas na kung ang NUMERATOR ng function ang hinahanap, iyon ang gamiting trigionometric function. Halimbawa, kung ang given side ay isang hypotenuse at ang unknown ay adjacent side, cosine ang gagamitin pero kung ang unknown ay opposite side at ang given ay hypotenuse, sine ang gagamitin.

F. Kung hindi ma-visualize ang isang problema, iguhit muna ito at lagyan ng label upang matiyak na tama ang iyong pagkakaunawa sa problema.

G. Sa ngayon, ang araling ito ay nakapokus lamang sa right triangle.

H. Alamin ang rules sa rounding-off para hindi magkamali sa pinal na sagot  kung ito ang hinihingi ng problema. 

PAGSASANAY

Use trigonometric functions to solve the following problems.

1. A surveyor wants to find the height of a tree. He measures the angle between his line of sight and the top of the tree and finds it to be 47°. The man is 5 feet (ft.) tall and he is 10 meters (m) away from the tree. What is the height of the tree?

2. From a lighthouse 35 m above sea level, the angle from the horizontal to the ship is 25°. How far is the boat from the top of the lighthouse?

3. Mike wants to find the height of a tree. He measures the angle from the horizontal to the top of the tree and finds it to be 47°. He is 1 m tall and 10 meters away from the tree. What is the height of the tree?

4. A scuba diver makes an angle of 46° with the vertical when diving into an ocean. How deep is he in the water if he swims 110 m?

5. A ladder is leaning against a wall at an angle of 54° with the ground. If the top of the ladder is 2 m from the ground, how long is the ladder?

6. A family picture is hung on the wall. Jack noticed that the angle of the picture from the horizontal is 25°. He is 1 m tall. If he is 3 m away from the wall, how high is the picture?

7. Mark is flying a kite, lies down on the ground and realizes that 300 feet of string are out. The angle of the string with the ground is 42.5°. How high is Mark's kite above the ground?

8. A 20 foot ladder rests against a wall. The ladder makes a 55° angle with the ground. How far from the wall is the base of the ladder?

9. The mast  is 9 meters high and a wire is stretched tight to form a straight line to the top of the mast at an  angle of 60°. How long is the wire in meters? (Source: https://brilliant.org)

10. A kite is flying at a height of 65 m attached to a string that is fixed at the base of a tree. If the inclination of the string with the ground is 31°, how far is the kite from the tree?

ANSWERS