Alternative Learning System

Tuesday, November 9, 2021

Request for copy of your ALS COR , Certificate or Diploma

Nawawala ba o naiwala ang iyong COR, Diploma at/o ALS Certificate?


Para po sa nanghihingi ng certified copy, photocopy, or original ng inyong Certificate of Rating (COR), Diploma o ALS Certficate, at anumang katanungan hinggil sa programa ng Alternative Learning System at/o Philippine Educational Placement Test (PEPT), makipag-ugnayan po lamang sa ibaba:

Bureau of Education Assessment (BEA)

Office Address: 2/F, Bonifacio Bldg., DepEd Complex, Meralco Ave., Pasig City

Tel. Nos.: (02) 631-2571 / 631-2588 / 631-2589 / 631-2591 / 687-6717 / 631-6921

Office Hours: Monday to Friday, 8:00am-5:00pm (except weekends, holidays, and work suspensions)


Maaari rin po kayong mag-email sa  Bureau of Education Assessment (BEA), sa  bea.ead@deped.gov.ph at/o bea.od@deped.gov.ph.

Dahil po sa daming kahilingan at katanungan sa BEA, mangyari pong maghintay ng kaukulang panahon. Sundin po ang sinasaad ng iyong email o ang sinagot sa iyong pagtawag sa telepono. 

Saturday, November 6, 2021

Answers to Exercises: Lesson 3 - Linear Equations In Two Variables

Narito ang mga sagot sa Pagsasanay ng Lesson 3 - Linear Equations In Two Variables | ALS Module Functions 1



Mga Sagot sa Pagsasanay

1. Determine whether (2, -1) is a solution of  y - 4 = 5(x – 3).

Ihalili lamang ang binigay na mga values sa equation at tingnan kung ang kaliwang panig ay katumbas o kapareho ang value ng kanang panig.

(2, -1) ==>  (x, y)

  y – 4 = 5(x – 3)
-1 – 4 ≟ 5(2 – 3)
      -5 ≟ 5(-1)
      -5 = -5 

     Samakatuwid, ang (2,-1) ay solution ng  y – 4 = 5(x – 3).


2. What are the coordinates of the point where the graph of the equation 
    2x + 3y = 3 crosses the y-axis?

Dahil nag-cross ang graph ng 2x + 3y = 3 sa y-axis, ibig sabihin, ang value ng x ay zero sa ordered pair na ito.

Kung gayon, kung x = 0, ang y ay:

2x + 3y     = 3
 2(0) + 3y = 3
            3y = 3
         3y/3 = 3/3
              y = 1

Samakatuwid, ang coordinates ng point kung saan tumawid ang equation 
2x + 3y = 3 ay (0, 1).


3. What are the coordinates of the point where the graph of the equation 
        y = -3x – 4 crosses the x-axis?

Dahil tumawid ang graph ng equation y = -3x – 4 sa x-axis, ibig sabihin ang value ng y sa coordinate na ito ay zero (0). Kung gayon, kung y = 0, ang value ng 
x ay:

    y = -3x – 4
    0 = -3x – 4
  3x = -4
       3x/3 = -4/3
    x = -4/3 or -1 1/3
Samakatuwid, ang coordinates ng point kung saan tumawid sa x-axis ang equation y = -3x – 4  ay (-4/3, 0) o (-1 1/3, 0).


4. Find the intercepts of the equation 3y = 5x, then draw its graph.

Upang mahanap ang x-intercept, ihalili ang zero (0) sa value ng y sa equation upang mahanap ang x.

3y     = 5x
3(0)  = 5x
    5x = 0
 5x/5 = 0/5
      x = 0

Upang mahanap ang y-intercept, ihalili ang zero (0) sa value ng x sa equation upang mahanap ang y.

3y = 5x
 3y= 5(0)
 3y = 0
      3y/3 = 0
   y = 0

Dahil parehong nasa origin ang ating x-intercept at y-intercept, hahanap pa tayo ng points upang ma-graph natin ang equation.

Maglalagay tayo ng arbirtrary o pansamantalang value ng x upang mahanap ang y.

Kung x = 3, ang y ay:

3y = 5x
3y = 5(3)
3y = 15
     3y/3 = 15/3
  y = 5

Ang ating dalawang points ay (0, 0) at (3, 5).

Maaari na nating i-drowing ang ating graph.


5. Write in slope-intercept form if the slope is ¾ and the y-intercept is – ½.

The slope-intercept form is y = mx + b, where m is the slope and b is the 
    y-intercept.

Dahil ang m = ¾ at ang y-intercept = -1/2, samakatuwid ang ating equation na nasa slope-intercept form ay:

        y = ¾ x½ 

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Friday, November 5, 2021

Lesson 3 - Linear Equations in Two Variables | ALS Module - Equations 1

Lesson 3 – Linear Equations In Two Variables

        Sa nakaraang aralin ay nalaman natin ang mga konsepto at terminong karaniwang ginagamit sa Algebra. Natuto tayong mag-solve ng linear equation na may iisang variable o unknown.

Sa araling ito ay pagtutuunan naman natin ang pagresolba ng linear equations na may dalawang variables o unknowns.

MATUTO TAYO

        Ang isang equation tulad ng 4x + 5 = 21 ay may isang value, 4, bilang solusyon nito. Ang isang equation na may dalawang variable tulad ng y = 2x + 1 ay may maraming solusyon na maaari nating isulat bilang ordered pairs ng mga numero.

         Para sa isang equation tulad ng y = 2x + 1, isinusulat natin ang ordered pairs sa ganitong anyo,   (x, y).

Halimbawa 1

Determine whether (3, 7) is a solution of y = 3x - 2. (Tukuyin kung ang (3,7) ay isang solusyon ng y = 3x – 2)

Para gawin ito, ihalili lamang ang ordered pair (x, y) ==> (3,7) sa equation, tulad nito:
y = 3x – 2
7 ≟ 3(3) – 2
7 ≟ 9 – 2
7 = 7 

Samakatuwid, ang equation ay tama. Ang ordered pair na (3,7) ay isang solution ng y = 3x -2.

        We can find solutions to equations in two variables by choosing a value for one variable, substituting it, and computing the equation to find the value of the other variable. (Maaari tayong makahanap ng mga solusyon sa mga equation na may dalawang variables/unknowns sa pamamagitan ng pagpili o paglapat ng isang value sa isang variable, ihalili ito at i-compute ang equation upang mahanap ang value ng ikalawang variable/unknown.)


Halimbawa 2

Find 3 solutions for the equation 3x + y = 9.

        Mas madaling masasagot ang problema kung kukunin muna natin ang value ng y. Gamit ang transposition, ang ating equation na 3x + y = 9 ay magiging ==>

y = -3x + 9

        Ngayon, maaari na tayong maghalili ng value ng x upang makuha ang value ng y.

Kung ang value ng x ay zero (0), ang value ng y ay:

          y = -3(0) + 9
          y = -3 + 9 
          y = 6

Kung gayon, ang ordered pair na (0,6) ay isang solution ng equation.

Kapag pinili natin ang 2 bilang value ng x, ang value ng y ay magiging:

y = -3(2) + 9 
                y = -6 + 9 
                y = 3

Ang ikalawang solution ng ating equation ay ang ordered pair na (2,3).

Ngayon, ihalili naman natin ang –1 bilang x.

     y = -3(-1) + 9 
             y = 3 + 9 
             y = 12

Ang ating ikatlong solution ay (-1, 12).

        P’wede nating itala sa isang table/talahanayan ang mga ordered pairs upang madaling makita ang ilang solutions sa ating equation na 3x + y = 9. 


        Alam na natin ngayon na ang solution ng isang  linear equation na may 2 variables ay isang ordered pair ng mga numero. Ang mga ordered pairs na ito ay maaaring i-graph sa isang coordinate plane. Ang graph ng isang equation ay isang  drowing na kumakatawan sa mga solutions nito.  


Halimbawa 3

Graph the equation 4x = 8 – 2y.

        Bago nating mai-graph ang equation, hanapin muna natin ang mga solutions o ordered pairs nito.

Step 1 Gawing simple ang equation (ayon sa natutunan natin sa lesson 2).

4x = 8 – 2y 
2y = - 4x + 8  
2y/2 = -4x/2 + 8/2
y = -2x + 4

Step 2 Maghalili ng value ng x para makuha ang value ng y.

y = -2x + 4

Kung ang x = 0, y = -2(0) + 4 =  0 + 4 = 4
        x = -1 y = -2(-1) + 4 =  2 + 4 = 6
        x = -2 y = -2(-2) + 4 = 4 + 4 = 8
        x = 2 y = -2(2) + 4 = -4 + 4 = 0
        x = 4 y = -2(4) + 4 = -8 + 4 = -4

Step 3 Ilagay sa table o talahanayan ang mga values.


Step 4 I-plot ang mga points na (0, 4), (-1, 6), (-2, 8), (2, 0), at (4, -4) sa                                 coordinate plane. 


        Mapapansin sa graph na ang lahat ng points o ordered pairs ay nakalapat sa isang straight line. Kung magpa-plot pa tayo ng iba pang solutions, ang mga ito ay lalapat din sa nasabing tuwid na linya. Ang linya ay ang graph ng ating equation na 4x = 8 – 2y. Ang lahat ng points sa linya ay solusyon sa equation na ito.

Matuto Tayo

        Sa pag-graph ng mga linear na equation, maaaring tumawid ang isang tuwid na linya sa x-axis sa ilang points at maaari ring tumawid sa y-axis sa iba pang mga points. Ang mga points na ito ay tinatawag na mga intercept ng isang graph.

Ang x-intercept ay ang x-coordinate (value ng x) ng point kung saan tumatawid ang isang graph sa x-axis. Ang y- intercept, sa kabilang banda, ay ang y-coordinate (value ng y) ng point kung saan ang isang graph ay tumatawid sa y-axis.

        Ang bawat point sa x-axis ay may y-coordinate na zero at ang bawat point sa 
y-axis ay may x-coordinate na zero. Kapag pinagsama ang mga obserbasyon na ito kasama ang ideya ng mga intercept, maaari nating ibuod ang mga sumusunod:

        1.  Ang x-intercept ng isang graph ay ang x-coordinate ng punto na may y-coordinate na zero (0).

2.  Ang y-intercept ng isang graph ay ang y-coordinate ng punto na may x-coordinate na zero (0).

        Sa madaling salita, upang mahanap ang x-intercept ng graph ng isang equation, dapat nating hayaan ang y = 0 sa equation. 

Upang mahanap ang y-intercept, dapat nating hayaan ang x = 0 sa equation. 

Kapag naghahanap ng mga intercept, hindi kinakailangang ihiwalay ang alinmang variable.


Halimbawa 4

Find the x-intercept and the y-intercept of the line for the equation 2x + y = 1.

Kapag ang x = 0, ang y ay:

2(0) + y = 1 
                    0 + y = 1 
                          y = 1

Samakatuwid, ang y-intercept ay 1.

Kapag ang y = 0, ang x ay:

2x + 0 = 1 
                      2x = 1 
                        x = ½

Samakatuwid ang x-intercept ay ½.

        Ang graph ng mga puntos na (0, 1) at (1/2, 0) ay makikita sa ibaba:      



Halimbawa 5

Graph the equation  x – y = 0.

Step 1 Hanapin ang x at y-intercept ng equation.

Kapag ang x = 0, ang y ay:

0 – y = 0 
                              y = 0

Samakatuwid, ang y-intercept ay 0.

Kapag ang y = 0, ang x ay:

x – 0 = 0 
                              x = 0

Samakatuwid, ang x-intercept ay 0 rin.

        Anong nangyari? Mukhang magkakaproblema tayo dito dahil ang x-intercept at y-intercept ay parehong zero (0) o nasa origin. Kapag nangyari ito, kailangan nating maghanap ng isa pang point sa graph sa pamamagitan ng pagpili ng arbitrary o pansamanalang  x-value at hanapin ang katumbas na y-value nito.

Ang una nating solution o point ay (0,0). Humanap pa tayo ng isa pang point.

Sa ating equation na x – y = 0, kung x = 2, ang y ay magiging: 

                    2 – y = 0 
                         -y = -2 
                          y = 2.

        Ang ating ikalawang puntos ay (2,2).

        Ang graph ng mga puntos na (0, 0) at (2, 2) ay makikita sa ibaba:      



Summary

A solution of an equation with two variables is an ordered pair of numbers that can be plotted on a coordinate plane.

A graph of an equation is a drawing that represents its solutions.

The intercepts of a graph are the points where a straight line crosses the x- and y-axes.

The x-intercept is the x-coordinate of the point at which a graph crosses the x-axis.

The y-intercept is the y-coordinate of the point at which a graph crosses the y-axis.

In the formula, y = mx + b, m is the slope of the line while b is the coefficient of y.

To graph a linear equation of the form ax + by = c using the intercept method, we follow these steps:

        1. Find the x-intercept by letting y = 0 in the original equation.

        2. Find the y-intercept by letting x = 0 in the original equation.

        3. Write the ordered pair corresponding to each intercept.

        4. Plot the intercept points and connect them with a straight line.

The equations for most lines may be given by the slope-intercept form 
                y = mx + b.

        Note that m is the slope of the line. The arbitrary constant, b, is the y-intercept.

        Without drawing a graph of the equation, determine the slope and y-intercept of the equation y = 2x + 5.

The coefficient of x is 2 and the constant term is 5. Thus m = 2 and b = 5.


Pagsasanay

1.    Determine whether (2, -1) is a solution of y - 4 = 5(x – 3).

2.   What are the coordinates of the point where the graph of the equation 
        2x + 3y = 3 crosses the y-axis?

3.   What are the coordinates of the point where the graph of the equation 
        y = -3x – 4 crosses the x-axis?

4.   Find the intercepts of the equation 3y = 5x, then draw its graph.

5.   Write in slope-intercept form if the slope is ¾ and the y-intercept is – ½.




Reference: ALS Module - Equations 1

Wednesday, November 3, 2021

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

 Lesson 2 – Linear Equations in One Variable



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 constantsarbitrary 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:

Tuesday, November 2, 2021

Lesson 1 - Mathematical Expressions: ALS Module - Equations 1

 ALS Module: Equations 1


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 1Mathematical Expressions 

Lesson 2Linear Equations in One Variable

Lesson 3Linear 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:

Saturday, October 30, 2021

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

AEither ... 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 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 + subjectbut also subject + verb.

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

    b. Not only + verb + subjectbut + 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 + subjectbut 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.

    cSubject 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)


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