ALS Module: Equations 1
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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.
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