Quick Math Methods ONE VARIABLE LINEAR EQUATION

Quick Math Methods ONE VARIABLE LINEAR EQUATION  

1. Understanding Similarities and Association of Settlement
Linear Equations in One Variable Note the open sentence x + 1 = 5.
Open sentence is linked by an equal sign (=). Next, open sentences connected by an equals sign (=) is called equality.
Equations with a single variable or a rank one linear equation of degree one is called a variable.
 

If the x in the equation x + 1 = 5 is replaced with x = 4 then the equation is true. As for if x is replaced numbers other than 4 then the equation x + 1 = 5 is false. In this case, the value of x = 4 is called the completion of the linear equation x + 1 = 5.
 

Furthermore, the completion of the set of the equation x + 1 = 5 is {4}.
Substitute the variable x resulting equation is true is called the completion of linear equations. Completion of the set of all sets of linear equations called linear equations settlement. Try to discuss with your friends instead of the settlement called linear equations.
One variable linear equation is an open sentence is linked by an equal sign (=) and only has one rank of one variable. General form of the equation variables linear one is ax + b = 0 with a X 0.

Quick Math Methods Example:
Of the following sentences, determine which is the linear equation of one variable.
a. 2x - 3 = 5
b. x ² - x = 2
c.1/3x = 5
d. 2x + 3y = 6
completion:
a. 2x - 3 = 5
Variables at 2x - 3 = 5 is x and rank 1, so the equation 2x - 3 = 5 is a linear equation of one variable.

b.x ² - x = 2
Variables in the equation x² - x = 2 is x rank 1 and 2. Since there are x rank 2 then the equation x ² - x = 2 is not a linear equation of one variable.

c. 1/3x = 5
Because the variables in the equation 1/3x = 5 is x and rank 1, then 1/3x = 5
a linear equation of one variable.

d. 2x + 3y = 6
Variables in the equation 2x + 3y = 6, there are two, namely x and y, so 2x + 3y = 6 is not a linear equation of one variable.

2. Association of Settlement One Variable Linear Equations by substitution
Completion of one variable linear equations can be obtained by substitution, is replacing variables with numbers corresponding to the equation into a sentence that is true.

Quick Math Methods example:
Determine the completion of the set of equations
x + 4 = 7, if x is a variable in the set of natural numbers.
If x is replaced natural numbers, is obtained
substitution x = 0, then 0 + 4 = 7 (one sentence)
substitution x = 1, then 1 + 4 = 7 (one sentence)
substitution x = 2, then 2 + 4 = 7 (one sentence)
substitution x = 3, then 3 + 4 = 7 (correct sentences)
substitution x = 4, then 4 + 4 = 8 (one sentence)
It turned out to x = 3, the equation x + 4 = 7 a correct sentence.
Thus, the set of the settlement equation x + 4 = 7 is {3}.

This series of Quick Math Methods ONE VARIABLE LINEAR EQUATION 

 

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Smart Math OPEN SENTENCE IN MATH Easy Methods

SMART MATH  >> Open Sentence in Math
1. Statement Smart Math
In everyday life we often see a variety of the following sentences.
a. Jakarta is the capital city of Indonesia.
b. Mount "Merapi" in Central Java.
c. 8> -5.
The third sentence above is a sentence that is true, because every person recognizes the truth of the sentence.
 

Then consider the following sentences. 

a. "Monas" is located in Jogjakarta.
b. 2 + 5 <-2
c. Sunset in the east.

The third sentence is a sentence that is false, because everyone does not agree with the sentence.
Sentences that can be determined truth value (true or false) statement is called.

Now consider the following sentences.
a. Apple fruit taste sweet.
b. Eat nutritious foods.
c. Learn diligently so you go up a class.
Can you determine the truth value of the sentence - the sentence above? Do you think the sentences are not statements? Why?

2. Open Sentences & Sentence Completion Association open Smart Math.

Can a sentence to answer the question "Indonesia is located on the continent of x". If x is replaced Asia then the sentence is true. As for if x is replaced Europe then the sentence is false. Sentences such as "Indonesia is located on the continent of x" is called open sentence.
a. 3 - x = 6, x member of the set of integers.
b. 12 - y = 7, y member of the set of natural numbers.
c. z * 5 = 15, z member of the set of natural numbers.

Sentence 3 - x = 6, x integer members would be true if x is replaced -3 and will be worth one if x is replaced numbers other than -3. Furthermore, x is called a variable, while the 3 and 6 are called constants.

Try to determine the variables and constants of sentence 12 - y = 7 and z * 5 = 15 in the above example.
Open sentence is a sentence that contains a variable and unknown truth value.

A variable is a symbol (symbol) on the open sentence that can be replaced by any member of the set that has been specified.
The constant is a fixed value (s) contained in an open sentence.
Now consider the sentence x2 = 9. If the variable x is replaced by -3 or 3 sentences then x2 = 9 will be true. In this case x = -3 or x = 3 is the completion of an open sentence x2 = 9. Thus, the completion of the sentence set x2 = 9 is {-3, 3}.
Completion of an open set is the set of all sentences replacement of variables in open sentences so that the sentence is true. That is SMART MATH Open Sentence in Math.

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Smart Math Quick Summary Algebra

Smart Math Quick Summary Algebra
 

1. Variables, constants, factors, as well as similar and not similar rates.
- A variable is a symbol that substitutes a number
its value is not known clearly.
- Constants are parts of an algebraic form
such as number and no load variable.
- The tribes are the kind that has a variable rate and
the rank of each variable are the same.
- Interest is not a type that has a variable rate and
the rank of each variable are not the same.
 

2. On the algebra, addition and subtraction operations
can only be done on similar tribes.
 

4. Multiplication between the two forms is expressed algebraically as follows.
(ax + b) (cx + d) = acx² + (ad + bc) x + bd
(ax + b) (cx² + dx + e) = acx³ + (ad + bc) x² + (ae + bd) x + be
(x + a) (x - a) = x² - a²

 

3. Multiplying a constant number k with the algebra
parts one and two tribes expressed as follows.
k (ax) = kax
k (ax + b) = kax + kb 

the two forms is expressed algebraically as follows.
(ax + b) (cx + d) = acx² + (ad + bc) x + bd
(ax + b) (cx² + dx + e) = acx³ + (ad + bc) x² + (ae + bd) x + be
(x + a) (x - a) = x² - a²
 

5. On the powers of the algebra of the two tribes, tribal tribes coefficient
determined by Pascal's triangle.
(a + b) ¹ = a + b
(a + b) ² = a² + 2ab + b²
(a + b) ³ = a³ + 3a²b + 3ab² + b³
and so on
 

6. Value of an algebraic form can be determined by
substituting any number on the variables
The algebraic form.
 

7. A fraction simplest form of algebraic say if
numerator and denominator have no factors fellowship
except 1 and the denominator is not equal to zero.
 

8. Results of operations of addition and subtraction on fractions
algebra obtained by equating the denominator,
then summing or subtracting the numerator.

 

That is there are 8 item Smart Math Quick Summary Algebra.

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Smart Math Methods "Algebra Arithmetic Operations" Easy way

Smart Math Methods algebra arithmetic operations 

1. Addition and reduction Algebra Shape On the algebra, addition and subtraction operations can only be performed on similar tribes. Totalizing or subtract the coefficients on similar tribes.Determine the sum and subtraction algebraic form following.a. -4ax + 7axb. (2x2 - 3x + 2) + (4x2 - 5x + 1)c. (3a2 + 5) - (4a2 - 3a + 2)Smart Math Solution:a. 7ax-4ax + = (-4 + 7) ax = 3ax 
b. (2x2 - 3x + 2) + (4x2 - 5x + 1) = 2x2 - 3x + 2 + 4x2 - 5x + 1= 2x2 + 4x2 - 3x - 5x + 2 + 1 = (2 + 4) x2 + (-3 - 5) x + (2 + 1)= 6x2 - 8x + 3c. (3A2 + 5) - (4A2 - 3a + 2) + 5 = 3A2 - 4A2 + 3a - 2= 3A2 - 4A2 + 3a + 5-2 = (3-4) a2 + 3a + (5-2) =-a2 + 3a + 32. 

Multiplication You need to recall that the integer multiplication effect / distributive properties of multiplication of the sum, ieax (b + c) = (axb) + (axc and distributive properties of multiplication the reduction, the ax (b - c) = (axb) - (axc),for any integers a, b, and c. This also applies to the nature of the algebra multiplication.a. Multiplying constants with the algebra of multiplication of a constant number k with the tribes and tribal algebraic form two stated as follows.k (ax) = kax k (ax + b) = kax + kb Hypotenuse angled triangle are (2x + 1) cm, while the length of the side of the elbow (3x - 2) cm and (4x - 5) cm. Find the area of ​​the triangle. Describe the following algebraic form, and then simplify it.a. 4 (p + q)b. 5 (ax + by)c. 3 (x - 2) + 6 (7x + 1)d. -8 (2x - y + 3z)

 Smart Math Solution:a. 4 (p + q) = 4p + 4Qb. 5 (ax + by) = 5ax + 5byc. 3 (x - 2) + 6 (7x + 1) = 3x - 42x + 6 + 6 = (3 + 42) x - 6 + 6 = 45xd. -8 (2x - y + 3z) =-16x + 8y - 24z

 b. Multiplication between two forms of algebraic as a constant multiplication by algebraic form, to determine the product of the two forms of algebra we can use the distributive properties of multiplication to addition and multiplication distributive nature of the reduction.Besides this way, to determine the product of the two forms of algebra, can use the following method.Note the multiplication of the algebra of two parts with the following two parts.(ax + b) (cx + d) = ax X cx + d + cx + b X b X d = acx2 + (ad + bc) x + bd

The above problem is to describe some of the workings of arithmetic operations in the form of algebraic subtraction, addition and multiplication, the ways solving already elaborated on smart math methods.

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Smart Math Quick Method "FORM AND ALGEBRAIC ELEMENTS"

SMART MATH Quick Method Form and Algebraic Elements

Consider the following illustration.
Many of her dolls more than 5 Rachel Angel doll. If a lot
Rachel stated stuffed with so many dolls Angel x expressed by x + 5. 

If Rachel doll 4 pieces then Angel doll as much as 9 pieces.
Forms such as (x + 5) is called the algebra.
The algebra is a mathematical form in presentation contains letters to represent numbers not yet known.

Algebraic form can be used to resolve Smart Math Quick Method
problems in everyday life. Things that are not known as much fuel it takes a
buses in each week, the distance covered in a certain time, or the amount of forage required within 3 days, can be found using algebra.
Examples of other forms like algebra :

2x,-3p, 4y + 5, 2x2 - 3x +7, (x + 1) (x - 5), and-5x (x - 1) (2x + 3). 

The letters x, p, and y on the algebra is called a variable. Furthermore, there is a form of algebra elements algebra, including variables, constants, factors, similar tribes, and tribal not similar.

In order for you to be more clear about the elements of the form algebra, study the following descriptions.
 

1. Variables, Constants, and Factor
Note the algebraic form 5x + 3y + 8x - 6y + 9.
On the algebraic form, the letters x and y are called variables.
A variable is a symbol that has not a substitute for a number known values ​​clearly.

Variable called variables. Variables are usually denoted with small letters a, b, c, ..., z.
As for the number 9 on the algebra of the above referred constants.
Is the rate constant of a form of algebra in the form numbers and no load variable.

If a number can be converted into a = p X q with a, p, q integers, then p and q are called the factors of a.
On the algebra above, 5x can be described as
5x = 5 X X X 1 or 5x = 5x.
So, the factors of 5x is 1.5, x, and 5x.

As for the meaning of the coefficients are constant factor a tribe in the form of algebra.
Note the coefficient of each term in the algebra
5x + 3y + 8x - 6y + 9. 5x the rate coefficient is 5, the tribe
3y is 3, at 8x rate is 8, and the tribe is-6y -6.

2. It kind of tribe and tribe Similar
a) The rate is variable or constant coefficients along the form of algebraic operations that are separated by a number or a difference.
The tribes are the kind that has a variable rate and the rank of each variable are the same.
Example: 5x and-2x, 3A2 and a2, y and 4y, ...

Tribe is not a type that has a variable rate and the rank of each variable are not the same.
Example: 2x and-3x2,-y and-x3, and 5x-2y, ...

b) the Tribe is a form of algebra that are not connected by sum or difference operations.
Example: 3x, 2a2,-4xy, ...

c) is the algebra of the two parts are connected by a single sum or difference operations.
Example: 2x + 3, a2 - 4, 3x2 - 4x, ...

d) is the algebra of three parts which are connected by two sum or difference operations.
Example: 2x2 - x + 1, 3x + y - xy, ...
Algebraic forms that have more than two parts called many tribes. so we can do with smart math Quick Method

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Smart Math exponential number and Operations powers of integer arithmetic

Smart Math in exponential number.
quadratic terms of a number. Squared or the square of a number is multiplying a number by the number itself.Furthermore, the powers of a number means multiplying repeated with the same number. 
Note the following powers of prime number 2. 

2 ² = 2 x 2 (2 ² read 2 squares or 2 to the power 2)= 42 ³ = 2 x 2 x 2 (2 ³ read 2 to the power 3)= 8 

Smart Math Solution:
a. 9 ² = 9 x 9 = 81b. (-6) ³ = (-6) x (-6) x (-6) = 36 x (-6) = -216c. -5 = - (5 x 5x 5 x 5) = -625d. (-10) 4 = (-10) x (-10) x (-10) x (-10) = 10,000 Properties exponential number.

a. Multiplicative nature of the exponential number Note the following rank of integer multiplication.3 ² x 3 ³ = [3x3] x [3x3 x3]Means [3x3] = 2 factors and [3x3x3] = 3 factors So there are 5 factors add up, so be 3x3x3x3x3 

b.Nature of the distribution of exponential number Note the following integer division rank.
5 ³: 5 ² = [5x5x5]: [5x 5] the meaning [5x5x5] = 3 factors and [5x 5] = 2 factors So there is 1 factor is reduced, so that a 5
Smart Math Operation Count Mixed Integer 
In completing integer arithmetic operations, there are two things to note, that1. sign arithmetic operations;2. brackets.If in a mixed integer arithmetic operations are bracketed, workmanship are in brackets should be done first. 

If in an integer arithmetic operation there is no parentheses, the process is based on the properties of the following arithmetic operations.1. Operations of addition (+) and subtraction (-) as strong, meaning that operations are located on the left side done first.2. Multiplication (x) and division (:) as strong, meaning that operations are located on the left side done first.3.  

Multiplication (x) and division (:) is stronger than the addition operation (+) and subtraction (-), multiplication means (x) and division (:) done first rather than the addition operation (+) and subtraction (-).
Example: 
a. 24 + 56 x 42-384 : 12= 24 + (56 x 42) - (384: 12)= 24 + 2352-32=       2376-32= 2,344 

b. 28 x (364 + 2,875): (9756-9742)= 28 x 3,239: 14= 90 692: 14= 6,478
c. 80: ((11-7) x (-4))= 80: (4 x (-4))= 80: (-16)= -5

d. (-8 + 5) x (36: (6-9))= -3 X (36 (-3))= -3 X (-12)= 36 

Thus example Smart Math exponential number and Operations of integer arithmetic, may be useful for the reader.

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SMART MATH Calculate The Numbers Round Operation

SMART MATH Calculate The Numbers Round Operation 
a. Summation with tools.

In calculating the sum of two integers, can used by using the number line. Numbers are summed illustrated with arrows corresponding to the direction with the number.If the positive numbers, arrows pointing the right direction. Conversely, if the negative number, the arrows pointing in the direction of left. 
1. 6 + (-8) 
2. (-3) + (-4) 

b. Summation without tools 
The sum of the numbers of little value can be done with the help of a number line. However, for the numbers of great value, it can not be done. Therefore,we should be able to add up the integers without tools.1) Both numbers marked with the same.

If both marked with the same numbers (both numbers positive or both negative numbers), totalizing both numbers. The result is the same mark with the mark the two numbers. 

SMART MATH Calculate The Numbers Round Operation

Example: 

a) 125 + 234 = 359

b) -58 + (-72) = - (58 + 72) = -1302) 

The two numbers opposite sign.

If the two numbers opposite sign (positive numbers and negative numbers), reduce the number of greater value with a smaller number of value without notice sign. The result, according tick numbers are worth greater.
 

SMART MATH Calculate The Numbers Round Operation

Example: 

a) 75 + (-90) = - (90-75) = -15
b) (-63) + 125 = 125-63 = 62
a. 6 + 5 = 5 + 6 = 11
b. (-7) + 4 = 4 + (-7) = -3
c. 8 + (-12) = (-12) + 8 = -4
d. (-9) + (-11) = (-11) + (-9) = -20

c.Has the identity element.

Number 0 (zero) is the identity element in summation. That is, for any integer when added to 0 (zero), the result is the number itself. It can be written as follows. For any integers a, always applya + 0 = 0 + a = a.

 d.Associative properties.

Associative properties of nature also called grouping. This trait can be written as follows.For any integers a, b, and c, apply(A + b) + c = a + (b + c).  

Properties Integer Additiona.Closed nature,On the sum of integers, always produces integers as well. It can be written as follows. For any integers a and b, apply a + b = cwith c is also an integer. 

a. -16 + 25 = 9-16 And 25 are integers.9 is also an integer. 

b. 24 + (-8) = 1624 and -8 are integers.16 is also an integer. 

b.Commutative propertiesCommutative properties called exchange properties. Addition two integers always obtained the same results although both numbers are exchanged place. It can be written as follows.For any integers a and b, always apply 

a + b = b + a.
 a. 6 + 5 = 5 + 6 = 11 
b. (-7) + 4 = 4 + (-7) = -3 
c. 8 + (-12) = (-12) + 8 = -4 
d. (-9) + (-11) = (-11) + (-9) = -20c. 

Has the identity element Number 0 (zero) is the identity element in summation. That is, for any integer when added to 0 (zero), the result is the number itself.It can be written as follows.For any integers a, always apply a + 0 = 0 + a = a.d.Associative properties Associative properties of nature also called grouping. This trait can be written as follows.For any integers a, b, and c, apply(A + b) + c = a + (b + c).

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Smart Math in Multiplication Algebra

Smart Math in Multiplication  Algebra ShapeNote back in shape algebra distributive properties. Distributive propertiesis the basic concept of multiplication in the algebra. For more details,learned the following description.The multiplication rate with Two PartsIn order for you to understand the multiplication rate of the two-rate form of algebra,learn about the following example. 

Use the distributive law to complete the following multiplication.a. 2 (x + 3)
b. -5 (9 - y) 

c. 3x (y + 5)

d. -9p (5p - 2q)Answer:a. 2 (x + 3) = 2x + 6
b. -5 (9 - y) + 5y = -45
c. 3x (y + 5) = 3xy + 15x

d. -9p (5p - 2q) =-45p2 + 18pq

Smart Math in factoring with Distributive Properties In elementary school, you would have learned how factoring a numbers. Remember you regarding the matter? Basically,factoring a number means a number expressed inform of multiplication factors. In this section, will be studied in ways factoring an algebraic form by using the distributive properties.With these properties, the form ax + ay algebra can be factored into a (x + y),where a is a factor of ax and ay fellowship. Therefore, study 

Example Problem 

Factor the following algebraic forms.a. 5ab 10b + c. +-15p2q2 10pqb. 2x - 8x2yAnswer:a. 10b + 5abFor factoring 5ab + 10b, specify fellowship factor of 5 and10, then from ab and b. Fellowship factor of 5 and 10 is 5.Factors fellowship of ab and b is b.So, 5ab factored into 5b 10b + (a + 2).b. 2x - 8x2y  

Fellowship factor of 2 and -8 is 2.Factor of x and x2y fellowship is x.Thus, 2x - 8x2y = 2x (1 - 4xy).c. +-15p2q2 10pqFactors fellowship of -15 and 10 is 5.Factors fellowship of p2q2 and pq is pq.So-15p2q2 10pq + = 5pq (-3pq + 2). 

Smart Math in Difference Two Squares Notice the multiplication (a + b) (a - b). This form can be written(a + b) (a - b) = a² - ab + ab - b ²= a² - b ²So, the form a2 - b2 can be expressed in terms of multiplication (a + b) (a - b).a² - b² = (a + b) (a - b)Forms a² - b ² is called the difference of two squares.Factor the following forms.a. p² - 4c. 16 m² - 9N ²b. 25x² - y ²d. 20p² - 5q²Answer:a. p2 - 4 = (p + 2) (p - 2)b. 25x² - y² = (5x + y) (5x - y)c. 16m2 - 9n2 = (4m + 3n) (4m - 3n)d. 20p ² - 5q ² = 5 (4p ² - q ²) = 5 (2p + q) (2p - q) 

Smart Math in factoring quadratic form ax ² + bx + c with a = 1Note the multiplication of the following two parts.(X + p) (x + q) = x² + px + qx + pq= X² + (p + q) x + pq So, the form x² + (p + q) x + pq can be factored into (x + p) (x + q).Supose, x² + (p + q) x + pq = ax ² + bx + c so a = 1, b = p + q,and c = pq.Of the example, it can be seen that p and q are factors of c. If p and q are summed, the result is b. Thus for factoring the form ax ² + bx + c with a = 1, specify two numbers a factor of c and if the two numbers are added together,the result is the same as b.So that you may better understand the material, study the following example problems:

Fraction following forms.

• a.) x² + 5x + 6 
• b.) ax ² + 2x - 8

Answer:a. x ² + 5x + 6 = (x + ...) (x + ...)Suppose, x ² + 5x + 6 = ax2 + bx + c, obtained a = 1, b = 5, and c = 6.To fill in the blank, specify two numbers is a factor of 6and if the two numbers are added together, the result is equal to 5.Factor of 6 is 6 and 1 or 2 and 3, which is 2 and a qualified So, x ² + 5x + 6 = (x + 2) (x + 3) 

b. x2 + 2x - 8 = (x + ...) (x + ...)By way as in (a), obtained by a = 1, b = 2, and c = -8.Factors of 8 are 1, 2, 4, and 8. Therefore c = -8, one of the two numbers is sought must be negative. Thus, the two Eligible numbers are -2 and 4, because -2 × 4 = -8 and-2 + 4 = 2.So, x ² + 2x - 8 = (x + (-2)) (x + 4) = (x - 2) (x + 4)Factoring Form ax ² + bx + c with a ≠ 1Previously, you had factored form ax2 + bx + c with a = 1.Now you will learn how to factor in the form ax2 + bx + c with a ≠ 1.Note the multiplication of the following two parts.(X + 3) (2x + 1) = 2x + x ² + 6x + 3= 2x ² + 7x + 3In other words, the form 2x2 + 7x + 3 be factored into (x + 3) (2x + 1).As for how factoring 2x ² + 7x + 3 is the reverse phase binomial multiplication above.2x ² + 7x + 3 = 2x ² + (x + 6 x) +3= (2x ² + x) + (6x + 3)= X (2x + 1) + 3 (2x + 1)= (X + 3) (2x +1)

From the description you can know how to factor in the forma x ² + bx + c with a ≠ 1 as follows.1) Describe bx be the sum of the two tribes that if the two tribes    The same result multiplied by (ax ²) (c).2) Factor the form obtained using distributive properties

This Mathematics in Smart Math in Multiplication  Algebra

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