Order of operation using algebra

Introduction of order of operation using algebra:
 The Order of Operation using algebra is
P          - Parentheses,
E          - Exponents,
M         - Multiplication,
D          - Division,
A          - Addition and
S          - Subtraction

1. Calculations have to be finished from left to right in algebra.
2. First complete the operations inside a parenthesis.
3. Next Complete the operations of  exponents.
4. Then do multiplication and division, from left to right.
5. Then  do addition and subtraction, from left to right.

Step by step process for order of operation:

Parenthesis

                In algebra the parenthesis, combining symbols are always completed from the innermost set outward.

Exponents

                In algebra, an exponent is unlike than just multiplying. 

Example: 2^4 means 2 x 2 x 2 x 2 = 16

Evaluate 2 + 5 x (4 + 3) ÷ 2 - 6 using the order of operations.

Solution: 

Step 1:	2 + 5 x (4 + 3) ÷ 5 - 6	=	2 + 2 x 9 ÷ 3 – 6	Parentheses
Step 2:	2 + 5 x 7 ÷ 5 – 6	=	2 + 18 ÷ 3 – 6	Multiplication
Step 3:	2 + 35 ÷ 5 – 6	=	2 + 6 – 6	Division
Step 4:	2 + 7 - 7	=	2+6 – 6	Addition
Step 5:	2	=	2	Subtraction

Multiplication and Division in algebra

                These operations are completed in the order they show from left to right.  They are finished together because they have same importance.

Addition and Subtraction in algebra

         These two operation have two operations of the same importance. These are finished in the order that they show from left to right. 

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Learn Greatest Common Factor

Introduction:
        Greatest Common Factor (GCF) is the greatest number that is a factor of each of two or more numbers. To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.
Let us study the GCF of two numbers 24 and 15.
The factors of 24 are 1,2,3,4,6,8,12 and 24.
The factors of 15 are 1,3,5 and 15.
From this we observe that the greatest factor common to both these numbers are 3. In other words we can say that the greatest common factor of 24 and 15 is 3.

Greatest Common Factor Learning

Another method of finding out  the GCF is by using prime factorisation. The prime factors of 8 are 1,2,2,2 and that of 12 are 1,2,2,3. The prime factors that is  common for both numbers are 1,2,2. Multiply together all these numbers and we get 4. We can say that 4 is the greatest common factor of these numbers.

Least Common Multiple

Introduction:
   In arithmetic number of theory is the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it was a multiple, it can be divided by a and b without a reminder. If either a or b is 0, so that number is no such positive integer, then LCM(a, b) is defined to be zero.

Example:

Multiples of 4 are
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, ......................
Multiple of 7 are
7, 14, 21, ................
(add 4 to each get to next multiple)

Applications

When adding and subtracting, or comparing vulgar fractions, Useful to find the least common multiple of the denominators, is  called the lowest common denominator. For instance,
{3\over15}+{1\over4}={12\over60}+{15\over60}={9\over20},
where the denominator of 60 was used to least common multiple of 15 and 4.

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How to simplify Negative Exponents

Introduction:
      Exponents are also called Powers or Indices. Exponents containing negative sign are called negative exponents. The exponent of a number  says how many times to use the number in a multiplication. Dividing is the inverse (opposite) of Multiplying. A negative exponent number says how many times to divide by the number.
     Calculate the positive exponent (aⁿ)
       Then take the Reciprocal (i.e. 1/aⁿ)
      To change the sign ( minus to plus or plus to minus) of the exponent, use the Reciprocal (i.e. 1/aⁿ).

Simplifying Negative Exponents

Simplifying Negative Exponents:

            Exponent  is a small number written near the top of another number which show how many times a number or a variable is multiplied by itself. Negative exponents  indicate the inverse of the corresponding positive exponent.

Simplifying Variables with Negative Exponents:

Consider a problem like the one shown below.
x^-4  = 1/x⁴

Simplifying Fractions with Negative Exponents:

To increase a fraction to a power, raise the numerator and denominator to that power.
            a^-3b-⁴/b^-2=a-³b^-2


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Arithmetic Problems

Definition:

Arithmetic involves the study of quantity, especially as the result of combining numbers. In common usage, Arithmetic refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.

1) Question: The arithmetic mean between two terms in an arithmetic sequence is 21. If one term is 13, then find the other term.

A ) 58
B ) 50
C ) 29
D ) 21

Steps to derive

1 (a+b)2
[The arithmetic mean of two numbers a and b.]

2 (13+b)2 = 21
[Substitute the values.]

3 13 + b = 42

4 b = 29

5 So, the second term is 29.



Hence the right answer is Option C

2) Question: If 6 arithmetic means are introduced between 1 and 15, then find their sum.

A ) 64
B ) 48
C ) 2
D ) 32

Steps to derive

1 Let x₁, x₂, x₃, x₄, x₅, x₆ be the six arithmetic means introduced between 1 and 15 respectively.

2 1, x₁, x₂, x₃, x₄, x₅, x₆, 15 are in arithmetic progression.

3 Sum of the 8 terms = 8 / 2[1 + 15]
[The sum of the n terms S_n = n2[a₁ + a_n] where n is the number of terms, a₁ is the first term and a_n is the last term.]

4 = 4 × 16 = 64

5 Therefore, the sum of the six arithmetic means = x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 64 - 16 = 48



Hence the right answer is Option B

Linear Equations and linear Functions

Definition:

  

Linear Equations: A linear equation is defined as an algebraic equation in which every expression is either a stable (constant) or the product of a stable and a single variable. Linear equations can have one or more than one variables.  

  Linear functions:  A linear function is defined as a function whose chart (graph) consists of sectors of one straight line during its domain.

We can establish the linear function and it acquires value f (a) at ‘a’ and f (b) at ‘b’ by the following formula:
                                 
f(x)=f(a) (x-b)/(a-b) + f(b) (x-a)/(b-a)
   Because the first term is Zero (0) when x is ‘b’ and is f (a) when x is ‘a’, while the second term is Zero (0) when x is ‘a’ and is   f(b) when x is ‘b’.
   Finally we get,     y= mx+b


1. Example of Linear Equation:
Solve 3[2m - (7 - 3m)] = m - 21.           
Solution:
First we can solve the left hand side.
3[2m-(7-3m)]
Here first solving the inside term.
3(2m-7+3m) and then,
3(5m-7)
15m -21
Now compare with right hand side.
15m- 21= m-21
15m-m= 21-21
14m= 0
Therefore m=0
Answer: m=0

2. Example of Linear Function:
Solve 3(2m - 1) = 4(m + 5).
Solution:
3(2m - 1) = 4(m + 5)
6m-3=4m+20
6m-4m= 20+3
2m= 23
  m =23/2
  m =11.5
Answer: m=11.5

What is PEMDAS rule? in Order of Operations

In Number Theory Tutorial ,an expression or string of symbols is intended to represent a numerical value; a properly-formed expression may be evaluated in an unambiguous way. But in practice, an expression with multiple terms and operators may be written without parentheses, in which case the intended value of the expression is determined by convention. When we evaluate the expression is both preceded and followed by an operator such as minus or times, a convention is needed to clarify which operator should be applied first; this rule is known as a precedence rule, or more informally order of operation. From the earliest use of mathematical notation[citation needed], multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3 + 4 × 5 = 5 × 4 + 3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. To change the order of operations, a vinculum (an overline or underline) was originally used. Today one uses parentheses (). Thus, if one wants to force addition to precede multiplication, one writes (3 + 4) × 5 = 35.We can use order of operations calculator as well.
Mnemonics are often used to help students remember the rules. One common strategy is to build an acronym from the names of the operations. For instance, in the United States, the acronym PEMDAS (for Parentheses, Exponentiation, Multiplication/Division, Addition/Subtraction) is used, sometimes expressed as the sentence "Please Excuse My Dear Aunt Sally" or one of many other variations. In other English speaking countries, Parentheses may be called Brackets, and Exponentiation may be called Indices, Powers or Orders. Also, as Multiplication and Division are of equal precedence, M and D may be interchanged in the mnemonic. (This does not mean that multiplication and division can be performed in any arbitrary order, just that it is wrong to say "M always comes before D", or "D comes before M". See below.) The same comments apply to Addition and Subtraction.

Question:-

Solve 2 - [ (7-6) + (9-19)]

Answer:-

2 - [ (7-6) + (9-19)]

Here we have many signs.

When we have more than one sign ,we have to apply PEMDAS rule

= 2 - [ (1) + (-10)]

We have to solve inside the parenthesis first

Here (+) (-) = -

= 2 - [ 1 - 10]

= 2 - [-9]

Here (-) (-) = +

= 2+9

= 11 is the Answer