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Variables and Expressions

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Variables are the symbols which are used to represent a value. Literally, a variable is that which varies. The quantity stored in a variable can change its value.

For example: $x$ = $10$, and $y$ = $4$.

Here, $10$ and $4$ are constant terms that are assigned in variables $x$ and $y$.
An expression is a well defined combination of algebraic operations variables, constants and functions.

An expression can contain algebraic operations and variables, like: $x + y$. It may have operations, variables and constants too, like: $x - 5y + 8$. It can be in the form of a numeric expression. $A$ numeric expression is made up of algebraic operations and numerical values, like: $2 + 5 \times 3 - 18$.

Order of Operations

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Operations in an algebraic expression are required to be solved in a particular manner. This particular way of solving algebraic operations is known as order of operations. The order of operations is a standard format which is to be followed while working with operators in algebra.

The operations in algebra follows the rule of “$PEMDAS$”. The letters in $PEMDAS$ stand for:

$P$ - Parenthesis.
$E$ - Exponents(roots and powers).
$M$ - Multiplication.
$D$ - Division.
$A$ - Addition.
$S$ - Subtraction.

While solving any algebraic expression, first solve parenthesis operation, followed by all exponents (roots and powers) operation. After this, multiplication and division has to be done from left to right. At last, addition and subtraction should be performed from left to right. If the above order of operation is not followed, then we would be left with an incorrect result.

For example, let us consider $6 \times 3^{2} + 7 \times (2 + 1)$

In order to solve above expression, we follow the order of operations.

First, parenthesis should be solved i.e. $(2 + 1)$ = $3$. Now, the expression becomes:

$6 \times 3^{2} + 7 \times 3$.

Now, exponents to be solved. 

So, $3^{2}$ = $3 \times 3$ = $9$.

Now, expression is in the following form:

$6 \times 9 + 7 \times 3$.

Then, operations of multiplication and division are to be solved, so we get:

$54 + 21$.

Lastly, we perform all additions and subtractions to get the final result. 

So, we add $54$ and $21$ to get $75$ as the result.

Hence, $6 \times 3^{2} + 7 \times (2 + 1)$ = $75$.

Distributive Properties

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Distributive property is also known as distributive law or distributivity.
Distributive Law states that multiplication of a number by the sum of two numbers is equal to the sum of multiplication of that number by two numbers separately.

There are two types of distributive properties as follows:
Left-Distributive Property:

$a \times(b + c)$ = $a \times b + a \times c$

Left-distributive property is also known as left distributivity.
Right-Distributive Property:

$(a + b) \times c$ = $a \times c + b \times c$

Right-distributive property is also known as right distributivity.

Let us understand this property with the help of examples.

Solved Examples

Question 1: Solve 3(x + 2).
Solution:
By distributive property,
3 (x + 2) = 3(x) + 3(2)
= 3x + 6

Question 2: Solve (x - 6)y.
Solution:
{x + (-6)} y
By distributive property,
{x + (-6)} y = xy + (-6)y
= xy - 6y

Associative Property

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Associative property is also known as associativity. It explains about the preference of performing addition and multiplication operation. Associative property is of two kinds.
Associative Property of Addition: 

It is also termed as additive associativity. It states that the order of performing addition operation does not matter. It means that the order of grouping the numbers with addition operation does not affect the result. Additive associativity can be expressed as:

$(a + b) + c$ = $a + (b + c)$
Associative Property of Multiplication: 

It is also termed as multiplicative associativity. It states that the order of performing multiplication operation does not matter. It means that the order of grouping the numbers with multiplication operation does not affect the result. Multiplicative associativity can be defined as:

$(a \times b) \times c$ = $a \times (b \times c)$
Associative properties allow the rearrangement of the parentheses for addition and multiplication in such a way that the expression does not change.

Identity and Equality Properties

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Given below are some of the identity and equality properties.

Identity Properties:
There are two types of identity properties - Additive identity and multiplicative identity. Identity of multiplication or addition refers to a number which if multiplied or added to a particular number, the result is that particular number only.

Additive Identity: 

Zero is said to be the additive identity, since if we add zero to any number, then we get that number. Therefore, we can write as:

$a + 0$ = $a$ = $0 + a$
Multiplicative Identity:

One is said to be the multiplicative identity, since if we multiply any number by 1, then we get that number. This can be expressed as:

$a \times 1$ = $a$ = $1 \times a$

In algebra, there are different properties of equality as well. These are discussed below:

Reflexive Property: It states that a number is equal to itself. It means that $a$ = $a$, for a real number "$a$".

Symmetric Property: Symmetric property says that for any real numbers $a$ and $b$, if $a$ = $b$, then $b$ = $a$.

Transitive Property: According to transitive property for all real numbers $a,\ b$ and $c$, if $a$ = $b$ and $b$ = $c$ then $a$ = $c$.

Substitution Property: If we have a = b, then '$a$' can be substituted by '$b$' at any place in an equation.

Addition Property: If two real numbers are equal, then we can add third real number to both the sides. If $a$ = $b$, then $a + c$ = $b + c$.

Subtraction Property: If two real numbers are equal, then we can subtract third real number from both the sides. If $a$ = $b$, then $a - c$ = $b - c$.

Multiplication Property: If two real numbers are equal, then we can multiply third real number to both the sides. If $a$ = $b$, then $a \times c$ = $b \times c$.

Division Property: If two real numbers are equal, then we can divide both the sides by third real number. If $a$ = $b$, then $\frac{a}{c}$ = $\frac{b}{c}$.

Commutative Property

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Commutative property states that if we exchange the order of operands, then it will not affect the result. In algebra, commutative property is applied to addition as well as multiplication operations.
Commutative Property of Addition: 

It states that while adding two quantities, if we exchange their places, then there will be no change in the result. So, we can write as follows:

$a + b$ = $b + a$
Commutative Property of Multiplication: 

It states that while multiplying two quantities, if we exchange their places, then there will be no change in its result. So, we can write as follows:

$a \times b$ = $b \times a$
Where, $a$ and $b$ belong to real number set.