In algebra, we initially learn about the variables, coefficients, terms, expressions and the operations on expressions. When we come across situation like, " the present age of a father and son are such that father is three times that of the son. After how many years father will be twice as old as the son? " How do we solve these types of questions? It will be too long by assuming the ages and check if they satisfy the condition.

These types of questions can be easily solved if we know the method of framing and solving equations. Equations are classified into linear, quadratic etc. In this section we shall learn the basic properties that are involved while solving equations.

The equality properties can be classified according to the addition, subtraction, multiplication and multiplication of real numbers.

1. Additive Property of Equality: If equals are added to equals the wholes are equal.
( or ) When same quantity is added on both sides of an equation, the equality is unaffected.
Let a, b, c be any three real numbers, then a = b => a + c = b + c

Example: 5 = 5
=> 5 + 2 = 5 + 2 [ 2 is added on both sides of the equality ]
=> 7 = 7

2. Subtractive property of Equality: If equals are subtracted from equals, the results are equal. ( or ) when same quantity is subtracted from both sides of an equation, the equality is unaffected.
Let a, b, c be ay three real numbers, then a = b => a - c = b - c

Example: 5 = 5
=> 5 - 2 = 5 - 2 [ 2 is subtracted from both sides of the equality ]
=> 3 = 3

3. Multiplicative property of Equality: If equals are multiplied to equals, the results will be equal.
( or ) when both sides of an equation is multiplied by the same non-zero number, the equality is unaffected.
Let a, b, c be any three real numbers such that c $\ne$ 0. Then a = b => a x c = b x c => ac = bc

Example: 5 = 5
=> 5 x 3 = 5 x 3 [ 3 is multiplied on both sides of the equality ]
=> 15 = 15

4. Division Property of Equality: If equals are divided by the same non-zero equals, the results will be equal.
( or ) when both sides of an equation is divided by the same non-zero number, the equality is unaffected.
Let a, b and c be any three real numbers such that c = $\ne$ 0, then a = b => $\frac{a}{c}$ = $\frac{b}{c}$,

Example: 10 = 10

=> $\frac{10}{2}$ = $\frac{10}{2}$ [ 2 is divided on both sides of the equality ]

=> 5 = 5

Identity Property:

1. Additive Identity: The real number when when added to the given real number results with the same real number, then the real number added is called the additive identity.
Example: For any real number a, a + 0 = 0 + a = a
( i. e ) 5 + 0 = 0 + 5 = 5
Therefore the additive identity is 0.

2. Multiplicative identity: The real number when multiplied by the given real number, results with the same real number, then the real number multiplied is called the multiplicative identity.
Example: For any real number, a, a x 1 = 1 x a = a
( i. e ) 5 x 1 = 1 x 5

$\frac{2}{3}$ x 1 = 1 x $\frac{2}{3}$ = $\frac{2}{3}$

Therefore the multiplicative identity is 1.

Equality Property: If a = c and b = c, then a = b

Example: If x = 5 and y = 5 then x = y = 5.

When two values are equivalent we can substitute the value of one in another.


Example: a = b and 2a + 5 = 15 then by substituting, a = b,
we get 2b + 5 = 15, where the values of a and b are equal.

Solved Examples

Question 1: If a = b and x = 2 a + 7 then express x in terms of b.
Solution:
 
We have a = b and x = 2 a + 7
Substituting b in the place of a, we get, x = 2 b + 7 which is the required expression.
 

Question 2: If x = 5, evaluate the expression 3x2 +  2
Solution:
 
We have the expression 3 x2 + 2 and x = 5
Substituting 5 in the place of x, we get,

              3 x2 + 2 =  3 ( 5 )2 + 2
                           = 3 ( 25 ) + 2
                           = 75 + 2
                           = 77
 

According to the reflexive property, for any real number a, we have a = a.


For Example, 5 = 5 and

$\frac{-2}{5}$ = $\frac{2}{-5}$

According to the symmetric property of equality, for any two real numbers, a = b => b = a


Example: If 5 + 4 = 9, then 9 = 5 + 4

Example: If x = 8 then 8 = x

According to the transitive property of Equality, For any three real numbers a, b and c, such that a = b and b = c then a = c.


Example: If x = y and y = 9 then x = 9

Example: If 5 = s and 5 = t then s = t
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