Algebraic Identities

Algebraic Identities are equations that hold true for all values of the variables involved. They are often used to simplify expressions and solve equations. Identifying and applying these identities can make complex algebraic manipulations more manageable.

Example: Implement the first Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
              L.H.S => (x+y)² = (4+3)²                           
                                      =  (7)²
                                                  = 49

             R.H.S => 4² +3²+ 2 (4) (3) = 49 

 As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the second Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
                 L.H.S => (x – y)² = (4 – 3)²                          
                                           =  (1)²
                                           =  1

               R.H.S => 4² + 3² – 2 (4) (3) = 1

As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the third Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
                 L.H.S => (x + y)(x – y) = (4 + 3)(4 – 3)                                                      
                                                    = (7)(1)  
                                                    = 7 

                 R.H.S :  x² – y²   = (4)^{2 –}  (3)²

                                                       = 16 – 9      

                                         = 7

 As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the fourth Identity on x = 3, y = 4, and z = 5.

Solution: 

Applying the identity:

                  L.H.S => (x + y + z)² = (3 + 4 + 5)² 

                                                   = (12) ² 

                                                   = 144                          
                                                    

                  R.H.S => x2 – y2   = (3)² + (4)² + (5)² + 2(3)(4) + 2(4)(5) + 2(5)(3)
                                               = 144

 As L.H.S = R.H.S, this identity is verified and true.

Using the above Identities we can derive many identities, some of the popularly used identities are written below:

Algebraic Expressions and Identities: In mathematics, An algebraic expression is a combination of variables and constants. The value of an algebraic expression can vary because the variables can take different values. On the other hand, an algebraic identity is an equation that holds true for all possible values of the variables involved.

To understand these terms, we need to have an understanding of terms, factors, and coefficients. In this article, we have discussed algebraic expressions and identities with their examples in detail.