IDENTIFYING EQUIVALENT EXPRESSIONS

Two expressions are equivalent, if they are equal in value, when we plug the same value for the variable which exists in the expressions. 

One way to test whether two expressions might be equivalent is to evaluate them for the same value of the variable.

Example 1 : 

Check whether the two expressions given below are equivalent for x  =  3. 

1 + 5x,  5x + 1

Solution : 

Expression : 1 + 5x

Substitute x  =  3. 

1 + 5(3)

Multiply 

1 + 15

Add

16 ------ (1)

Expression : 5x + 1

Substitute x  =  3. 

5(3) + 1

Multiply 

15 + 1

Add

16 ------ (1)

From (1) and (2), we get equal values for the given two expressions, when x  =  3. 

Hence, the given two expressions are equivalent. 

Example 2 : 

Check whether the two expressions given below are equivalent for x  =  3. 

5x + 65,  5(13 + x)

Solution : 

Expression : 5x + 65 

Substitute x  =  3. 

5(3) + 65

Multiply 

15 + 65

Add

80 ------ (1)

Expression : 5(13 + x)

Substitute x  =  3. 

5(13 + 3)

Add 

5(16)

Multiply

80 ------(2)

From (1) and (2), we get equal values for the given two expressions, when x  =  3. 

Hence, the given two expressions are equivalent. 

Example 3 : 

Check whether the two expressions given below are equivalent for x  =  3. 

5(x + 1),  5x + 5

Solution : 

Expression : 5(x + 1) 

Substitute x  =  3.

5(3 + 1)

Add 

5(4)

Multiply

20 ------ (1)

Expression : 5x + 5

Substitute x  =  3.

5(3) + 5

Multiply  

15 + 5

Add

20 ------(2)

From (1) and (2), we get equal values for the given two expressions, when x  =  3. 

Hence, the given two expressions are equivalent. 

Example 4 : 

Check whether the two expressions given below are equivalent for all real values. 

2x,  x²

Solution : 

When we plug x  =  2, 

2x  =  2(2)  =  4

and 

x²  =  2²  =  4

From the above working, it is clear that the given two expressions are equivalent when x  =  2.

Now, let us try some other different real value for x, say x  =  3. 

Substitute x  =  3 into the given two expressions.

2x  =  2(3)  =  6

and 

x²  =  3²  =  9

For the real value "2", the given two expressions are equivalent.

But, for the real value "3", the given two expressions are not equivalent.  

Therefore, the two expressions 2x and x² are not equivalent for all real values.

Note : 

Two expressions are equivalent for all real values, if they are equal in value for all real values. 

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