Question

Show that the following equations are not identities.

(1) 5y − 3 = 17
(2) 3a(a − 2) = 3a² − 6a + 10
(3) 18 − 5b + b² = 2b
(4) 12m + (2m − 3)² = 4m² + 2m + 9
(5) (n − 2) (n + 5) = 3n − 16 + n²
(6) a³ + b³ = 3ab
(7) 3x²y + 3xy² = x³ − y³

Answer 1

An equation is said to be an identity if the left-hand and right-hand sides are equal.

(1) 5y − 3 = 17: The left-hand side is 5y − 3 and the right-hand side is 17.
The two sides will be equal only if we take the value of y as 4; for any other value of y the two sides will not be equal.
∴ 5y − 3 =17 is not an identity.

(2) 3a (a−2) = 3a² − 6a: The left-hand side is 3a (a−2), while 3a² − 6a + 10 forms the right-hand side.
The left-hand side is not equal to the right-hand side.
So, 3a (a−2) = 3a² − 6a+ 10 is not an identity.

(3) 18 − 5b + b² = 2b: The left-hand side is not equal to the right-hand side.
So, this equation is not an identity.

(4) 12m + (2m − 3)² = 4m² + 2m + 9
The left-hand side can be solved as
12m +4m² +9-12m
= 4m² +9
Since the left-hand side is not equal to the right-hand side, the equation is not an identity.

(5) (n − 2) (n + 5) = 3n − 16 + n²
The left-hand side of the equation can be expanded as n² + 3n − 10.
As the left-hand side is not equal to the right-hand side, the given equation is not an identity.

(6) a³ + b³ = 3ab: The left-hand side of the equation can be written as (a + b)³ − 3ab² − 3a²b. Since it is not equal to the right-hand side, the equation is not an identity.

(7) 3x²y + 3xy² = x³ − y³
The right-hand side of the equation can be written as (x − y)³ − 3xy² + 3x²y. Since it is not equal to the left-hand side, the given equation is not an identity.