Final answer:
An equation is an identity if, after simplifying, it can be rewritten as 0 = 0, indicating it is true for all variable values, hence it has an infinite number of solutions option c is correct.
Step-by-step explanation:
An equation is an identity if it is true for all values of the variable involved. This means that after simplifying the equation, if both sides of the equation are identical, then the equation is an identity. The correct answer to how you know whether an equation is an identity and how many solutions it has is C, which states, 'An equation is an identity if it can be rewritten as 0 = 0. Such an equation has an infinite number of solutions because it reduces to a true statement regardless of the value of the variable.'
An equation is considered an identity if it holds true for all values of the variables involved. In other words, substituting any valid values for the variables into an identity will result in a true statement. Unlike regular equations, which may have specific solutions, identities encompass an infinite set of solutions. This is because every value that satisfies the given expression or equation makes it universally true.
For example, the equation
�
2
−
�
2
=
(
�
+
�
)
(
�
−
�
)
x
2
−y
2
=(x+y)(x−y) is an identity because, when expanded, it simplifies to
�
2
−
�
2
x
2
−y
2
, demonstrating that it holds true for all values of
�
x and
�
y.
In summary, an equation is an identity if it is universally true for all valid values of its variables. Identities have an infinite number of solutions, making them distinct from regular equations with specific solution sets.