Identities
One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true that
However, equations can also be correct for only certain values of the variables. In this case, they can be solved to find the values that satisfy the equality. For example, consider the following.
The equation is true only for two values of x, the solutions of the equation. In this case, the solutions are and .
Many mathematicians reserve the term equation exclusively for the second type, to signify an equality which is not an identity. The distinction between the two concepts can be subtle; for example,
is an identity, while
is an equation with solutions and . Whether a statement is meant to be an identity or an equation can usually be determined from its context. In some cases, a distinction is made between the equality sign for an equation and the equivalence symbol for an identity.
Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand, while letters from the end of the alphabet, like ...x, y, z, are usually reserved for the variables, a convention initiated by Descartes.
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