Definition
Identity equation
An identity equation is true for every allowed value of its variable or variables.
What it means
Try any allowed value and the equality keeps working. That happens because the two sides are different ways to write the same quantity. A conditional equation works only for particular values; an identity works throughout its allowed domain.
Remember it
Identity: every allowed value keeps the equation true.
Example
For 5(t − 2) = 5t − 10, use t = 4: both sides are 10. Use t = −1: both sides are −15. Distributing 5 turns the left side into 5t − 10, so the equation is true for every real t.
Watch for
x + 2 = 7 is not an identity; it is true only when x = 5.
Common mix-up
“Every allowed value” matters. For example, x ÷ x = 1 excludes x = 0 because the left side is undefined there.
Exact meaning
An identity equation is an equation whose two sides have equal values for every input in the common domain where both sides are defined. It records two equivalent expressions rather than selecting only certain solutions.