Which property requires the same operation to be performed on both sides of an equation?

The property that requires the same operation to be performed on both sides of an equation is the equivalence property. This property is fundamental in algebra, as it allows us to maintain the equality of an equation while performing manipulations to isolate a variable or simplify an expression.

When you apply the equivalence property, if you add, subtract, multiply, or divide both sides of an equation by the same non-zero value, the equality remains true. This consistency ensures that the solutions derived from the equation are valid and accurate.

The other properties mentioned serve different purposes in mathematics. The distributive property pertains to distributing multiplication over addition or subtraction, allowing parentheses to be removed. The commutative property refers to the ability to change the order of numbers in addition or multiplication without changing the result. The additive property is closely related but typically refers to adding zero to a number, highlighting that it does not change the value.