Final answer:
The transformation in the question is erroneous; however, it seems to imply the use of the Division Property of Equality, which allows for dividing both sides of an equation by the same non-zero number without changing the equality.
Step-by-step explanation:
The step used between the original equation 9(22 - 10) - 81 = 81.2x + 81 and the first step (22 – 10) – 9 = 9x² + 9 appears to involve a mistake; however, if we ignore that and focus on the properties of equality, the most relevant property is the Division Property of Equality. This property allows us to divide both sides of an equation by the same non-zero number and maintain equality. An example of this property would be if we had an equation such as 2x = 10 and we divided both sides by 2 to isolate the variable x, resulting in x = 5.
However, the transformation shown in the question does not actually represent a correct use of any property, as it appears there is an error when trying to simplify the equation. The right application would be to first simplify inside the parentheses, subtract 81 on both sides using the Subtraction Property of Equality, and then apply the Division Property of Equality if needed, dividing each term by 9 to solve for x.