Long ago, and in a guide far, far away, we learned the properties of numbers: commutative, associative, distributive, inverse, and identity. These properties also apply to adding and multiplying with variables, and they even have the same names.
This table provides a quick review.
The Commutative Property states that we can add or multiply numbers in any order. For example: 7xy is equal to y7x and yx7. In Algebra we almost always put the coefficient in front of the variables, but just for consistency, not because it needs to be that way mathematically.
The Associative Property allows us to move the parentheses to a different pair of numbers as long as everything is being multiplied or everything is being added. For example: 7 + (x + 10) is the same thing as (7 + x) + 10.
Notice that none of these examples have subtraction or division in them. The properties above do NOT work with subtraction and division.
The Identity Property: The identity for addition, or the additive identity, is 0. This is the number that we can add anything to and it won't change. For example: x + 0 = x.
The multiplicative identity is 1. This is the number that we can multiply anything by and it won't change. For example: (x)(1) = x.
The Inverse Property states that a number added to or multiplied by its inverse equals the identity. This works for variables too. When we add a variable to the same variable with the opposite sign, we get zero (the additive identity).
Examples:
x + (-x) = 0
-4xy + 4xy = 0
When we multiply a variable by its reciprocal, we get 1 (the multiplicative identity). Remember the reciprocal of a number (or variable) has the same sign. We don't change the sign; just flip the fraction.
Examples:
Note: both of these are true only when x does not equal zero, because we can't have a zero in the denominator.