PROPERTIES OF ALGEBRA

Additive Identity  

Multiplicative Identity a · 1 = a Note: the value remains the same Example: 7 · 1 = 7

Additive Inverse a + (-a) = 0 Note: the terms cancel each other out and equal the identity. Example: 2 + (-2) = 0

Multiplicative Inverse  

Property of Zero (Multiplication) Note: any term multiplied by 0 is equal to 0 Example: 5 · 0 = 0

Commutative Property  

Associative Property  

Reflexive Property of Equality a = a Note: the term is equal to itself Example: 5x + 1 = 5x + 1

Symmetric Property of Equality If a = b, then b = a Note: there are two equations and the left and right sides are switched Example: if x + 3 = 7, then 7 = x + 3

Transitive Property of Equality If a = b and b = c, then a = c Note: there are three equations that follow a pattern Example: if x = 3 + 5 and 3 + 5 = 8, then x = 8

Substitution Property of Equality If a = b then “a” can replace “b” Note: substitution means replacement Example: if x = 2, then 5x + 3 = 5(2) + 3

Distributive a (b + c) = ab + ac Note: a coefficient is multiplied by at least two terms. Example: 2 (x + 5) = 2x + 10

Addition Property of Equality If a = b, then a + c = b + c Note: the same thing is added to both sides of the equation Example: if x – 10 = 15, then x = 25 (added 10 to each side)

Subtraction Property of Equality If a = b, then a – c = b – c Note: the same thing is subtracted to both sides of the equation Example: if y + 5 = 70, then y = 65 (5 is subtracted from both sides)

Multiplication Property of Equality If a = b, then a · c = b · c Note: the same thing is multiplied to both sides of the equation Example: If ½ x = 10, then x = 20 (each side is multiplied by 2)

Division Property of Equality If a = b, then a/c = b/c Note: the same thing is divided to both sides of the equation Example: If 5x = 20, then x = 4 (each side is divided by 5)

Closure Property If a & b are integers, then a+b is an integer. If you perform an operation on any two numbers of a set, the solution is still in the set. 5 * 2 = 10 is closed for integers (5, 2 and 10 are all integers)   Division is NOT closed for integers because (5 ÷ 2 = 2.5, and 2.5 is NOT an integer)