1. PROPERTIES OF ALGEBRA

2. Additive Identity

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

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

5. Multiplicative Inverse

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

7. Commutative Property

8. Associative Property

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

10. 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

11. 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 = and = 8, then x = 8

12. 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

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

14. 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)

15. 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)

16. 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)

17. 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)

18. 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)