Algebra Honors (Periods 6 & 7)

More on Properties 
There are 2 types of Properties- Axioms and Theorems

Axioms = properties we accept as obvious and so we don’t need to prove them

Theorems =  properties that need to be proved USING the Axioms

Examples of Axioms 
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse

Example of Theorem:
The DP in reverse (a + b) c = ac + bc

Property of Negative 1 and the Inverse Property of a Sum

Both of these properties have to do with multiplying  -1 to another factor

Property of Negative 1

When you multiple any term by -1  you always get the opposite sign

-1(3a) = -3a  and   -1(-3a) = 3a

Both of these examples can be written with just the negative sign—without a 1

-(3a) = -3a  and –(-3a) = 3a

Inverse Property of a SUM

When you multiply by -1 by more than 1 term—you always get the opposite sign of each term

-1(3a + 5) = -3a – 5

-1(-3a – 5) = 3a + 5

Both of the above examples can be rewritten with just a negative sign ( without the 1)

Remember when you see a negative sign it is just a shortened for of -1.

-(3a + 5) = -3a – 5  and  -(-3a -5) = 3a + 5 

Properties of Equality (these are axioms)

Reflexive
a= a
 it looks the same – identical--on both sides !

Symmetric
a= b then b = a
3 + 5 = 8 the 8 = 3 + 5  You can switch the sides of an equation.

Transitive

a = b  b = c then a = c

3 + 5 = 8 and 2 + 6 = 8, then 3 + 5 = 2 + 6

If two things both equal a third, then we can say that they equal each other as well.

The Reflexive Property only has ONE equation

The Symmetric Property only has TWO equations

The Transitive Property only has THREE Equations