Algebra Period 4

REVIEW OF THE PROPERTIES AND SOME NEW ONES

Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac

Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN

INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES

EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b

Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]

Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]

Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]

Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21

Simplify by combining like terms:
12x - 6y -17

Number properties and Proofs 2-10
MORE NEW FRIENDS! (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 WE ACCEPT AS FACT!

EXAMPLES OF AXIOMS:
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse

EXAMPLE OF A THEOREM:
Distributive Property in REVERSE (a + b)c = ac + bc


PROPERTIES OF EQUALITY
(these are AXIOMS)
” Prop = “
REFLEXIVE PROP =:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.


SYMMETRIC PROP =:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5
In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12

TRANSITIVE PROP =:
a = b and b = c then a = c
3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.

I've got a pattern that will help you recognize the difference between these 3 properties specifically.
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations