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

Math 6H Period 3, 6 & 7

Distributive Property 1-4

I like to eat...eat...eat..eat Apples and Bananas... now what did that have to do with the distributive property???
The Distributive Property of Multiplication with Respect to Addition
For any whole numbers , a,b, and c,

a(b + c) = ab + ac
and
(b + c)a = ba + ca

so the reason we want to use the Distributive Property is to make our lives easier
for example
13 X 15 could be easily multiply using the distributive property 13(10 + 5) =
13(10) + 13(5) = 130 + 65 = 195

and if you were given (11 X4) + (11 X 6) you could do the reverse and say 11(4 +6) or 11(10) = 110


Similarly, we have the Distributive Property of Multiplication with Resepect to Subtraction

For any whole numbers a, b, and c
a(b - c) = ab - ac
and
(b-c)a = ba - ca

Since multiplication is distributive with respect to both addition and subtraction, we refer to both properties as the DISTRIBUTIVE PROPERTY

99(43) yes, you could multiply it out but if you used the distributive property and thought of two numbers whose difference was 99 -- you could easily do this in your head..
That is,
(100 - 1)43 = 100(43) - 1(43) = 4300 -43 = 4257

51(27) could become (50 + 1)27 = 50(27) + 1(27) = 1350 + 27 = 1377

The distributive property is especially helpful with mental math!!