Algebra (Period 1)

Chapter 1 Review ... continued...
CHAPTER 1-2: COMMUTATIVE PROPERTY

PROPERTIES ARE OUR FRIENDS! (mathematically speaking)

YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!


COMMUTATIVE PROPERTY (works for all multiplication or all addition)

You can SWITCH THE ORDER and still get the same sum or product.

This is the property YOU CAN HEAR because you've switched the order.

a + b = b + a [you can abbreviate as C+]
OR
ab = ba [ you can abbreviate as Cx]

Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS

SO WHY SHOULD YOU CARE????

Because it makes the math easier sometimes!

Which would you rather multiply:

(2)(543)(5) OR
(2)(5)(543) ???



TWO MORE FRIENDS: 
THE IDENTITY PROPERTY OF
 ADDITION AND
THE IDENTITY PROPERTY OF MULTIPLICATION

For addition, we know that adding zero to anything will not change the IDENTITY of what you started with:
a + 0 = a (what you started with)

0 is known as the ADDITIVE IDENTITY.


For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with:
(1)(a) = a (what you started with)

1 is known as the MULTIPLICATIVE IDENTITY.

Sometimes 1 is "incognito" (disguised!)

We use this concept all the time to get EQUIVALENT FRACTIONS.

Say we have 3/4 but we want the denominator to be 12

We multiply both the numerator and the denominator by 3 and get 9/12


We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3

ANYTHING OVER ITSELF = 1 (except zero because dividing by zero is UNDEFINED!)

a + b - c = 1

a + b - c



We also use this property to SIMPLIFY fractions.
We "cross cancel" all the parts on the top and the bottom that equal 1
(your parents would say that we are reducing the fraction)

6abc/2a = 3bc since both the numerator and denominator can be divided by 2a.
                                 

CHAPTER 1-4: ASSOCIATIVE PROPERTY

ANOTHER FRIEND!

This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!

a + (b + c) = (a + b) + c
 [ you can abbreviate as A+]
a(bc) = (ab)c
 [you can abbreviate as Ax]
Why? TO MAKE THE MATH EASIER OF COURSE!

This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.

EXAMPLE: [(543)(5)](2)

Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]

But our friend the Associative Property allows us to simply move the [ ]

[(543)(5)](2) = (543)[(5)(2)] which is so much easier to multiply in your head!!!



CHAPTER 1-5: DISTRIBUTIVE PROPERTY

A new friend! It's a property. It has NO counterexamples.

a( b + c) = ab + ac

or

(b + c)a = ba + ca (What other property justifies this corollary?)

You don't need to show the "middle step" once you understand this property.

EXAMPLE: -5(3y - 4) = (-5)(3y) + (-5)(-4) = - 15y + 20


You can't combine the y term with the 20 because they are UNLIKE TERMS.

LIKE TERMS:

1. Same variable
(s)
2. Same exponent(s)

Constants are like terms because they all have no variables

You can only combine (add or subtract) like terms.


BUT YOU CAN MULTIPLY UNLIKE TERMS!

3a(7y) = 21ay

BUT

3a + 7y cannot be simplified