ADDITION PROPERTY OF EQUALITY

  Add the same value (expressions, integers) to BOTH sides of anequation.

If    a      =  b

                + c      + c

then  a + c = b + c

SUBTRACTION PROPERTY OF EQUALITY

Subtract the same value from BOTH sides of an equation

If    a      =  b

          -  c      - c

then  a - c = b – c

Multiply the same value to BOTH sides of an equation

If      a      =  b,

then c(a) = c(b)

You must multiply every term on both sides by that same value

Ifa=b + c ,

then d(a) = d(b +c)

ad = bd +cd

A value equals itself     OR       Mirrors (reflects) itself

a = a

a + b + 2c = a + b + 2c

Switch (swap) Both ENTIRE sides of an equation

If     a      =  b

then   b   =  a

If     a + 2b +3 =  4 – 2b

then     4 – 2b = a + 2b +3

If  a = b, then a can be substituted for b or b for a  in any equation or expression.   

If     a  =  b  and  2a + 3 b = 9

     Then 2a + 3 a = 9 or            2b+ 3 b = 9 

If  a = b AND b = c  THEN    a =c

You have 2 equations. One side of one is the exact same as one side in the other.   The remaining  sides in each equation will equal each other.

x + 2 = y + 2z  &   y + 2z = 12

then x + 2 = 12

COMMUTATIVE PROPERTY OF ADDITION

You can  commute or move terms in a sum

 2 + 3  = 3 + 2

 Note:  2 - 3 ¹ 3 - 2  

 although 2 - 3 = - 3 + 2 if you do not consider it subtraction, but addition with the sign being part of the term.

COMMUTATIVE PROPERTY OF MULTIPLICATION

You can  commute or move FACTORS in a PRODUCT

  2 (3)  = 3 ( 2)

Note:  2 ÷ 3 ¹ 3 ÷ 2  

x ∙2 = 2x     We use  this often.  The constant real number factor is normally placed in the front.  The variables are normally put in alphabetical order

ASSOCIATIVE PROPERTY OF ADDITION

You can  ASSOCIATE OR GROUP terms DIFFERENTLY in a sum

 (2 + 3)  + 7  =  2 + (3 + 7)  

PROPERTY OF OPPOSITES OR ADDITIVE INVERSE

For every real number a, there is a unique real number - a such that

a + (  - a)  = 0   and ( - a) + a = 0

IDENTITY PROPERTY OF MULTIPLICATION

The identity element for multiplication is 1.

a ∙ 1 = a           or         1∙ a  = a           

DISTRIBUTIVE PROPERTY OF MULTIPLICATION

a (b + c) =  ab + ac

Distribute entire term outside to every term inside  (   ).

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 (- 2x²)(3x²) + (- 2x²)( - 11x)+ (- 2x²)( 4)

  - 6x⁴ + 22x³ - 8x²

Collecting like terms, using order of operations on ONE (1) side of an equation Or  in an expression.

– 4x – 5x = – 2  

– 9x =  – 2    

You simplified.   You did what problem instructed. You did not use the ADDITION PROPERTY OF EQUALITY. 

You can divide every term on both sides of an equation by the same term.

IF  a = b

then

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You cannot divide by 0, so c can be any number except 0.