These are the basic rules used when solving problems in Algebra. They are also used for higher levels of mathematics, like Trigonometry and Calculus.
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property 
Example 

Commutative Property of Addition  a + b = b + a  3x + x^{2} = x^{2} + 3x  
Commutative Property of Multiplication  ab = ba  (3  x)x^{2} = x^{2}(3  x)  
Associative Property of Addition  (a + b) + c = a + (b + c)  (x + 3) + x^{2} = x + (3 + x^{2})  
Associative Property of Multiplication  (ab)c = a(bc)  (3x • 2)(5) = (3x)(2 • 5)  
Distributive Properties  a(b + c) = ab + ac
(a + b)c = ac + bc 
3x(5 + 2x) = (3x• 5) + (3x • 2x)
(y + 5)4 = (y • 4) + (5 • 4) 

Additive Identity Property  a + 0 = a  7x^{2} + 0 = 7x^{2}  
Multiplicative Identity Property  a • 1 = a  8y • 1 = 8y  
Additive Inverse Property  a + (a) = 0  5x^{2} + (5x^{2}) = 0  
Multiplicative Inverse Property 


Because subtraction is defined as "adding the opposite", the Distributive Properties are also true for subtraction. For example, the "subtraction form" of a(b + c) = ab + ac is a(b  c) = ab  ac
Let a and b be real numbers, variables, or algebraic expressions.
Property 
Example 
1. (1)a = a  (1)7 = 7 (1)3x =  3x 
2. (a) = a  (6) = 6 ( x^{2}) = x^{2} 
3. (a)b =  (ab) = a(b)  (4)3 =  (4 •3) = 4(3) 
4. (a)(b) = ab  (3)(x) = 3x 
5. (a + b) = (a) + (b)  (x + 5) = (x) + (5) = x  5 
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property 
Explanation 
1. if a = b, then a + c = b + c  add c to each side 
2. if a = b, then ac = bc  multiply each side by c 
3. if a + c = b + c, then a = b  subtract c from each side 
4. if ac = bc and c ≠ 0, then a = b  divide each side by c 
Let a and b be real numbers, variables, or algebraic expressions.
Property 
Explanation 

1. a + 0 = a and a  0 = a  zero added to or subtracted from anything equals itself  
2. a • 0 = 0  zero multiplied by anything equals zero  

zero divided by anything (except zero) equals zero  

you can't divide by zero  
5. Zero Factor Property: if ab = 0, then a = 0 or b = 0 or both = 0 
when the product of two or more things is zero at least one of the things must equal zero 
Let a, b, c and d be real numbers, variables, or algebraic expressions such that b ≠ 0 and d ≠ 0.
Property 
Explanation 

1. Equivalent Fractions 

cross multiply  
2. Rules of Signs 

one negative equals negative, two negatives is positive, you can put the negative sign anywhere in the fraction  
3. Generate Equivalent Fractions 

if you multiply top and bottom of a fraction by the same thing, then they are still equal  
4. Add or Subtract with Like Denominators 

if the denominators are equal, just add or subtract the top of the fraction  
5. Add or Subtract with Unlike Denominators 

find the common denominator  
6. Multiply Fractions 

top times top 

7. Divide Fractions 

to divide, multiply the divisor by the reciprocal 
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