Basic Rules of Algebra

These are the basic rules used when solving problems in Algebra. They are also used for higher levels of mathematics, like Trigonometry and Calculus.


Algebra Properties

Let a, b, and c be real numbers, variables, or algebraic expressions.

Property

Example

Commutative Property of Addition a + b = b + a 3x + x2 = x2 + 3x
Commutative Property of Multiplication ab = ba (3 - x)x2 = x2(3 - x)
Associative Property of Addition (a + b) + c = a + (b + c) (x + 3) + x2 = x + (3 + x2)
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 7x2 + 0 = 7x2
Multiplicative Identity Property a 1 = a 8y 1 = 8y
Additive Inverse Property a + (-a) = 0 5x2 + (-5x2) = 0
Multiplicative Inverse Property
a  

1

 = 1
a
(x2 + 3)  

1

 = 1
(x2 + 3)

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


Properties of Negation

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
-(- x2) = x2
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

Properties of Equality

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

Properties of Zero

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
3.  

0

 = 0     a ≠ 0
a
zero divided by anything (except zero) equals zero
4.  

a

 is undefined
0
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

Properties and Operations of Fractions

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

a

 = 

c

    if and only if  ad = bc
b d
cross multiply
2.  Rules of Signs

a

 = 

-a

 = 

a

b

b

-b

and

  

-a

 = 

a

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

a

 = 

ac

   c ≠ 0
b bc
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

a

 + 

c

 = 

a + c

b b

b

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

a

 + 

c

 = 

ad + bc

 
b d

bd

find the common denominator
6.  Multiply Fractions

a

  

c

 = 

ac

b d bd

top times top
bottom times bottom

7.  Divide Fractions

a

  

c

 = 

a

  

d

 = 

ad

   c ≠ 0
b d b c bc
to divide, multiply the divisor by the reciprocal
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Last Updated: 11/24/2008
K. A. Shearer 2005-2007
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