# 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 Ģ 2)(5) = (3x)(2 Ģ 5)
Distributive Properties a(b + c) = ab + ac

(a + b)c = ac + bc

3x(5 + 2x) = (3xĢ 5) + (3x Ģ 2x)

(y + 5)4 = (y Ģ 4) + (5 Ģ 4)

Additive Identity Property a + 0 = a 7x2 + 0 = 7x2
Multiplicative Identity Property a Ģ 1 = a 8y Ģ 1 = 8y
Additive Inverse Property a + (-a) = 0 5x2 + (-5x2) = 0
Multiplicative Inverse Property
 a Ģ 1 = 1 a
 (x2 + 3) Ģ 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 Ģ3) = 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 Ģ 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 Ģ c = ac b d bd

top times top
bottom times bottom

7.  Divide Fractions
 a „ c = a Ģ d = ad c ≠ 0 b d b c bc
to divide, multiply the divisor by the reciprocal
Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 11/24/2008
® K. A. Shearer 2005-2007

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