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Basic Rules of Algebra

There are basic properties in math that apply to all real numbers. When working with variables in algebra, these properties still apply. We will apply most of the following properties to solve various Algebraic problems.

Algebraic Properties

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

Commutative Property of Addition

We can add numbers in any order.

Commmutative Property of Multiplication

We can also multiply numbers in any order.

Associative Property of Addition

We can group numbers in a sum any way we want and get the same answer.

Associative Property of Multiplication

We can group numbers in a product any way we want and get the same answer.

Distributive Property

When we are adding and multiplying with a parenthesis, we can distribute the multiplication through the addition.

For an in depth discussion, see Distributive Property

Additive Identity Property

If we add 0 to any number, we will end up with the same number.

Multiplicative Identity Property

If we multiply 1 to any number, we will end up with the same number.

Additive Inverse Property

If we adda number by the opposite of itself, we will end up with 0.

Multiplicative Inverse Property

If we multiply a number by its reciprocal, we will end up with 1.



Keep in mind that subtraction is also considered addition, but with a negative number. Similarly, divison can be thought of as inverse multiplication, but with a restriction that the denominator cannot be equal to 0.

Properties of Negation

We must be careful not to make arithmetic mistakes when dealing with negative signs and subtraction.

Properties of Equality


Add c to each side


Multiply both sides by c


Subtract c from both sides


Divide both sides by c

Properties of Zero


0 added or subtracted to anything equals itself


0 multiplied by anything equals 0


0 divided by anything equals 0


We cannot divide by 0

Zero Product Property



If the product of two or more things equals 0, at least one of the values must be 0

Properties and Operations of Fractions

Let a, b, c and d be real numbers, variables, or algebraic expressions such that b and d do not equal 0.

Equivalent Fractions


cross multiply

Rules of Signs


the negative can go anywhere in the fraction and two negatives equal a positive

Generate Equivalent Fractions


multiplying the top and bottom by the same thing keeps the fraction the same value

Add/Subtract with Like Denominators


if the denominators are the same, add or subtract the top of the fraction

Add/Subtract with Unlike Denominators


find a common denominator

Multiply Fractions


top times the top and bottom times the bottom

Divide Fractions


when dividing two fracitons, multiply the divisor by the reciprocal

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