# 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*