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.

We can add numbers in any order.

Commmutative Property of Multiplication

We can also multiply numbers in any order.

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

For an in depth discussion, see Distributive 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.

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

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.

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

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