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.
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.
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.
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
top times the top and bottom times the bottom
when dividing two fracitons, multiply the divisor by the reciprocal