# Algebra

In arithmetic, you come across problems with numbers in them. In algebra, you come across problems with letters in them. In algebra, variables are letters that represent numbers.

For example: a + 2 = b

The letter a represents an unknown number, but when you know a and add 2 to it, you can figure out what b is.

Variables: A variable is a symbol that represents a value that can vary (hence the name variable). It stands for a number you don't know.

Constants: The opposite of a variable is a constant, which has a fixed value. Sometimes a constant used in multiplication is called a coefficient or an index.

Expressions: An expression is a combination of symbols. An algebraic expression includes variables.

Equations: An equation is a combination of symbols, like an expression. The difference is that it has an equal sign (=) to show that two expressions are equal.

Terms: A term is any one part of an expression or equation, separated from other terms by an addition (+) or subtraction (-) sign. A term can be a constant, a variable, or the product of constants and variables.

For example: x + y + 3a - ab - xy + 3abxy

The terms are x, y, 3a, ab, xy and 3abxy. The term 3a is a combination of 3 and a, and it means "three times the amount of a". The term xy is a combination of x and y, and it means that x and y are multiplied together.

### Simplifying Variables

You can change the order of terms without changing the result. This property is called commutativity. For example, these two expressions are the same. The terms are just in a different order.

x + y = y + x

In algebra, as in arithmetic, an equation has two sides. These sides are called the left side and the right side. For example: a + b = c + d. Here, the left side contains a + b and the right side contains c + d.

You can add, subtract, multiply, or divide both sides of an equation. Also, just like numbers, you can add, subtract, multiply, and divide variables. These operations are how you reduce equations to their simplest terms.

Adding variables: 3a + 2b + 4c + 2a + b + 4c = 5a + 3b + 8c

Subtracting variables: 3a + 2 - 2a - 1 = a + 1

Multiplying variables: (am)(an) = a(m+n)

Dividing variables: am/an = a(m-n)

### Formulas

A formula uses variables (letters) and may have constants (numbers). A formula is usually expressed as an equation, so it contains an equal (=) sign.

A simple example of a formula with a constant is the one that shows the relationship of the radius of a circle to the diameter. That formula is d = 2r. You say this formula as "the diameter is equal to two times the radius." The radius may vary (being a variable), but the 2 is constant (being an unchanging number).

Formulas have three properties you can use to manipulate them. The properties are associativity, commutativity, and distributivity.

Property A: Associativity

In a formula, how you group the terms doesn’t matter as long as the sequence doesn’t change. Rearranging parentheses doesn’t affect the value of an expression. This property works for addition and multiplication.

For addition, (1 + 2) + 3 = 1 + (2 + 3)

For multiplication, (1 × 2) × 3 = 1 × (2 × 3)

The property doesn't apply to subtraction or division.

Property C: Commutativity

Commutativity means that changing the order of the elements in an equation doesn’t change the result. Commutativity applies to addition and multiplication. The property doesn’t apply to subtraction or division. For example,

1 + 2 + 3 = 3 + 2 + 1 = 2 + 1 + 3 = 6

4 × 5 × 6 = 6 × 5 × 4 = 5 × 4 × 6 = 120

Property D: Distributivity

When you multiply one term inside parentheses by another term outside the parentheses, you must distribute the multiplication to each of the terms inside the parentheses. Distributivity is a common way to break down formulas for solutions and making the complex become simple.

For example,  6a(b + 2c) = 6ab + 12ac