# Elementary algebra

**Elementary algebra**is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, -, ×, ÷) occur, in algebra one also uses symbols (such as

*a*,

*x*,

*y*) to denote numbers. This is useful because:

- It allows the general formulation of arithmetical laws (such as for all
*a*and*b*), and thus is the first step to a systematic exploration of the properties of the real number system - It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number
*x*such that ) - It allows the formulation of functionalal relationships (such as "if you sell
*x*tickets, then your profit will be dollars")

In algebra, an "expression" may contain numbers, variables and arithmetical operations; examples are and . An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as ); these are also known as "identities". Other equations contain symbols for unknown values and we are then interested in finding those values for which the equation becomes true: . These are the "solutions" of the equation.

As in arithmetic, in algebra it is important to know precisely how mathematical expressions are to be interpreted. This is governed by the order of operations rules.

It is then necessary to be able to simplify algebraic expressions. For example, the expression

- .

*x*. For the above example, if we subtract 3 from both sides, we obtain

Expressions or statement may contain many variables, from which you may or may not be able to deduce the values for some of the variables. For example:

However, if we had another equation where the values for x and y were the same, we could deduce the answer in a process known as systems of equations. For example (assume x and y are the same values in both equations):

Now choose one of the equations from the beginning.

## Laws of elementary algebra

- Addition is a commutative operation.
- Subtraction is the reverse of addition.
- To subtract is the same as to add a negative number:

- Multiplication is a commutative operation.
- Division is the reverse of multiplication.
- To divide is the same as to multiply by a reciprocal:

- Exponentiation is not a commutative operation.
- Distributive property of multiplication with respect to addition: .
- Distributive property of exponentiation with respect to multiplication: .
- How to combine exponents: .
- If
*a*=*b*and*b*=*c*, then (Transitivity of Equality). - (Reflexivity of Equality).
- If then (Symmetry of Equality).
- If and then .
- If then for any
*c*, due to Reflexivity of Equality.

- If then for any
- If and then = .
- If then for any
*c*due to Reflexivity of Equality.

- If then for any
- If two symbols are equal, then one can be substituted for the other at will.
- If and then (Transitivity of Inequality).
- If then for any c.
- If and then .
- If and then .

**See also:**binomial, distributivity, vulgar fraction.