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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 functional relationships (such as "if you sell x tickets, then your profit will be dollars")

These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a much more advanced topic generally taught to college seniors.

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

can be written in the equivalent form

.

The simplest equations to solve are the linear ones, such as

The central technique is add/subtract/multiply or divide both sides of the equation by the same number, and by repeating this process eventually arrive at the value of the unknown x. For the above example, if we subtract 3 from both sides, we obtain

and if we then divide both sides by 2, we get our solution

Equations like

are known as quadratic equations and can be solved using the quadratic formula.

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:

After some algebraic steps (not covered here), we can deduce that x = 1, however we cannot deduce what the value of y is. Try some values of x and y (which may lead to either true or false statements) to get a feel for this.

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, multiply the second by 2, and you have the following equations:

Because we multiplied the entire equation by 2, it actually represents the same statement. Now we can combine the two equations:

You can see that since we multiplied the second equation by 2, we can cancel out y when combining the equations, and then we can solve for x, which is 2. Note that you can multiply by negative numbers, or multiply both equations to get to a point where a variable cancels out (you can also cancel out x).

Now choose one of the equations from the beginning.

Substitute in 2 for x.

Simplify.

And solve for y, which equals 3. The answer to this problem is and , or .

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:

Example: if then .

• MultiplicationArithmetic In its simplest form, multiplication is a quick way of adding identical numbers. The result of multiplying numbers is called a product''. The numbers being multiplied are called coefficients or factors and individually as the multiplicand and m is a commutative operation.
  • Division is the reverse of multiplication.
  • To divide is the same as to multiply by a reciprocalIn mathematics, the reciprocal or multiplicative inverse of a number x is the number which, when multiplied by x, yields 1. Zero does not have a reciprocal. Every complex number except zero has a reciprocal that is a complex number. If it is real then so:

• ExponentiationIn mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. The next operation after exponentiation is sometimes called tetration; repeatin is not a commutative operation.
  • Therefore exponentiation has a pair of reverse operations: logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are r and exponentiation with reciprocal exponents (e.g. square rootIn mathematics, the square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is. For example, since. This example suggests how square roots can arise whes).
    • Examples: if then . If then .
  • The square root of negative one is iIn mathematics, the imaginary unit i allows the real number system R to be extended to the complex number system C . Its precise definition is dependent upon the particular method of extension. The primary motivation for this extension is the fact that no.
• DistributiveIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: : 4 · (2 + 3) (4 · 2) + (4 · 3) In the left-hand side of the above equatio property of multiplication with respect to addition: .
• Distributive property of exponentiation with respect to multiplication: .
• How to combine exponents: .
• If and , then ( TransitivityIn mathematics, a binary relation R over a set X is transitive if it holds for all a b and c in X that if a is related to b and b is related to c then a is related to c''. In notation, this is: : For example, "is greater than" and "is equal to" are transi of Equality).
• ( Reflexivity of Equality).
• If then ( Symmetry of Equality).
• If and then .
  • If then for any c, due to Reflexivity of Equality.
• If and then = .
  • If then for any c due to Reflexivity of Equality.
• 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.