Equation

Jump to: navigation, search
This article is about equations in mathematics. For the chemistry term, see chemical equation.

The first equation to ever be written, by Robert Recorde, who invented the equality sign, in its original form and in modern mathematic syntax.

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in

2 + 3 = 5\,.
9 - 2 = 7\,.

The equations above are examples of an equality: a proposition which states that two constants are equal. Equalities may be true or false.



While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are called variables. This is useful because:

* It allows the generalization of arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
* It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.
* It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").

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

In elementary algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with 'higher-power' terms on the left (see polynomial); a few examples are:

x + 3\,

y^{2} + 2x - 3\,

z^{7} + a(b + x^{3}) + 42/y - \pi.\,

In more advanced algebra, an expression may also include elementary functions.
A typical algebra problem.

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called "identities". "Conditional" equations are true for only some values of the involved variables: x2 − 1 = 4. The values of the variables which make the equation true are called the "solutions" of the equation.


example :
x/4 = 3
x = 4 x 3
x = 12