In Mathematical, a equation is an equality which binds various quantities, generally to pose the problem of their identity. Résoudre the equation consists in determining all the ways of giving to some quantities which appear there, the unknown , of the values which make the statement true. These possible values are called solutions equation. An equation is often used to refer to the whole of its solutions, and in Géométrie in particular, one calls equation of a mathematical object his definition like whole of the solutions of an equation. In a broader direction, the equation term is synonymous with equality , with frequently the insinuation which the equality is not always true.

An equation is generally presented in the form of a formula of the $A=B$ type. (However, any mathematical problem can be seen like an equation, without always being expressed explicitly in this form.) Both members $A$ and $B$ separated by the sign $=$ depend on variables (unknown or parameters) whose values are not specified. One adds to it, sometimes implicitly, the data of the units where one seeks values for the unknown factors. For example, the equation

$x+1=0,\; \backslash \; quad\; X\; \backslash \; in\; \backslash \; mathbb\; R$

of unknown factor X admits for single solution reality $-1$.

One holds names particular to certain types of equations. Thus, when she is written as the “combination” of several simpler equations which must be checked simultaneously, one speaks about system of equations or simply of system.

Certain necessary informations with the comprehension and the solution of an equation are sometimes implied. In particular, a usual convention of notation wants that letters of the beginning of the alphabet ( has , B , has , B …) parameters represent, whereas those of the end of the alphabet (mainly X , there , Z , X ) indicate unknown factors. Thus “To solve the equation $ax^2+bx+c=0$, of unknown factor $x\; \backslash \; in\; \backslash \; mathbb\; R$, for any value of the parameters $a,\; B,\; C\; \backslash \; in\; \backslash \; mathbb\; R$” will be able to shorten without too much ambiguity of “To solve in $\backslash \; mathbb\; R$: $ax^2+bx+c=0$”.

Certain categories of equations are the subject of general theories. One thus manages to solve certain classes of equations by expressing their solutions in a form more explicit than the equation itself. In the less favorable cases, one is satisfied to study the conditions of existence of the solutions and their properties.

Properties

If an algebraic equation is true, then the following operations can be applied there and the new equation will be still true:

1. Any quantity can be added on each side of the equality.

2. Any quantity can be withdrawn on each side of the equality.
3. Any quantity can multiply each side of the equality.
4. Any quantity different from zero can divide each side of the equality.
5. Generally, any function can be applied on each side of the equality. However, the number of solutions can change, which can not be wished.

The algebraic properties (1-4) imply that the equality is a relation of Congruence in a body. In practice, it is only. The whole of the real numbers, being a body, allows these transformations. However, if the starting equation are valid in the whole of the natural numbers, division and the subtraction do not make it possible to maintain the veracity of the new equation.

Functions 1 to 4 being injective (except while multiplying by 0 on the two sides of the equation for 3.), they do not modify the number of solutions. If a function not Injective is applied to the two sides of a true equation, then the resulting equation can still be true, but it will be less useful. Formally, it is about a logical Implication, not of a equivalence, which can carry out to increase the solution unit.

Types

Because of their relative importance, certain equations are gathered in classes with share:

• linear equation
• polynomial equation
• differential equation
• partial derivative equation.

Parametric

A parametric equation arises in the form

$x\; =\; x\_i\; +\; has\; T\; \backslash ,$,

$y\; =\; y\_i\; +\; B\; T\; \backslash ,$,

$z\; =\; z\_i\; +\; C\; T\; \backslash ,$,

$\backslash \; ldots\; \backslash ,$,

which represents an algebraic object depend on the parameter $t\; \backslash ,$, passing by the Point $(x\_i;\; y\_i;\; z\_i;\; \backslash \; ldots)\; \backslash ,$ and of directing Vecteur $(has;\; B;\; C;\; \backslash \; ldots)\; \backslash ,$, all constants in the equation being able to be real numbers or complexes.

Let us suppose that a line $(D)\; \backslash ,$ in $\backslash \; R^3$ passes by the points $(3;\; 4;\; 5)\; \backslash ,$ and $(1;\; 9;\; 12)\; \backslash ,$. One of its directing vectors is $(-\; 2,\; 5,7)\; \backslash ,$, and its parametric equation is

$x\; =\; 3\; -\; 2\; T\; \backslash ,$,

$y\; =\; 4\; +\; 5\; T\; \backslash ,$,

$z\; =\; 5\; +\; 7\; T\; \backslash ,$,

Symmetrical

A symmetrical equation arises in the form

$\backslash \; frac\; \{X\; -\; x\_i\}\; \{has\}\; =\; \backslash \; frac\; \{there\; -\; y\_i\}\; \{B\}\; =\; \backslash \; frac\; \{Z\; -\; z\_i\}\; \{C\}\; =\; \backslash \; cdots\; \backslash ,$,

which represents an algebraic object passing by the Point $(x\_i;\; y\_i;\; z\_i;\; \backslash \; ldots)\; \backslash ,$ and of directing Vecteur $(has;\; B;\; C;\; \backslash \; ldots)\; \backslash ,$, all constants and all variables in the equation being able to be real numbers or complexes.

Let us suppose that a line $(D)\; \backslash ,$ in $\backslash \; R^3$ passes by the points $(3;\; 4;\; 5)\; \backslash ,$ and $(1;\; 9;\; 12)\; \backslash ,$. One of its directing vectors is $(-\; 2,\; 5,7)\; \backslash ,$, and its symmetrical equation is

$\backslash \; frac\; \{3\; -\; X\}\; \{2\}\; =\; \backslash \; frac\; \{there\; -\; 4\}\; \{5\}\; =\; \backslash \; frac\; \{Z\; -\; 5\}\; \{7\}\; \backslash ,$.

Vectorial

A vectorial equation utilizes at the same time Vecteur S and Scalaire S:

$(X;\; there;\; Z;\; \backslash \; ldots)\; =\; (x\_i;\; y\_i;\; z\_i;\; \backslash \; ldots)\; +\; has\; (k\_0;\; k\_1;\; k\_2;\; \backslash \; ldots)\; +\; B\; (l\_0;\; l\_1;\; l\_2;\; \backslash \; ldots)\; +\; \backslash \; cdots\; \backslash ,$,

which represents an algebraic object passing by the Point $(x\_i;\; y\_i;\; z\_i;\; \backslash \; ldots)\; \backslash ,$ and containing the vectors $(k\_0;\; k\_1;\; k\_2;\; \backslash \; ldots)\; \backslash ,$, $(l\_0;\; l\_1;\; l\_2;\; \backslash \; ldots)\; \backslash ,$,…, each vector being multiplied by the scalars $a,\; B,\; \backslash \; ldots\; \backslash ,$. All the scalar constants and all variables in the equation can be real numbers or complexes. If the vectors are neither colinéaires, nor orthogonal, then their number in the equation indicates the number of dimensions of the object: 1 vector implies that it is about a line; 2 vectors, a plan; 3 vectors, a volume; etc

Let us suppose that an object $(G)\; \backslash ,$ in $\backslash \; R^3$ passes by the points $(3;\; 4;\; 5)\; \backslash ,$ and that it contains the vectors $(1;\; 9;\; 12)\; \backslash ,$ and $(3;\; 2;\; 4)\; \backslash ,$. Its vectorial equation is

$(X,\; there,\; Z,\; \backslash \; ldots)\; =\; (3;\; 4;\; 5)\; +\; has\; (1;\; 9;\; 12)\; +\; B\; (3;\; 2;\; 4)\; \backslash ,$, with $a,\; B\; \backslash \; in\; \backslash \; R\; \backslash ,$

Since the two vectors are neither parallel (one is not multiple other) nor orthogonal (the scalar product of both is not null), then $(G)\; \backslash ,$ is a plan.

See too

Related articles

• Equation (elementary mathematics)
• Equations: of the first degree | quadratic (second degree) | cubic (third degree) | quartic (fourth degree) | quintic (fifth degree)
• Inequation
• Change of variable (algebraic technique of simplification)

External bonds

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• Equation Wizard , finds the solutions algebraic
• EqWorld , information on the solutions of various types of equations
• EquationSolver , calculates the solution of systems of linear Equation

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