mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and particularly in

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, a

system of equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single ...

(either

linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when

substituted A substitution reaction (also known as single displacement reaction or single substitution reaction) is a chemical reaction during which one functional group in a chemical compound is replaced by another functional group. Substitution reactions ar ...

into each of the equations, they make each equation hold true as an

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

. In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations. If a system of equations is inconsistent, then it is possible to manipulate and combine the equations in such a way as to obtain contradictory information, such as , or

x^3 + y^5 = 5

and

x^3 + y^3 = 6

(which implies ). Both types of equation system, consistent and inconsistent, can be any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined.

Simple examples

Underdetermined and consistent

The system :

\begin
x+y+z &= 3, \\
x+y+2z &= 4
\end

has an infinite number of solutions, all of them having (as can be seen by subtracting the first equation from the second), and all of them therefore having for any values of and . The nonlinear system :

\begin
x^2+y^2+z^2 &= 10, \\
x^2+y^2 &= 5
\end

has an infinitude of solutions, all involving

z=\pm \sqrt.

Since each of these systems has more than one solution, it is an

indeterminate system In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may b ...

Underdetermined and inconsistent

The system :

\begin
x+y+z &= 3, \\
x+y+z &= 4
\end

has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible . The non-linear system :

\begin
x^2+y^2+z^2 &= 17, \\
x^2+y^2+z^2 &= 14
\end

has no solutions, because if one equation is subtracted from the other we obtain the impossible .

Exactly determined and consistent

The system :

\begin
x+y &= 3, \\
x+2y &= 5
\end

has exactly one solution: . The nonlinear system :

\begin
x+y &= 1, \\
x^2+y^2 &= 1
\end

has the two solutions and , while :

\begin
x^3+y^3+z^3 &= 10, \\
x^3+2y^3+z^3 &= 12, \\
3x^3+5y^3+3z^3 &= 34
\end

has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of can be chosen and values of and can be found to satisfy the first two (and hence the third) equations.

Exactly determined and inconsistent

The system :

\begin
x+y &= 3, \\
4x+4y &= 10
\end

has no solutions; the inconsistency can be seen by multiplying the first equation by 4 and subtracting the second equation to obtain the impossible . Likewise, :

\begin
x^3+y^3+z^3 &= 10, \\
x^3+2y^3+z^3 &= 12, \\
3x^3+5y^3+3z^3 &= 32
\end

is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction .

Overdetermined and consistent

The system :

\begin
x+y &= 3, \\
x+ 2y &= 7, \\
4x+6y &= 20
\end

has a solution, , because the first two equations do not contradict each other and the third equation is redundant (since it contains the same information as can be obtained from the first two equations by multiplying each through by 2 and summing them). The system :

\begin
x+2y &= 7, \\
3x+6y &= 21, \\
7x+14y &= 49
\end

has an infinitude of solutions since all three equations give the same information as each other (as can be seen by multiplying through the first equation by either 3 or 7). Any value of is part of a solution, with the corresponding value of being . The nonlinear system :

\begin
x^2-1 &= 0, \\
y^2-1 &= 0, \\
(x-1)(y-1) &= 0
\end

has the three solutions .

Overdetermined and inconsistent

The system :

\begin
x+y &= 3, \\ 
x+2y &= 7, \\
4x+6y &= 21
\end

is inconsistent because the last equation contradicts the information embedded in the first two, as seen by multiplying each of the first two through by 2 and summing them. The system :

\begin
x^2+y^2 &= 1, \\
x^2+2y^2 &= 2, \\
2x^2+3y^2 &= 4
\end

is inconsistent because the sum of the first two equations contradicts the third one.

Criteria for consistency

As can be seen from the above examples, consistency versus inconsistency is a different issue from comparing the numbers of equations and unknowns.

Linear systems

A linear system is consistent

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

its

coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations. Coefficient matrix In general, a system with ''m'' linear e ...

has the same

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

as does its

augmented matrix In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices and , where ...

(the coefficient matrix with an extra column added, that column being the

column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...

of constants).

Nonlinear systems

References

{{reflist Equations Algebra