mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and particularly in

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, a system of equations (either

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

nonlinear In mathematics and science, a nonlinear system (or a non-linear 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, mathe ...

) 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 into each of the equations, they make each equation hold true as an identity. 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 the equations cannot be true together leading to contradictory information, such as the false statements , or

x^3 + y^3 = 5

and

x^3 + y^3 = 6

(which implies ). Both types of equation system, inconsistent and consistent, 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 .

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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 linear ...

has the same

rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...

as does its

augmented matrix In linear algebra, an augmented matrix (A \vert B) is a k \times (n+1) matrix obtained by appending a k-dimensional column vector B, on the right, as a further column to a k \times n-dimensional matrix A. This is usually done for the purpose of p ...

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

column vector In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of 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 , c ...

of constants).

Nonlinear systems

References

{{reflist Equations Algebra