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{{Short description|Whether or not there exists a set of values to satisfy a given system of equations}}
In [[mathematics]] and particularly in [[algebra]], a [[system of equations]] (either [[linear equation system|linear]] or [[nonlinear equation system|nonlinear]]) 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 [[substitution (algebra)|substituted]] into each of the equations, they make each equation hold true as an [[identity (mathematics)|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.<ref>{{Cite web|title=Definition of INCONSISTENT EQUATIONS|url=https://www.merriam-webster.com/dictionary/consistent+equations|access-date=2021-06-10|website=www.merriam-webster.com|language=en}}</ref><ref>{{Cite web|title=Definition of consistent equations {{!}} Dictionary.com|url=https://www.dictionary.com/browse/consistent-equations|access-date=2021-06-10|website=www.dictionary.com|language=en}}</ref>
If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements {{math|1=2 = 1}}, or <math>x^3 + y^3 = 5</math> and <math>x^3 + y^3 = 6</math> (which implies {{math|1=5 = 6}}).
Both types of equation system, inconsistent and consistent, can be any of [[overdetermined system|overdetermined]] (having more equations than unknowns), [[underdetermined system|underdetermined]] (having fewer equations than unknowns), or exactly determined.
==Simple examples==
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===Underdetermined and consistent===
The system
:<math>
x+y+z &= 3, \\
x+y+2z &= 4
\end{align}</math>
has an infinite number of solutions, all of them having {{math|1=''z'' = 1}} (as can be seen by subtracting the first equation from the second), and all of them therefore having {{math|1=''x'' + ''y'' = 2}} for any values of {{mvar|x}} and {{mvar|y}}.
The nonlinear system
:<math>\begin{align}
x^2+y^2+z^2 &= 10, \\
x^2+y^2 &= 5
\end{align}</math>
has an infinitude of solutions, all involving <math>z=\pm \sqrt{5}.</math>
Since each of these systems has more than one solution, it is an [[indeterminate system]] .
===Underdetermined and inconsistent===
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The system
:<math>
x+y+z &= 3, \\
x+y+z &= 4
\end{align}</math>
has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible {{math|1=0 = 1}}.
The non-linear system
:<math>\begin{align}
x^2+y^2+z^2 &= 17, \\
x^2+y^2+z^2 &= 14
\end{align}</math>
has no solutions, because if one equation is subtracted from the other we obtain the impossible {{math|1=0 = 3}}.
===Exactly determined and consistent===
The system
:<math>\begin{align}
x+y &= 3, \\
x+2y &= 5
\end{align}</math>
has exactly one solution: {{math|1=''x'' = 1, ''y'' = 2}}
The nonlinear system
:<math>\begin{align}
x+y &= 1, \\
x^2+y^2 &= 1
\end{align}</math>
has the two solutions {{math|1=(''x, y'') = (1, 0)}} and {{math|1=(''x, y'') = (0, 1)}}, while
:<math>\begin{align}
x^3+y^3+z^3 &= 10, \\
x^3+2y^3+z^3 &= 12, \\
3x^3+5y^3+3z^3 &= 34
\end{align}</math>
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 {{mvar|z}} can be chosen and values of {{mvar|x}} and {{mvar|y}} can be found to satisfy the first two (and hence the third) equations.
===Exactly determined and inconsistent===
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The system
:<math>
x+y &= 3, \\
4x+4y &= 10
\end{align}</math>
has no solutions; the inconsistency can be seen by multiplying the first equation by 4 and subtracting the second equation to obtain the impossible {{math|1=0 = 2}}.
Likewise,
:<math>\begin{align}
x^3+y^3+z^3 &= 10, \\
x^3+2y^3+z^3 &= 12, \\
3x^3+5y^3+3z^3 &= 32
\end{align}</math>
is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction {{math|1=0 = 2}}.
===Overdetermined and consistent===
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The system
:<math>
x+ 2y &= 7, \\
4x+6y &= 20
\end{align}</math>
has a solution, {{math|1=''x''
The system
:<math>
x+2y &= 7, \\
3x+6y &= 21, \\
7x+14y &= 49
\end{align}</math>
has an infinitude of solutions since all three equations
The nonlinear system
:<math>\begin{align}
x^2-1 &= 0, \\
y^2-1 &= 0, \\
(x-1)(y-1) &= 0
\end{align}</math>
has the three solutions {{math|1=(''x, y'') = (1, –1), (–1, 1), (1, 1)}}.
===Overdetermined and inconsistent===
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The system
:<math>
x+2y &= 7, \\
4x+6y &= 21
\end{align}</math>
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
:<math>\begin{align}
x^2+y^2 &= 1, \\
x^2+2y^2 &= 2, \\
2x^2+3y^2 &= 4
\end{align}</math>
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
===Linear systems===
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===Nonlinear systems===
{{Main|System of polynomial equations#What is solving?}}
==References==
{{reflist}}
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