Consistent and inconsistent equations: Difference between revisions

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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&mdash;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 CONSISTENT 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 it is possible to manipulate and combine the equations in such a way as to obtain contradictory information, such as {{math|1=2 = 1}}, or <math>x^3 + y^53 = 5</math> and <math>x^3 + y^3 = 6</math> (which implies {{math|1=5 = 6}}).

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