Consistent and inconsistent equations: Difference between revisions

Revision as of 20:47, 24 January 2019

Revision as of 19:13, 5 March 2019

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has no solutions, because if one equation is subtracted from the other we obtain the impossible 0 = 3.

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===Exactly determined and consistent===

The system

:<math>x+y=3,</math>

:<math>x+2y= 5</math>

has exactly one solution: ''x'' = 1, ''y''= 2.

The nonlinear system

:<math>x+y=1,</math>

:<math>x^2+y^2=1</math>

has the two solutions (''x, y'') = (1, 0) and (''x, y'') = (0, 1), while

:<math>x^3+y^3+z^3=10,</math>

:<math>x^3+2y^3+z^3=12,</math>

:<math>3x^3+5y^3+3z^3=34</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 ''z'' can be chosen and values of ''x'' and ''y'' can be found to satisfy the first two (and hence the third) equations.

===Exactly determined and inconsistent===