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