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→Simple examples: nonlinear examples |
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has an infinitude of solutions, all of them having ''z'' = 1 (as can be seen by subtracting the first equation from the second), and all of them therefore having ''x+y'' = 2 for any values of ''x'' and ''y''.
The nonlinear system
:<math>x^2+y^2+z^2=10,</math>
:<math>x^2+y^2=5</math>
has an infinitude of solutions, all involving <math>z=\pm \sqrt{5}.</math>
===Underdetermined and inconsistent===
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has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible 0 = 1.
The nonlinear system
:<math>x^2+y^2+z^2=10,</math>
:<math>x^2+y^2+z^2=12</math>
has no solutions, because if one equation is subtracted from the other we obtain the impossible 0 = 2.
===Exactly determined and consistent===
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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 infinitude 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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