Your question is very general, including linear and non-linear equations of any number and/or degree. In the general case you're asking about, there will be ...
no solutions -- no values of the variables satisfy all equations
a finite number of solutions -- all equations are satisfied for some finite set of variable values
an infinite number of solutions -- equations are dependent (or periodic*) so the set of solutions is an infinite set
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For 2 linear equations in 2 unknowns, there will be 0, 1, or infinite solutions.
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* Aperiodic equations may also have an infinite number of solutions. 1/sin(x)=0 is an example.
