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It depends. Partly on the domain over which your system of equations is defined - are they integer solutions? Reals or complex numbers? Are the equations linear or more complicated?

In any case, there can be none, one or many - including infinitely many.

If the system is inconsistent eg

a + b = 3

a + b = 2

then there are no solutions.

If the system is incomplete (the relevant matrix is singular), you have an infinite number:

a + b + c = 1

a + 2b + 3c = 2

has an infinite number of solutions.

A set of n independent linear equations in n unknowns will have a unique solution.

A single equation such as (a-2)2 + (b-3)2 + (c-7.5)2 = 0

has a unique real-number solution since each on the brackets MUST be zero.

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Q: How many solutions does an independent system of equations have?
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