Theorem (over an infinite field, such as the reals or complexes): If a consistent linear system has more variables than equations ($n > m$), then it has infinitely many solutions.
Why: The echelon form of the coefficient matrix $A$ ($m \times n$) can have at most $m$ pivot positions (one per row). Since $n > m$, there must be at least $n - m > 0$ free variables. A consistent system with at least one free variable has infinitely many solutions.
Example:
$$\begin{cases} x + y + z = 3 \\ 2x - y + z = 1 \end{cases}$$
2 equations, 3 variables ($n = 3 > m = 2$) → at least 1 free variable → infinitely many solutions.
Note: This applies to both homogeneous and nonhomogeneous systems, as long as the system is consistent.
Over a finite field with $q$ elements, the same free-variable argument gives $q^{n-r}$ solutions rather than infinitely many.
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