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# A consistent system with more variables than equations has infinitely many solutions.

Put content here**Theorem:** If a consistent linear system has more variables than equations ($n > m$), then it has infinitely many solutions.
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**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.
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**Example:**
$$\begin{cases} x + y + z = 3 \\ 2x - y + z = 1 \end{cases}$$
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2 equations, 3 variables ($n = 3 > m = 2$) → at least 1 free variable → infinitely many solutions.
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**Note:** This applies to both homogeneous and nonhomogeneous systems, as long as the system is consistent.

# Parents

* The number of solutions to a linear system