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# Definition of inconsistent linear system

Put content here**Definition:** A linear system is **inconsistent** if it has no solutions.
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In echelon form of the augmented matrix, an inconsistent system always contains a row of the form:
$$[0 \ 0 \ \cdots \ 0 \ | \ b] \quad \text{where } b \neq 0$$
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This represents the equation $0 = b$ with $b \neq 0$, which is impossible.
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**Example:**
$$\begin{cases} x + y = 2 \\ x + y = 5 \end{cases}$$
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Row reducing: subtract the first from the second gives $0 = 3$, which is inconsistent. Geometrically, these are parallel lines that never intersect.

# Parents

* Linear systems of equations