# Definition of consistent linear system

Put content here**Definition:** A linear system is **consistent** if it has at least one solution.
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Equivalently, the system $Ax = b$ is consistent if and only if the vector $b$ lies in the column space of $A$ (i.e., $b$ can be expressed as a linear combination of the columns of $A$).
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In echelon form of the augmented matrix, a system is consistent if and only if there is **no row** of the form $[0 \ 0 \ \cdots \ 0 \ | \ b]$ where $b \neq 0$.
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**Example:**
$$\begin{cases} x + y = 3 \\ 2x + 2y = 6 \end{cases}$$
is consistent (infinitely many solutions: the equations are multiples of each other).

# Parents

* Linear systems of equations