# Equation operations on a linear system give an equivalent system.

Put content here**Definition:** Two systems of linear equations are **equivalent** if they have exactly the same solution set.
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Equivalently, systems $A_1 x = b_1$ and $A_2 x = b_2$ are equivalent if:
$$\{x : A_1 x = b_1\} = \{x : A_2 x = b_2\}$$
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Systems can be equivalent even if they look very different. For example:
$$\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases} \quad \text{and} \quad \begin{cases} x + y = 2 \end{cases}$$
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are equivalent (both have the same infinite solution set).
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**Key fact:** Systems connected by a sequence of equation operations are always equivalent.

# Parents

* Linear systems of equations