Equation operations on a linear system give an equivalent system.

Definition: Two systems of linear equations are equivalent if they have exactly the same solution set.

Equivalently, systems $A_1 x = b_1$ and $A_2 x = b_2$ are equivalent if:
$$\{x : A_1 x = b_1\} = \{x : A_2 x = b_2\}$$

Systems can be equivalent even if they look very different. For example:
$$\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases} \quad \text{and} \quad \begin{cases} x + y = 2 \end{cases}$$

are equivalent (both have the same infinite solution set).

Key fact: Systems connected by a sequence of equation operations are always equivalent.