From Hoffman and Kunze, Linear Algebra, page 7, Theorem 3:
If $A$ and $B$ are row-equivalent $m\times{}n$ matrices, the homogeneous systems of linear equations $AX=0$ and $BX=0$ have exactly the same solutions.
Proof: Suppose we pass from $A$ to $B$ by a finite sequence of elementary row operations: $A=A_0\to{}A_1\to{}\ldots{}\to{}A_{k}=B$. It is enough to prove that the systems $A_jX=0$ and $A_{j+1}X=0$ have the same solutions, i.e., that one elementary row operation does not disturb the set of solutions.
So suppose that $B$ is obtained from $A$ by a single elementary row operation. No matter which of the three types the operation is (row exchange, multiplication of a row by a non-zero scalar $c$, or row replacement), each equation in the system $BX=0$ will be a linear combination of the equations in the system $AX=0$. Since the inverse of an elementary row operation is an elementary row operation, each equation in $AX=0$ will also be a linear combination of the equations in $BX=0$. Hence these two systems are equivalent, and by Theorem 1 they have the same solutions.
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Note:the authors define "equivalent" systems of linear equations as those for which all equations of the first system are linear combinations of the equations in the second system, and vice versa.
Question: why restrict ourselves to homogeneous systems? I see no reason to do this. I think the result applies perfectly well to non-homogeneous systems.