In Hoffman and Kuntze's Linear Algebra, it is stated that given $2$ systems of equations $A$ and $B$ where each equation of $B$ is a linear combination of the equations of $A$, then "then every solution of $A$ is a solution of $B$. Of course it may happen that some solutions of $B$ are not solutions of $A$. This clearly does not happen if each equation in $A$ is a linear combination of the equations of $B$."
Now, my question: Is the only reason that $B$ might contain more solutions than $A$ because each equation in $B$ might have a $0$ for some specific equation of $A$? And thus could we equally well say that $A$ and $B$ have have exactly the same solution set if $B$ is a set of equations all of which are linear combinations of $A$ and such that every equation in $A$ is multiplied by a non-zero scalar in at least one of the equations of $B$?