Consider the equation $\frac{1}{4}x=3$. If you multiply both sides by $0$ to eliminate the fraction, does this operation preserve the original solution set? Justify your answer.

Yes. Multiplying by zero is a valid algebraic operation as long as you later perform the inverse operation of dividing by zero.

No; multiplying by zero transforms the equation into the identity $0=0$, which incorrectly suggests all values of $x$ are solutions.

Yes. The operation is valid because $0=0$ is a true statement, confirming that the equality remains balanced.

No. Multiplying by zero is only permitted if the variable $x$ is already known to be zero; otherwise, the equation becomes a contradiction.