The purported reason for the penalty is that “The student made an error by not manipulating expressions independently in an algebraic proof”. It’s unclear what, if anything, this means, but there is no requirement that expressions be manipulated independently in an algebraic proof. This is an artificial criticism.
I suspect the complaint has to do with multiplying both sides of the equation by some quantity. I have occasionally heard teachers argue that, when proving an identity, you can’t multiply both sides of an equation by the same thing. Their reasons vary, but the most common explanation is that in doing so you are assuming the sides are equal, which is what you are trying to prove.
This is faulty mathematics. For the most part, there is no issue with multiplying both sides of a purported identity by the same quantity: if the original equation is true, the new equation will be true, and if the original equation is false, the new equation will be false. In general, the equations are logically equivalent, that is, true and false under exactly the same circumstances.