Logic implication with first degree equation and a sentence My sister got a great math test back but had some errors with the logic equivalences. For example: $x - 17 = 2 \iff x = 2 + 17 \iff x = 19$ Now, the teacher took some points for forgetting the last equivalence. That's correct. But another question: Search the number which subtracted with $23$, equals $-8$ She wrote: $\iff$ $x - 23 = -8 \iff x = -8 + 23 \iff x = 15$ He did the same thing to take some points for forgetting the last equivalence. But he also took points for adding the first equivalence. In my eyes, the equation she wrote and the questions, implicate each other so that quivalence is not wrong. What are you thoughts on this one?

Note that if this is what your sister wrote for the second answer:

$$\iff x - 23 = -8 \iff x = -8 + 23 \iff x = 15$$

...then indeed, the instructor look off points for having used the first "$\iff$" symbol at the start of the string of equivalences, since it is meaningless, and at worse states that $$\text{nothing} \iff x - 23 = -8$$

An equation is true or false, depending on $x$ in this case. The left hand side ("nothing") is neither true nor false. The right hand side is either true or false, depending on $x$. The two cannot be equivalent.