In my class we've been asked to complete an exercise and choose whether to use implication or equivalence arrows: "The equation $2x−4=2$ is fulfilled only when $x=3$."
I understand that we can use the implication arrow to state $2x-4=2 ⇒ x=3,$ but I am unsure as to why we cannot use the equivalence arrows: when $x=3,$ doesn't the equation $2x-4=2$ still compute?
On
"The equation $2x−4=2$ is fulfilled only when $x=3.$"
I understand that we can use the implication arrow to state $2x-4=2 ⇒ x=3,$ but I am unsure as to why we cannot use the equivalence arrows: when $x=3,$ doesn't the equation $2x-4=2$ still compute?
While both $$2x-4=2 \iff x=3$$ and $$2x-4=2 \implies x=3$$ are true statements, only the latter is an accurate translation of the quoted English-language sentence.
Analogously, "the plane takes off only when it has sufficient fuel" means $$\text{the plane takes off at time }t\implies\text{the plane has sufficient fuel at time }t$$ and does not additionally claim that $$\text{the plane takes off at time }t\;\Longleftarrow\;\text{the plane has sufficient fuel at time }t\\\text{(the plane takes off}\textbf{ whenever }\text{it has sufficient fuel)}.$$
In the original statement, "only when" can be replaced by "only if", which is half of "if and only if" and in your case that does mean $P ⇒ Q$, if $P$ represents the equation and $Q$ represents $x=3$. In this case, you can say more. It is also the case that $Q ⇒ P$, in other words
$2x-4=2$ if and only if $x=3$.
So while more is true in this case that just goes beyond what you were asked and the reverse implication may not always be true.