Understanding implications

First one:

Sufficiency ($\Rightarrow)$: Suppose $x/(x+1) = 0$. This expression already implies $x\neq -1$. So we can multiply it with $x+1$ to get $x = 0$. (True)

Necessity ($\Leftarrow)$: Suppose $x = 0$. It is a trivial result that $x/(x+1) = 0$. (True)

Therefore, the first one is an N-S condition.

Second one:

In set-builder notation, 'the point is on the line' if and only if $$ (a,b) \in \{(x,y) \in \mathbb{R}^2~|~y = 2x -1\} $$ and it is equivalent to say that $b = 2a - 1$ given that $(a,b) \in \mathbb{R}^2$ due to Axiom schema of specification. That is, the second one is also an N-S condition.