Problem
Suppose $v, w \in V$. Explain why there exists a unique $x \in V$ such that $v + 3x = w$
Solution
Suppose that $x,x^\prime \in V$ satisfies $v + 3x = w = v + 3x^\prime$, then adding the additive inverse of $v$ to both sides will reduce the equality to $3x = 3x^\prime$, dividing by $3$ on both sides results in $x = x^\prime$, thus $x$ is unique.
Now, let $x = \frac{1}{3}(w - v)$, then: $$v + 3x = v + 3\left(\frac{1}{3}(w - v)\right) \stackrel{P1A13}{=} v + \left(3 \cdot \frac{1}{3}\right)(w - v) = v + 1(w - v) \stackrel{D1.19}{=} w$$ Proving existence of such an $x$.