Given V = $\mathbb{R}$ with the following addition and scalar multiplication:
$$x+y = x+y+7$$
$$a \cdot x=ax+7(a-1)$$
where a $\in \mathbb{R}$, while proving that V is a vector space, how come the correct way to verify axiom #5* is as follows:
$$x+y=7$$
$$x+y+7=-7$$
$$x+y=-14$$
$$y=-14-x$$
and not:
$$x+y+7=0$$
$$y=-x-7$$
Since the axiom states that $u+(-u)=0$ then why would $x+y=7$ and not $x+y=0$
* Axiom #5 states that for each u in V, there exists another vector "-u" such that $u+(-u)=(-u)+u=0$
I'm going to use $\oplus$ for vector addition. The zero vector is $-7$, since for all $x$ we have
$$x \oplus -7 = x + (-7) + 7 = x.$$
So, to find the inverse of any $x$, we need a $y$ such that $x \oplus y = -7$:
$$x \oplus y = -7\\ x + y + 7 =-7\\ y = -x - 14$$
Without distinguishing $+$ from $\oplus$, this would get very confusing! How is it actually written in your notes (or textbook), I wonder?