Show that $x=v$ is the only solution to the equation $x+x=2v$ in a vector space. Cite all axioms used

Question

I'm not sure how to go about this question... I feel like it is not the only solution so I'm confused. Any help would be appreciated. Thanks in advance.

Answer 1

$$x+x=2x=2v\iff 2x-2v=0\iff2\cdot (x-v)=0\iff x-v=0\iff x=v$$

Answer 2

Yes, of course!

It's $$2x=2v$$ or $$\frac{1}{2}(2x)=\frac{1}{2}(2v)$$ or $$\left(\frac{1}{2}\cdot2\right)x=\left(\frac{1}{2}\cdot2\right)v$$ or $$1x=1v$$ or $$x=v.$$