Motivation for the definition of a group

We know that $x+1=2$ implies $x=1$. When solving such an equation, we are actually working with the properties of a group. $$\begin{align}x+1&=2\quad &\text{integers under +}\\x+1+(-1)&=2+(-1)\quad &\text{inverses}\\x+1+(-1)&=1\quad &\text{closure under +}\\x+[1+(-1)]&=1\quad &\text{associativity}\\x+0&=1\quad &\text{identity}\\x&=1\end{align}$$ As it turns out, the definition of a group is the simplest definition that will let you solve a basic equation. How amazing is that!