If $x$ and $y$ are non-zero numbers such that $x=1+\frac{1}{y}$ and $y=1+\frac{1}{x}$, then $y$ equals
$\text{(A) } x-1\quad \text{(B) } 1-x\quad \text{(C) } 1+x\quad \text{(D) } -x\quad \text{(E) } x$
We see after multiplying the first equation by $y$, that 
$xy=y+1.$
Similarly, we see that after multiplying the second equation by $x$, we get that 
$xy=x+1.$
Thus $x+1=y+1 \implies x=y$, giving us our final answer of $\fbox{E}.$
~SirAppel