Since $10x+y=75$ and $10y+x=57$, then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$. Dividing by 11, we get $x+y=12$. (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$.)