Subtract team rates to isolate one worker

Working at a steady rate, the work done is the rate times the days, so a team that finishes the whole job in $6$ days does $\frac{1}{6}$ of it each day.

When people work side by side, the team's daily work is just each person's share added together, so $x+y+z=\frac{1}{6}$ and $y+z=\frac{1}{12}$.

Taking $y+z$ away from $x+y+z$ undoes adding it and leaves $x$, so $x=\frac{1}{6}-\frac{1}{12}$.