Find the point of intersection of the line
\[\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}\]and $x - y + z = 5.$
Let
\[t = \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}.\]Then $x = 3t + 2,$ $y = 4t - 1,$ and $z = 12t + 2.$  Substituting into $x - y + z = 5$ gives us
\[(3t + 2) - (4t - 1) + (12t + 2) = 5.\]Solving, we find $t = 0.$  Thus, $(x,y,z) = \boxed{(2,-1,2)}.$