Haskell

Pen and Paper solved these equations for me.

import Control.Arrow

import qualified Data.Char as Char
import qualified Data.List as List
import qualified Data.Maybe as Maybe


window6 :: [Int] -> [[Int]]
window6 [] = []
window6 is = List.splitAt 6 
        >>> second window6
        >>> uncurry (:)
        $ is

parse :: String -> [[Int]]
parse s = window6 . map read . words . List.filter ((Char.isDigit &&& Char.isSpace) >>> uncurry (||)) $ s

solveEquation (ax:ay:bx:by:tx:ty:[]) transformT
        | (aNum `mod` aDenom) /= 0   = Nothing
        | (bNum `mod` bDenom) /= 0   = Nothing
        | otherwise                  = Just (abs $ aNum `div` aDenom, abs $ bNum `div` bDenom)
        where
                tx' = transformT tx
                ty' = transformT ty
                aNum   = (bx*ty')  - (by*tx')
                aDenom = (ax*by)   - (bx*ay)
                bNum   = (ax*ty')  - (ay*tx')
                bDenom = (ax*by)   - (bx*ay)

part1 = map (flip solveEquation id)
        >>> Maybe.catMaybes
        >>> map (first (*3))
        >>> map (uncurry (+))
        >>> sum
part2 = map (flip solveEquation (+ 10000000000000))
        >>> Maybe.catMaybes
        >>> map (first (*3))
        >>> map (uncurry (+))
        >>> sum

main = getContents
        >>= print
        . (part1 &&& part2)
        . parse

(Edit: coding style)