Haskell, 14 ms. The hardest part was the parser today. I somehow thought that the buttons could have negative values in X or Y too, so it's a bit overcomplicated.

import Text.ParserCombinators.ReadP

int, signedInt :: ReadP Int
int = read <$> (many1 $ choice $ map char ['0' .. '9'])
signedInt = ($) <$> choice [id <$ char '+', negate <$ char '-'] <*> int

machine :: ReadP ((Int, Int), (Int, Int), (Int, Int))
machine = do
    string "Button A: X"
    xa <- signedInt
    string ", Y"
    ya <- signedInt
    string "\nButton B: X"
    xb <- signedInt
    string ", Y"
    yb <- signedInt
    string "\nPrize: X="
    x0 <- int
    string ", Y="
    y0 <- int
    return ((xa, ya), (xb, yb), (x0, y0))

machines :: ReadP [((Int, Int), (Int, Int), (Int, Int))]
machines = sepBy machine (string "\n\n")

calc :: ((Int, Int), (Int, Int), (Int, Int)) -> Maybe (Int, Int)
calc ((ax, ay), (bx, by), (x0, y0)) = case
        ( (x0 * by - y0 * bx) `divMod` (ax * by - ay * bx)
        , (x0 * ay - y0 * ax) `divMod` (bx * ay - by * ax)
        ) of
    ((a, 0), (b, 0)) -> Just (a, b)
    _                -> Nothing

enlarge :: (a, b, (Int, Int)) -> (a, b, (Int, Int))
enlarge (u, v, (x0, y0)) = (u, v, (10000000000000 + x0, 10000000000000 + y0))

solve :: [((Int, Int), (Int, Int), (Int, Int))] -> Int
solve ts = sum
    [ 3 * a + b
    | Just (a, b) <- map calc ts
    ]

main :: IO ()
main = do
    ts <- fst . last . readP_to_S machines <$> getContents
    mapM_ (print . solve) [ts, map enlarge ts]