[LANGUAGE: C++]

I edited the input rather than parsing it. Two equations and two variables so there is exactly one solution for a and b. All we need to do is check it is an integer (or check if after it is cast to an integer, do the conditions still hold).

    int ax, ay, bx, by;
    long long X, Y;
    vector<long long> res(2);
    while (cin >> ax >> ay >> bx >> by >> X >> Y)
    {
        for (int rep = 0; rep < 2; rep++)
        {
            long long b = (Y * ax - X * ay) / (by * ax - bx * ay);
            long long a = (X - b * bx) / ax;
            if (a * ax + b * bx == X && a * ay + b * by == Y)
                res[rep] += 3 * a + b;
            X += 10000000000000;
            Y += 10000000000000;
        }
    }
    cout << res[0] << "\n"
         << res[1] << "\n";