Rate of A: \(\frac{1}{A}\)
Rate of B: \(\frac{1}{B}\)
\(\frac{1}{A} + \frac{1}{B} = \frac{1}{8}\)
\(\frac{1}{A} = - \frac{1}{B} + \frac{1}{8}\)
\(\frac{12}{A} + \frac{6}{B} = 1\)
\(12(-\frac{1}{B} + \frac{1}{8})+ \frac{6}{B} = 1\)
\( \frac{6}{B} = \frac{3}{2}-1\)
\(B = 12\)
\(\frac{1}{A} + \frac{1}{12} = \frac{1}{8}\)
\(\frac{1}{A} = \frac{1}{24}\)
\(A = 24\)
A would paint the house in 24 days alone and B in 12 days alone.