Let  "a"  be the rate of work of A  (part of work he does per day)

let  "b"  be the rate of work of B  (part of work he does per day).


Then from the condition you have these 2 equations


 8a + 8b = 1     (1)    ("When A and B both work, they can paint a certain house in 8 days.")
12a + 6b = 1     (2)    ("Also, they could paint this house if A worked 12 days and B worked 6 days.)


Divide equation (1) by 8 (both sides).  Divide equation (2) by 12 (both sides). You will get


a +    b =      (1')
a + 0.5b =     (2')


Now subtract eq(2') from eq(1'). You will get

b - 0.5b =  -  =  -  = ,


0.5b =   ====>  b =  = .


Then from eq(1')  a =  -  =  = .


Answer.  "A" needs 24 days to complete the job working alone.

         "B"  needs 12 days to complete the job working alone.


Check.  Equation 1:   +  =  +  = 1.   ! Correct !


        Equation 2:   +  =  +  = 1.   ! Corret !