(x+3a)/(x-2a)=(ax-1)/(4x-8a) Solve for x?

(x + 3a) / (x - 2a) = (ax - 1) / (4x - 8a)

Look at the denominator on the right side of the equation:
(4x - 8a) = 4(x - 2a)
which just so happens to be four times the denominator of the left side:

(x + 3a) / (x - 2a) = (ax - 1) / 4(x - 2a) ## multiply by 4(x - 2a)
4(x + 3a) (x - 2a) / (x - 2a) = ax -1 ## (x - 2a) / (x - 2a) just cancels out
4(x + 3a) = ax -1 ## expand the left side and add 1 to the equation
4x + 12a + 1 = ax ## subtract 4x
12a + 1 = ax - 4x ## factorize the right side
12a + 1 = (a - 4) x ## divide by (a - 4)
(12a + 1) / (a - 4) = x

It looks nicer the other way round:

x = (12a + 1) / (a - 4)