This is driving me insane, I must be doing the arithmetic wrong. I just have to solve the equation (1), but I don't think I get the right answer
You say that you're "solving an equation", but there is no variable. In what you've posted, one line does not follow from the previous (such as where denominators disappear and then later reappear).
Are the various lines different exercises and you need to "simplify"...? Assuming so, I think the following is what you did, with the missing steps filled in:
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
But I'm afraid I can't figure out how you arrived at your later lines...?