Does that mean that the next stage is to multiply the top and bottom of n/n+3 by 4, (giving 4n/n + 12) and the top and bottom of 7/n+4 by 3 (giving 21/n + 12)? We can then multiply both fractions by n + 12 leaving us with 4n + 21 = 1?
Am I on the right track?
No, we're not dealing with 1/3 and 1/4 in your equation. Post #5 was just an example. However you must use the same method. Perhaps it is instructive if you tell me exactly how you converted 1/3+ 1/4 into 7/12 (step by step). Then copy that method for denominators n+3 and n+4 instead of 3 and 4.
Where does n+3+n+4 come from? You don't claim that the common denominator for 1/4+1/3 is 3+4 so why would it be for the exercise at hand. The objective is to find a common denominator so you can add the two fractions together. For numbers you're doing it correctly, however n is just a number therefore n+3 is just a number and n+4 is just a number. The rules of mathematics don't suddenly change when you pick another number.
Perhaps an intermediate step. lets define a=n+3 and b=n+4. Can you add 1/a+1/b together in terms of a and b?
Gringo, the least common MULTIPLE of two numbers is a PRODUCT.
Since "n + 3" and "n + 4" are relatively prime, we must multiply them together to get their LCM. This is analogous to LCM(3,4) = 12 (NOT SEVEN!!!)
If you multiply the entire equation by the LCM/LCD, there will be no more fractions.
(But there will be a quadratic equation to solve...)