You correctly multiply each numerator by what's needed to get a common denominator, but then you ignore the common denominator! Since the RHS is not zero, your denominator matters in this problem.
Here's a useful trick: when solving algebraic equations, you never need to check an answer key, because it's quite easy to check the answer yourself. All you have to do is plug your answer into the original equation, and see if it satisfies the equality. So what do you get when you plug your answer into the original equation?