Strength: It's not math. It's a puzzle.
Weakness: Dealing with negatives is a real pain in the butt.
Guess and Check
I actually really like this method. Guess and check is probably my most under-used problem solving strategy, but using it to solve equations has been really helpful. I've noticed a greater understanding of rate(s) of change, using information from wrong answers to help find right ones and checking answers--something most kids don't want to do--is embedded in the process.
We've gotten to the point where we can nail the answer on the third guess by using the information gained in the first two--even for equations with non-integer solutions.
Strength: Students understand that simplified expressions on each side of the equal sign end up looking the same every time (ax + b = cx + d). Rate of change is very useful. Being wrong helps you to become right. Did I mention they are checking their answers?
Weakness: Leave 'em in the comments.
From Construction to Deconstruction
We spend a lot of time teaching kids how to break things down whether it's reducing fractions, simplifying radicals or solving equations but we rarely (read: I rarely) have taught them how to construct things that may eventually need to be deconstructed.
Constructing a more complicated equation from a simple equation has helped my students understand that, no kidding, the two expressions on opposite sides of the equal sign are equivalent. I've used the just-unwrap-the-present illustration many times, but we really need to teach the students to wrap one up first. Having them list their steps for construction makes the process for actually solving the equation seem much more natural. When I say, "just use the inverse order of operations" --or whatever completely abstract thing I've been known to throw out there in order to make myself feel better when they keep screwing it up--it makes no sense to them. This helps.
Strength: Kids get a grasp of which operation to tackle first while solving for x.
Weakness: Very complicated equations with variables on both sides don't seem so natural when you begin with x = 2.
I've heard rumors that there are some teachers who actually teach solving equations by graphing. Never seen it in the wild, though.