actually america is not entirely backing up israel this time and will not have support if the IDF moves on syria.
israel is at war with the lebanese government because the lebanese government refuses to reign in the hezbolla,stating they can't.
the hezbolla have representation (23 seats) with the lebanese government,so therefore the actions of hezbolla militiamen are a reflection on the lebanese government.
israel has been patient with violations of it's nothern borders and pulled out earlier.
the solution?
the lebanese people voice it's desire to become a peaceful neighbour of israel and refuse to endorse the policies of the hezbolla (who are supported by syria and iran).