As you said, set both equations equal to 0. Then take one of the equations and solve it for a variable. For instance, you could say (1.) . Then plug this in for y in the second equation and you get . Solve this equation for x and you get . Plugging in to (1.), you get and .
When the system involves partial derivatives it seems to make people hesitate, but you can essentially just treat and like you would any other set of two equations involving two independent variables. It's just the interpretation of the results that makes the system special.