There are four unknowns and two conditions, so only one condition controls two unknowns, and then partial F and partial G control X, Y and U, V respectively. This is the most basic method to solve multivariate unknowns in multivariate differential calculus. That's all you need to understand. You can think that the number of equations corresponds to the number of unknowns, otherwise there will only be countless solutions (for example, the number of unknowns is more than the number of equations), so you can only choose a set of unknowns instead of a set of derivatives.