Can relationship between two variables in maxima and minima be solved using partial derivatives?

Question

In obtaining the maxima and minima of the volume of solids, it is common to express the equation in terms of one variable and then apply derivatives. Like for example, a rectangular prism with volume $ V = x^2 y $ and $ SA = 2x^2 + 4xy $ has its volume and surface area constant. When manipulating to form a single equation in terms of one variable and involving both constants, the relation $ x = y $ is produced. I was wondering if in maxima and minima, there is an alternative solution solving the relationship between two variables that avoids substitution. Like for example, partial derivatives are used with the volume and surface area equations I mentioned.