To minimize F(x,y,z), subject to G(x,y,z)= constant, use \(\displaystyle \nabla F= \lambda \nabla G\) where \(\displaystyle \lambda\) is the "Lagrange multiplier". Setting components equal gives three equations in the four unknowns, x, y, z, and \(\displaystyle \lambda\). The constraint G(x,y,z)= constant is a fourth equation.

Since the value of \(\displaystyle \lambda\) is not part of the solution, I find that **dividing** one equation by another to eliminate \(\displaystyle \lambda\) is often a good first step.