Problems with Lagrange Multipliers

[jean28]

Does anyone have any tips for solving the system of equations formed while trying to find Lagrange Multipliers? I have searched for videos online (patrickjmt and the MIT lecture on Lagrange Multipliers) but I still find it a bit confusing.

 

[HallsofIvy]

It is, as you can imagine, impossible to give anything that will work for all Lagrange multiplier problems but here is something I have found useful: since the equations can all be put into the form "$f= \lambda g$" and a value of $\lambda$ is not part of the solution, start by eliminating $\lambda$ by dividing equations.