Lagrange Multipliers

I do not have one specific question that needs answering. Rather, it is about Lagrange multipliers in general.

So for certain minimization/maximization questions (ie find the shortest distance from some point to some plane) it seems that one could solve the question using lagrange multipliers and not using lagrange multipliers (I have done it both ways). One could set both partials equal to 0 and solve for x and y without using lagrange (as long as a function of x and y is substitued for z). Or, one could not substitute for z and use lagrange multipliers to find the distange.

Is this right? Or am I missing something? Also, is it ever 100% necessary to use lagrange multipliers? How can one tell when it is better to use lagrange multipliers and when it is better not to?

Thanks
DR13

Ok, you do the same steps required by Lagrange multipliers, but you don't call it Lagrange multipliers. :)

Using Lagrange is the only way for constrained maximum/minimum.

Oh I get it. By substituting for z before taking the partials you are just doing the step earlier rather than later. Is that right?

I'm not sure I follow you completely.
Do you have an example ?

Its fine. I worked it out myself and get it now. Thanks!