Let x = (x, y ,z). Just to be sure you get the methodology of using Lagrange multipliers, the general theorem is that if you are studying values of a function f(x) restricted to a level curve g(x) = C, we note that when a value x on g is an extremum for f parametrized over the level curve, the gradient of g at x is parallel to the gradient of f at x. Mathematically, we say
where is some scalar. So the value of is not actually important, it is just a way of saying the gradients are parallel.
Rearranging the equation above gets you
which is the form you seem to be familiar with. Can you work out what you need to do from here to find x?