When should one use Lagrange Multipliers as opposed to calculating the ratio of Marginal Utilities for MRS? As far as I could see in the examples I found, they led to the same result through relatively similar processes.  I have only looked at 2 variable utility functions, so is does the use of Lagrange become evident beyond that?  I.e. U(x,y,z) and if so is that the only real use for Lagrange Multipliers?

Edit:  For clarity, I was wondering about why Lagrange Multipliers are used instead of simply calculating partial derivatives individually to find the MRS.  What MRS-related situations are Lagrange Multipliers used in?  Why are they used in those situations?
Lagrange multipliers not only incorporate constraints on a maximization/minimization problem, but the multiplier itself can have a meaningful interpretation. For example, say we have the problem:

$$\max_{x, m}: U(x, m) = m + \phi(x) \quad s.t. m + px = y$$

Where $m$ is a numeiare good, $p$ is the price of good $x$, and $y$ is income.

Now you could just rewrite the constraint as

$$m = y - px$$

and substitute that into the original maximization problem. But you can't always do this easily. So you can also solve with a Lagrangian.

$$\mathcal{L} = m + \phi{x} - \lambda(m + px - y)$$

Solving for $\lambda$ will give you the shadow price of income, or the marginal price of income. That is, it represents the rate of increase in maximized utility as income is increased. You also need to find $\lambda$ to get a nice full Marshallian demand, or to relate indirect utility to Marshallian demand in Roy's identity.

Even for problems that aren't utility maximization, $\lambda$ is still representing the rate of change of the quantity being optimized as a function of the constraint variable.

Here's a useful example with the firm maximization problem and bribing problem.