Algorithms to solve for an arbitrary multivariate equation?

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I am wondering what the algorithm or approach is to find the unique solution for a single arbitrary multivariable equation? For example what approach does wolfram alpha or matlab use when I enter in an equation with unique solutions (composed of non-elementary functions) for them to solve?

Also can someone tell me the official name of this type of problem as I couldn't find anything online when I googled "equation solver algorithm"

It seems like this problem can be cast into a nonlinear programming problem, but I'm not sure if these techniques are what's used.