How does a system like Wolfram Alpha or Mathematica solve equations?

I dealt with such questions myself some time ago. I then found this document about simplification of expressions. It is titled Rule-based Simplification of Expressions and shows some details about simplification in Mupad, which later became a part of Matlab.

According to this document, your assumption is correct. There is a set of rules for manipulation of expressions. A heuristic quality metric is is used as a target function for simplification.

Wolfram alpha is developed by Mathematica

• mathematica is stephen wolphram's brainchild. Mathematica 1.0 was released in 1988. mathematica is much like maple and they both rely heavily on older software libraries like LaPack.
• The libraries that these programs are, based on, and often simply, legacy software. They've been around, and modified, for a very long time.

If you would like to know about the background programs running, sagemath is a free open source alternative; you could possible reverse engineer the solutions to your questions:

SageMath.org