If you apply general relativity, you get the Friedmann equations, which govern the expansion of the universe.
http://en.wikipedia.org/wiki/Friedmann_equations
Using the equation, you can find an expression for the age of the universe that depends on the current day density of matter, vacuum energy, and rate of expansion (Hubble constant). These quantities can all, in principle, be measured in a number of ways. Plug them into the equation, and it gives you 13.7 billion years.