Solve system of equations x^2 + y^2 = 24x - 28y -340, 2y = x^2 - 24x +140 by avoiding quartic

Natural substitution leads to gnarly(Read Rachel Crowell article!) fourth degree equation in this Modified ARML 2010 Team Problem 1,T-1. American Regions Math League problems are substantively more difficult than most of the problems that are assigned homework for high school students. In fact, most high school and college math teachers would not be able to answer many of the ARML contest problems.
The key to solving this particular ARML team problem was to rewrite one of the equations where expression for 2y appeared yielding a quadratic in the y variable. It was unnecessary to state the solutions are Gaussian Integers.