Clever System of Equations Problem

Algebra Level 3

Find all solutions to the system of equations below. x+y+xy=19x+y+\frac{x}{y}=19 x(x+y)y=60\frac{x(x+y)}{y}=60 Now, let (x,y)(x,y) be one solution where xx and yy are integers. Then, let (ab,cd)(\frac{a}{b},\frac{c}{d}) be the other solution where a,ba,b and c,dc,d are pairwise coprime positive integers. Find a+b+c+d+x+ya+b+c+d+x+y.

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