How would one solve or reduce an equation like this:
$\frac{exp(-(-v_{11}+x_1)^2-(-v_{12}+x_2)^2)}{\sum_i^N exp(-(-v_{i1}+x_1)^2-(-v_{i2}+x_2)^2)}=\frac{exp(-(-v_{11}+y_1)^2-(-v_{12}+y_2)^2)}{\sum_i^N exp(-(-v_{i1}+y_1)^2-(-v_{i2}+y_2)^2)}$?
When $N=2$, Mathematica is able to simplify it to
$x_1=\frac{1}{v_{11}-v_{21}}(-v_{12}x_2+v_{22}x_2+v_{11}y_1-v_{21}y_1+v_{12}y_2-v_{22}y_2)$,
but fails with any bigger N. What would be the process to simplify that, and is it possible to generalise it to higher values of N?
Thank you.