How would you find the solution to the following equation?

First, evaluate x in the first equation with y = 1 on the left, call this x_1. Then evaluate y in the second equation with x_1, call this y_1. Iterate over this step and pray for its convergence.

With initial value of y=0.05, 1, it converges to the same solution, but with y = 2, it converges to a different solution.

y = 1

1.8393972058572117 1.112392324266727
1.643857457460797 1.3526324971948598
1.2927935443605563 1.9215201252661318
0.7319213524275276 3.366887734134954
0.17248416881620013 5.891001282468265
0.013821037791172457 6.9039182397468535
0.005019222050473703 6.9649534723748445
0.004722034378070397 6.967023678290299
0.004712268906324325 6.967091714895387
0.004711948310451926 6.967093948516591
0.004711937785756024 6.9670940218431365
0.004711937440245915 6.967094024250339
0.004711937428903327 6.967094024329364

y =2 


0.6766764161830634 3.5581250432278053
0.14246098150124253 6.070549739162265
0.011549515086734801 6.919618471743886
0.004941034487355336 6.965498066404045
0.004719463486452072 6.967041589776102
0.004712184503342602 6.967092302938727
0.004711945539622921 6.967093967821218
0.004711937694793822 6.96709402247688

y = 0.05


4.756147122503569 0.06019072709803884
4.707924653479178 0.06316439515230665
4.693945643104534 0.06405357130916164
4.689773753604319 0.06432135392210456
4.6885180818655 0.06440217115776774
4.688139184105709 0.06442657761962998
4.6880247646118 0.06443394969778218
4.6879902042542465 0.06443617659660332
4.687979764586011 0.06443684929242072
4.687976611002692 0.06443705249971418

This is to a simple example that helps understanding the underlying mechanism of Self-consistent field approximation.