That system has 8 solutions. None of the solutions are real-valued in even one of the components (but y might possibly have a purely-imaginary solution). Are you expecting real-valued solutions?

Your system of equations is very sensitive to the precision of your 0.046328 and your .65116 . Are we to take it that .65116 is 65116/100000 exactly, or are we to understand it as 65116/100000 +/- 5/1000000 ? If we say that the "real" value is 65116/100000 + delta2 for some unknown delta2 in the range +/- 5/1000000, then the solution to the equations involves the root of an expression involving delta2^20 multiplied by large numbers.

I have done a bunch more investigating. If we take the values as being approximate, then it is possible to determine some of the conditions under which real-valued solutions can be found. If we call the error on the 0.046328 as "delta1" and the error on the .65116 as "delta2", then:

If we assume that delta2 is close to 0, then delta1 needs to be at least +5.8114 -- very large in comparison, making it unlikely that the .65116 is fundamentally correct.

If we assume that delta1 is close to 0, then delta2 needs to be at least +0.6595228062 or -0.7587797008 (actually that's a double root, so safer would be -2.177082208). Adding +0.6595228062 to .65116 is slightly more than a factor of 2, suggesting perhaps a relatively minor miscalculation in determining it.

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