Is it possible to solve and get x and y using luau?
m, n, r, and S are all known number constants
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If you can somehow get your set of equations down to linear equations you can solve them using matrix operations like this page does:
http://www.math.odu.edu/~bogacki/lat/
It’s not an API but it is nice in that it shows each operation done to reach a solution. It helped me with some linear algebra homework in the past 
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I’m pretty sure the second equation is impossible to be converted to a linear equation because the terms are squared.
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Can you provide one set of constants for the two equations? It will make solving and checking equations a lot easier.
I forgot graphing calculators where a thing…
I converted the equations to lua code here:
local x1 = (mS - math.sqrt(n^2(m^2 * r^2 + n^2 * r^2 - S^2)))/(m^2 + n^2)
local y1 = (n^2 * S + m * math.sqrt(n^2 * (m^2 * r^2 + n^2 * r^2 - S^2)))/(m^2 * n + n^3)
local x2 = (mS + math.sqrt(n^2(m^2 * r^2 + n^2 * r^2 - S^2)))/(m^2 + n^2)
local y2 = (n^2 * S - m * math.sqrt(n^2 * (m^2 * r^2 + n^2 * r^2 - S^2)))/(m^2 * n + n^3)
thx dawg that works, thanks to @SevenRingsOfSaturn too