I have the following equation:
eq 1:
$f=100 \left(\left(\frac{z^{c/252}}{y^{b/252}}\right)^{\frac{252}{c-b}}-1\right)-100 \left(\left(\frac{y^{b/252}}{x^{a/252}}\right)^{\frac{252}{b-a}}-1\right)$
Given [a, b, c, x, y, z], solving for f is trivial, e.g.:
eq 2:
$100 \left(\left(\frac{1.1030^{1188/252}}{1.1015^{937/252}}\right)^{\frac{252}{1188-937}}-1\right)-100 \left(\left(\frac{1.1015^{937/252}}{1.0979^{687/252}}\right)^{\frac{252}{937-687}}-1\right) = -0.283604$
The problem is solving for y given [a, b, c, x, z, f].
I tried WolframAlpha and inputing the values does give me a numerical solution for y so I know it's possible. The following for example returns y $\approx$ 1.10150000054474...
eq 3:
$100 \left(\left(\frac{1.1030^{1188/252}}{y^{937/252}}\right)^{\frac{252}{1188-937}}-1\right)-100 \left(\left(\frac{y^{937/252}}{1.0979^{687/252}}\right)^{\frac{252}{937-687}}-1\right) = -0.283604$
I've read about Newton–Raphson but I'm not sure how to implement it since I can't isolate y and neither can WolframAlpha aparently. Asking it to solve for y results in a timeout.
Any ideas on an alternative method or maybe some way to isolate the y variable on eq 1?
UPDATE Limitless' comments below seem to confirm that there is no closed-form way to solve for y, so the focus is on finding an appropriate numerical method to achieve that.
UPDATE 2:
a,b,c are non-zero positive integers
x,z may be positive or negative, but not zero. Though I will accept solutions that assume they are positive only.