I have the following expression $$z=\alpha-\beta q$$ where $\alpha$, $\beta$ are known parameters and $q=1-F_X(z)$, where $X\sim N(\mu, \sigma^2)$ [*footnote1]. I want to solve for $z$ and have struggled to isolate it.
I have tried playing with the algebra of some known approximations of $\Phi$ (e.g. wikipedia or these), with no luck. Although I would prefer it, I don't need an exact answer, it would be enough to have a good approximation (e.g. to a couple decimal places $z \approx$...).
For an approximation, I unfortunately cannot assume that z is small.
Thanks for your time!
[*footnote1]: with $\mu= c(b-a)+da, \sigma^2 =e((b-a)^2+a^2)$, a combination of some known parameters