these equations can only be applied for subsonic speeds to get an exact answer
Unless a real gas model is constructed, including quantum-mechanical effects, the solution will be an approximation regardless of the regime. Within the static temperature limits of the troposphere/stratosphere (in ISA) the assumption of a calorically perfect gas is justified up to, shall we say, around Mach 2 or 3 or higher.
And this gives you the "q-factor" or does it give you the stagnation temperature? I don't see any temperature figures in there...
A correction term, in whatever form is chosen, can be included to account for discrepancies between expected values and those measured in practice. The Rayleigh supersonic pitot equation given above by Blackmail does not include any.
BTW: How do you determine total air-pressure? Can I just set it as an "x-value" and then fill in the rest, then solve for it?
Leave the Mach number as the independent variable to avoid iteration. Use the post-shock total pressure and pre-shock static pressure in the Rayleigh equation. A full derivation is given in Anderson (Fundamentals of Aerodynamics) which PBL has previously advised you to consult. Alternatively look up P. Balachandran (Fundamentals Of Compressible Fluid Dynamics), chapter 9, on Google Books. For the hypersonic regime refer to Anderson, J. (2000).
Hypersonic and High Temperature Gas Dynamics. AIAA, or Fletcher, D.G. (2004).
Fundamentals of Hypersonic Flow-Aerothermodynamics. NATO RTO AVT Lecture Series:
RTO-EN-AVT-116.