Relationship Between Specific Impulse and Expansion Ratio Equation

I am trying to find an equation with both Specific Impulse and Expansion Ratio (throat area is constant with exit area changing) of a rocket. This assumes the following constants (all of which I know): Local speed of sound, gamma, chamber pressure, mass flow rate, throat area, ambient pressure. The problem I kept running into, was creating a function of the exit pressure in respect to the expansion ratio. In other words, I tried my best to isolate the exit pressure in the following equation – which proved to be challenging.

If I was to have the exit pressure isolated, I can plug it into this equation that has specific impulse to create this function.

Any help would be greatly appreciated!