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I was reading this article and everything made sense until the equivilant circuit for the rotor was divided by the slip s. I understand that the current in the circuit would remain the same, but it seems that both the power delivered and how its distributed would fundamentally change. Without dividing by s, the power in being delivered would be $$sEI$$and we wouldn't be able to rewrite $$R/s$$ as $$R+\frac{(1-s)}{s}R$$

Since the resistor would remain as R. Why do we divide by slip and what makes the two situations equivalent?

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R/s accounts for the change in rotor current and power with different values of slip. The power dissipated in R/s is the electrical power converted to heat in the rotor plus the electrical power converted to mechanical power. Separating R/s into R and R(1-s)/S separates the rotor losses from the mechanical power. You should probably find some other presentations that are similar to the one you linked. One of them may explain in a way that is easier for you to understand.

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