Confused about proving series and parallel resistance equations
You need to set up conditions that mean series and parallel, I will do a bit of the series proof, you can do the parallel roughly in the same manner.







simulate this circuit &ndash; Schematic created using CircuitLab

so your first equation from KVL and Ohm's law(Your sign convention can be the opposite doesn't matter as long as it is consistent)

$$
IR1 +IR2 + IR3 - V = 0
$$
$$
I(R1+R2+R3)  = V
$$
$$
I(Rtotal) = V
$$
$$
Rtotal= R1+R2+R3
$$

this means you can add resistances together to form an equivalent resistance if the same physical current flows through them and their voltages are in the same voltage loop.

You can repeat this same line of thinking with the parallel connection and using KCL and you will run into the formula for the equivalent of many resistors connected together in parallel.