Opamp’s (ideal) are simple…
Before going to the "standard equations", consider the following.
1) They have zero output impedance
2) They have infinite input impedance
3) Both inputs want to be at the same potential
4) Whichever input is “higher” wins and the output will swing in that direction.
5) The output will do whatever it can to make both inputs equal
Apply the above to a simple inverting amplifier. As an example, take a circuit where the input resistor to the “-“ input is 1K from a 1V source, the “+” input is tied to gnd and a 10K resistor is connected between the “-“ input and the output.
According to #3, both inputs want to be the same… since the “+” input is tied to gnd, the “-“ input will want to be at gnd potential. So you have a 1V source connected to a 1K resistor which is connected to the “-“ input which wants to be at gnd potential (virtual ground), so that’s 1V/1K=1mA. Per #2, since the input has infinite input impedance, no current can flow into or out of an input, so the 1mA of current has to go somewhere… it goes thru the 10K “feedback” resistor to the output. Well, if 1mA flows thru 10K, that would be a 10V drop. If the “-“ input is at gnd potential and you have 10V across the 10K and current flows from a higher potential to a lower potential, that means that the output has to be at -10V for all the “rules” above to be true. You will find that the “rules” will apply to any opamp circuit… try applying them to a non-inverting circuit to convince yourself that they work. Now you know where the equations came from and have a better understanding of how opamps work and how to apply them.