the golden rules
Two short rules turn scary-looking op-amp circuits into one line of algebra. Think of the op-amp as a tireless thermostat: it keeps nudging its output until the room (the two inputs) reaches the setpoint. Once you trust that behavior, you barely have to think about the chip at all, just the resistors around it.
Rule 1: no current flows into either input, because the ideal input resistance is infinite. Rule 2: the output does whatever it must, through the feedback network, to make the two inputs equal, V+ equals V-. To solve any op-amp circuit, write down V+ equals V-, declare the input currents zero, and then just apply Ohm's law and Kirchhoff's current law to the resistors. The whole circuit collapses into simple node equations.
This is the heart of the entire field. Combined with the resistor laws you already know, the golden rules let you derive every standard op-amp circuit in two lines. The honest caveat is that Rule 2 is not a law of nature, it is something negative feedback achieves. With positive feedback, or open-loop as a comparator, or once the output hits a rail or the device runs out of bandwidth or slew rate, Rule 2 is simply false.
Inverting amp: the plus input is grounded, so Rule 2 says the minus input is also at 0 V. Rule 1 says no current enters that input, so the input current must equal the feedback current. Equate them and you get Vout over Vin equals minus Rf over Rin, in a single step.
Two rules plus Ohm's law solve the canonical circuit in one move.
Rule 2 (the inputs are equal) is not a law of nature, it is what negative feedback produces. With positive feedback or open-loop it is false.