the summing amplifier
The summing amplifier is an inverting amplifier with several input resistors all feeding the one virtual-ground node. Because that node holds steady at 0 V and sinks each input's current independently, the currents simply add up by Kirchhoff's law, and the output becomes a weighted sum of all the inputs at once.
Walk it through. The minus node is a virtual ground at 0 V, so each input contributes a current Ii equals Vi over Ri. None of that current enters the input pin, so the total, the sum of all the Ii, flows through the feedback resistor Rf, giving Vout equals minus Rf times (V1 over R1 plus V2 over R2 plus and so on). If every input resistor is the same value R, this simplifies to Vout equals minus (Rf over R) times (V1 plus V2 plus ...). For example, all resistors 10 kilohm and inputs of 1 V, 2 V, and 3 V give an output of minus 6 V.
This is how an audio mixer combines channels and how a binary-weighted digital-to-analog converter sums bits. The key reason it works cleanly is the virtual ground: each input sees only its own resistor into a fixed 0 V, so the channels do not interact or crosstalk. Give a channel a different resistor value and you give it a different weight, which is exactly how a DAC scales each bit by a power of two.
An audio mixer sums three line inputs through three 10 kilohm resistors into a virtual ground, with a 10 kilohm feedback resistor. The output is the inverted sum of the three signals, mixed without one channel pulling on another.
The virtual ground keeps the inputs from interacting, so signals add cleanly.
It works because the summing node is a virtual ground that keeps the inputs independent. Lose the virtual ground (open-loop) and the inputs fight each other.