This loaded-node-feeds-sample lesson makes the sampler capture the node voltage that remains after the source resistance and input load divide the source. The 12 V source with 3 ohm internal resistance and 9 ohm input load gives a 9 V held sample only for the stated ideal divider and closed-switch row. Real front ends add source impedance variation, sampling capacitance, switch resistance, leakage, settling time, noise, finite precision, and measurement uncertainty.
A front-end sampler captures the node voltage left after the source is loaded. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The input load sets the node voltage
A 12 volt source with 3 ohms inside and 9 ohms at the input gives 9 volts at the node.
12V/(3ohm+9ohm)=1A
The sampler captures the loaded node
With the sample switch closed, the held voltage is the loaded 9 volt node.