One of the primary advantages of an opamp is that its output is not affected by noise or changes on the supply pins. The spec to look for on the data sheet is the PSRR Power Supply Rejection Ratio. For a good opamp, that number is > 90db, meaning if the 9V positive supply rail suddenly jumps by one volt from 9V to 10V, the output of the opamp will change by less than 30uV PROVIDED THAT none of the opamp's inputs are referenced to the positive rail!!!
Compare the outputs of these two gain of -5 amplifiers. The intrinsic PSRR for the LT1001 is over 100db. Look what happens if you do the dumb thing of referencing the non-inverting input to the positive supply rail, and then let that rail jump by one volt!