A.) The value of the floating leg of an IRS is only the payment at the next settlement date + notional amount discounted back to today.

2.) The value of the fixed leg, however, is the present value of all of the fixed payments + the notional amount.

In solving for the fixed rate that would set the value of a swap at origination to zero, why do we equate A & B when A only uses a singular payment whereas B uses the aggregate?

At inception (t = 0), we set the PV of fixed leg equals to the PV of the floating leg.

PV of fixed leg is the PV of all fixed payments and notional amount.

PV of floating leg is the PV of all floating payments and notional amount. The PV of the floating leg at inception is equals to the swap notional.

For example, if you have a 1-year swap with semiannual payments, referenced to 6-month LIBOR, and given that swap notional = $100 and the spot rate and forward rates:

The assumption is that the floating rate will reset to the market rate, and that the market rate is the appropriate discount rate for the swap payments. Given those assumptions, the value of the floating-rate bond used in the swap equation will reset to par at the next coupon date. That (par) value is the present value of all future coupon plus principal payments, so the value of the floating leg does, in fact, include the aggregate of all payments; it simply does so sneakily.

@S2000magician Thanks so much for the response magician! I went through some mental gymnastics and I can see how par value actually captures the present value of all future coupon + principal payments!