The final answer works out, but we shouldn't have #y=y_0=0# in our equation (at least not in good practice).
It’s true that we can define #y_0=0#, but then #y# must be the maximum altitude of the projectile, which is unknown; we have to consider the projectile's rise and fall separately (could have different launch and landing altitudes).
When we have the same launch and land altitudes, we can find the flight time for the rise or fall and simply multiply by two—but this requires us to have the maximum altitude in either case to use the kinematic equation which was applied here to find #Deltat#. It would be better to use #v_f=v_i+aDeltat# for the perpendicular components and then using twice this value for time, find the projectile’s range using the parallel components with the second equation which was used.
The final answer works out because it's as if you added the equations describing the rise and fall together: #y_"max"-y_"min"+(y_"min"-y_"max")=v_"iy"Deltat+0+2(1/2a_y(Deltat)^2)#, which then reduces to