vd + vc = 1 ( i think this is what you were trying to help me understand and get)
More or less.
One thing is slightly wrong in your prof approach IMHO. Volumes are not additive, so you can't assume
2.60% (vd) + 8.50% (vc) = 4.85% 1 gal vd + vc = 1
are right at the same time. Final volume of mixture is a function of initial concentrations. Final differences are usually pretty small, but they do exist.
In this particular case you need to mix 620.7 mL of diluted (2.60%) acid with 379.4 mL of concentrated one (8.50) to get 1000 mL of 4.85% solution. Sum of volumes is 620.7mL+379.4mL = 1000.1mL, but there is a small volume contraction when mixing and the final volume will be exactly 1000 mL. Difference is small, but you don't know how large it is beforehand. On teh other hand, if you will use masses of the solutions instead of volumes your final result will be exact. After you know masses you have to convert them to volumes using densities taken from the density tables.