I don't think you made an error, those are the same results I got.

Its a methodology thing. Since, as nehmer pointed out, there is no correct answer. It is a line, and all points on that line satisfy the equation.

We solve for y, and graph it.

It can be graphed using y = -8/22x + 1818.18, as you solved.

Then its a simple matter of finding a data point within acceptable reasonable ranges. This can easily be done on any graphing calculator, but apparently nothing on the internets can do it. We can also "brute force" it.

Say we wanted to play 1000 hands on the weekends, exactly how many hands do we need to play on weekdays?

well, x=1000.
8x+22y=40000
8000+22y=40000
y=1454.54

so the point 1000,1454.54 satisfys the equation.

FWIW, I think these are the exact ranges I want, but lets experiment.

let y=1400 -- 1400 hands during weekdays
8x+22(1400)=40000
solve for x, x=1150 hands during weekends.

Actually, those might be the ideal numbers. Guess I might as well try y=1300.

8x+22(1300)=40000
x=1425 ... but if one of our requirements be x<y -- we're playing less hands on weekends than weekdays -- this has to be ignored

Alright one last equation, y=1500
8x+22(1500)=40000
x=875

Looks like the ideal is one of these three:

(1150,1400), (1000,1454), or (875,1500)

Well. Guess I figured out my own answer(s). Now it doesn't matter which one of these I use, and in fact I can substitute one for the other on a weekly basis. One week I can do 1150 hands on the weekends and 1400 hands on weekdays, and one week I can do 875 hands on the weekends and 1500 hands during the week. All depends on my forseeable availability.

Actually, thats incredibly useful. If I think I'll be very busy next weekend, I know exactly how much during the week I need to compensate.