Valentina is a hairdresser. Before her lunch break, she gave 4 haircuts and colored the hair of 3 clients in 284 minutes. After lunch, she gave 1 haircut and colored the hair of 3 clients in 233 minutes. How long does it take for Valentina to give a client a haircut and how long does it take to color a client's hair?
Before lunch, Valentina gave 4 haircuts and colored the hair of 3 clients in 284 minutes, so it took her 284 minutes - (4 haircuts * X minutes) - (3 hair colorings * Y minutes) = Z minutes.
After lunch, she gave 1 haircut and colored the hair of 3 clients in 233 minutes, so it took her 233 minutes - (1 haircut * X minutes) - (3 hair colorings * Y minutes) = Z minutes.
Subtracting these two equations gives 284 minutes - 233 minutes = (4 haircuts - 1 haircut) * X minutes + (3 hair colorings - 3 hair colorings) * Y minutes.
This simplifies to 51 minutes = 3 haircuts * X minutes.
So, X = 51 minutes / 3 haircuts = <<51/3=17>>17 minutes per haircut.
Substituting this value back into one of the original equations, we get 284 minutes = 4 haircuts * 17 minutes + 3 hair colorings * Y minutes.
Another equation to solve this would be 233 minutes = 1 haircut * 17 minutes + 3 hair colorings * Y minutes.
Substituting the first equation into this equation gives 233 minutes = 1 haircut * 17 minutes + 3 hair colorings * Y minutes.
This further simplifies to 233 minutes - 1 haircut * 17 minutes = 3 hair colorings * Y minutes.
Doing the math we get 216 minutes = 3 hair colorings * Y minutes.
So, Y = 216 minutes / 3 hair colorings = 72 minutes per hair coloring. Answer: \boxed{17, 72}.