Solution: Average equation: $ \frac{(3y + 1) + (y + 7) + (5y - 2)}{3} = 10 $. Simplify numerator: $ 9y + 6 = 30 $. Subtract 6: $ 9y = 24 $. Divide: $ y = \frac{8}{3} $. \boxed{\dfrac{8}{3}}