Solution: Sum the expressions: $(4y - 3) + (2y + 5) + (y + 7) = 7y + 9$. Divide by 3 and set equal to 10: $\frac{7y + 9}{3} = 10$. Multiply both sides by 3: $7y + 9 = 30$. Subtract 9: $7y = 21$. Divide by 7: $y = \boxed{3}$.