Step 1: Find a common denominator.
The denominators are 3y, 4, and 8. The least common multiple (LCM) of these is 24y.
Step 2: Adjust each term to have a denominator of 24y.
To do this, we multiply the numerator and denominator of each term by the appropriate factor:
For the term 8/3y, multiply numerator and denominator by 8: (8/3y) * (8/8) = 64/24y.
For the term 5y/4, multiply numerator and denominator by 6y: (5y/4) * (6y/6y) = 30y^2/24y.
For the term 5/8, multiply numerator and denominator by 3: (5/8) * (3/3) = 15/24y.
Step 3: Combine the terms.
Now that all terms have a common denominator of 24y, we can add them together:
(64/24y) + (30y^2/24y) - (15/24y)
Step 4: Combine the numerators.
Add the numerators of the fractions:
(64 + 30y^2 - 15) / 24y
Step 5: Simplify the numerator.
Combine like terms in the numerator:
(49 + 30y^2) / 24y
Thus, the simplified expression is (49 + 30y^2) / 24y, considering all permissible values of y.