Step 1: Understand how the classes are formed.
Two pairs $(x, y)$ and $(x', y')$ sit in the same class exactly when their coordinate sums $x + y$ are equal. So each distinct value of the sum $s = x + y$ gives one equivalence class. Counting classes means counting how many different sums are possible.
Step 2: Note the ranges.
Here $x$ runs over $1$ to $15$ and $y$ runs over $1$ to $20$, both whole numbers.
Step 3: Find the smallest possible sum.
The least sum is when both are smallest: $x = 1, y = 1$, giving $s = 2$.
Step 4: Find the largest possible sum.
The greatest sum is when both are largest: $x = 15, y = 20$, giving $s = 35$.
Step 5: Check every value in between is reachable.
As $x$ and $y$ are independent integers covering full ranges, every integer sum from $2$ up to $35$ can be made. For example, to reach any $s$ in this range we can keep $y$ free between $1$ and $20$ and pick a matching $x$ between $1$ and $15$.
Step 6: Count the distinct sums.
The sums run over all integers from $2$ to $35$, which is $35 - 2 + 1 = 34$ values. So there are $34$ equivalence classes.
\[ \boxed{34} \]