We have the equation \(2_9 \times (Y3)_9 = 3_5 \times (Y3)_5\).

First, let's calculate the values of \(2_9\) and \(3_5\) in base-10:

\(2_9 = 2 \times 9^0 = 2\)

\(3_5 = 3 \times 5^0 = 3\)

Now, we have the equation \(2 \times (Y3)_9 = 3 \times (Y3)_5\).

Next, let's consider the digits in the base-9 and base-5 numbers:

In base-9, the digit \(Y\) can represent any value from 0 to 8.

In base-5, the digit \(Y\) can represent any value from 0 to 4.

To make the equation hold true, the possible values of \(Y\) are limited to those that satisfy both sides of the equation.

Let's check the possibilities:

For \(Y = 0\):

\(2 \times (03)_9 = 2 \times 3 = 6\)

\(3 \times (03)_5 = 3 \times 3 = 12\)

For \(Y = 1\):

\(2 \times (13)_9 = 2 \times 12 = 24\)

\(3 \times (13)_5 = 3 \times 8 = 24\)

For \(Y = 2\):

\(2 \times (23)_9 = 2 \times 21 = 42\)

\(3 \times (23)_5 = 3 \times 13 = 39\)

For \(Y = 3\):

\(2 \times (33)_9 = 2 \times 30 = 60\)

\(3 \times (33)_5 = 3 \times 18 = 54\)

For \(Y = 4\):

\(2 \times (43)_9 = 2 \times 39 = 78\)

\(3 \times (43)_5 = 3 \times 23 = 69\)

Based on the calculations, we can see that only when \(Y = 1\), the equation holds true:

\(2 \times (13)_9 = 24\)

\(3 \times (13)_5 = 24\)

So, the correct value of Y is 1.