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To simplify the expression \( 9x + 3y - 6 - 5x + 4y - 3 \), we can combine like terms.
1. Combine the \( x \) terms:
\[
9x - 5x = 4x
\]
2. Combine the \( y \) terms:
\[
3y + 4y = 7y
\]
3. Combine the constant terms:
\[
-6 - 3 = -9
\]
Putting it all together, we have:
\[
4x + 7y - 9
\]
Thus, the simplified expression is:
\[
\boxed{4x + 7y - 9}
\]