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