To subtract the expression \((3b - 2c) - (6c + 2b) - (b + c)\), we will distribute the negative sign across each term in the parentheses and then combine like terms.
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Start with the original expression: \[ (3b - 2c) - (6c + 2b) - (b + c) \]
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Distribute the negative sign: \[ 3b - 2c - 6c - 2b - b - c \]
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Combine like terms:
- Combine the \(b\) terms: \(3b - 2b - b = 0b\)
- Combine the \(c\) terms: \(-2c - 6c - c = -9c\)
The expression simplifies to: \[ 0b - 9c \] Since \(0b\) is just 0, we can write the final result as: \[ -9c \]
Thus, the result of the subtraction is: \[ \boxed{-9c} \]