To determine which expressions are equivalent to \( 12 - \frac{14}{3}m + \frac{2}{3}m \), we first simplify the original expression.
Combine the terms involving \( m \):
\[ -\frac{14}{3}m + \frac{2}{3}m = \left(-\frac{14}{3} + \frac{2}{3}\right)m = -\frac{12}{3}m = -4m \]
Now, substitute this back into the expression:
\[ 12 - 4m \]
Now we have the simplified form of the original expression as \( 12 - 4m \).
Next, we will check the other expressions to see if they are equivalent to \( 12 - 4m \):
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\( 12 - 4m \): This is equivalent since it is the same expression.
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\( -4(m - 3) \): This expands to \( -4m + 12 \), which can be rewritten as \( 12 - 4m \). Therefore, it is equivalent.
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\( 12 - \frac{16}{3}m \): We need to see if this expression can be rewritten to match \( 12 - 4m \).
The expression \( 12 - 4m \) can be rewritten as \( 12 - \frac{12}{3}m \).
Since \( \frac{16}{3} \) is greater than \( \frac{12}{3} \), \( 12 - \frac{16}{3}m \) is not equivalent to \( 12 - 4m \).
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\( -4(3 + m) \): This expands to \( -12 - 4m \). This does not match \( 12 - 4m \).
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\( 8m \): This is not equivalent to \( 12 - 4m \), as it only has a term with \( m \) and no constant term.
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\( 4(m - 3) \): This expands to \( 4m - 12 \), which is not equivalent to \( 12 - 4m \).
In summary, the equivalent expressions to \( 12 - \frac{14}{3}m + \frac{2}{3}m \) are:
- \( 12 - 4m \)
- \( -4(m - 3) \)
Thus, the equivalent expressions are:
- \( 12 - 4m \)
- \( -4(m - 3) \)
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