To simplify the expression \( \frac{1}{3}(6 - 12y) + \frac{1}{4}(24y + 14) \), start by distributing \( \frac{1}{3} \) to \( (6 - 12y) \) and \( \frac{1}{4} \) to \( (24y + 14) \). This gives us:
\[ \frac{1}{3} \cdot 6 - \frac{1}{3} \cdot 12y + \frac{1}{4} \cdot 24y + \frac{1}{4} \cdot 14 \]
Which simplifies to:
\[ 2 - 4y + 6y + \frac{7}{2} \]
Next, combine like terms. The \( y \) terms \( -4y + 6y \) combine to give \( 2y \). For the constant terms, we combine \( 2 \) and \( \frac{7}{2} \). To do this, we can convert \( 2 \) into a fraction:
\[ 2 = \frac{4}{2} \]
Now, adding \( \frac{4}{2} + \frac{7}{2} \) results in:
\[ \frac{11}{2} \]
Putting it all together, the simplified expression is:
\[ 2y + \frac{11}{2} \]
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