To simplify the expression \( \frac{1}{3}(6 - 12y) + \frac{1}{4}(24y + 14) \), we first need to distribute \( \frac{1}{3} \) to \( (6 - 12y) \) and \( \frac{1}{4} \) to \( (24y + 14) \).
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Distributing \( \frac{1}{3} \):
- \( \frac{1}{3} \times 6 = 2 \)
- \( \frac{1}{3} \times -12y = -4y \)
So, \( \frac{1}{3}(6 - 12y) \) becomes \( 2 - 4y \).
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Distributing \( \frac{1}{4} \):
- \( \frac{1}{4} \times 24y = 6y \)
- \( \frac{1}{4} \times 14 = \frac{7}{2} \)
So, \( \frac{1}{4}(24y + 14) \) becomes \( 6y + \frac{7}{2} \).
Now, we put both parts together: \[ 2 - 4y + 6y + \frac{7}{2} \]
Next, we combine the like terms:
- For the \( y \) terms: \( -4y + 6y \) gives us \( 2y \).
- For the constant terms: \( 2 + \frac{7}{2} \).
To combine \( 2 \) and \( \frac{7}{2} \), we can change \( 2 \) into a fraction that has the same denominator as \( \frac{7}{2} \). This means turning \( 2 \) into \( \frac{4}{2} \).
So, now we have: \[ \frac{4}{2} + \frac{7}{2} = \frac{11}{2} \]
Putting it all together, the simplified expression is: \[ 2y + \frac{11}{2} \]
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