To simplify the expression \( \frac{1}{3}(6 - 12y) + \frac{1}{4}(24y + 14) \), we will first distribute the constants and then combine like terms.
-
Distribute \( \frac{1}{3} \) to \( (6 - 12y) \): \[ \frac{1}{3}(6) - \frac{1}{3}(12y) = 2 - 4y \]
-
Distribute \( \frac{1}{4} \) to \( (24y + 14) \): \[ \frac{1}{4}(24y) + \frac{1}{4}(14) = 6y + \frac{14}{4} \] Simplifying \( \frac{14}{4} \) gives: \[ \frac{14}{4} = \frac{7}{2} \]
So we have: \[ \frac{1}{4}(24y + 14) = 6y + \frac{7}{2} \]
-
Now combine the two results: \[ 2 - 4y + 6y + \frac{7}{2} \]
-
Combine like terms: For the \( y \) terms: \[ -4y + 6y = 2y \]
So the expression now looks like this: \[ 2y + 2 + \frac{7}{2} \]
-
To combine the constant terms \( 2 \) and \( \frac{7}{2} \): Convert \( 2 \) into a fraction with a denominator of 2: \[ 2 = \frac{4}{2} \]
Now add \( \frac{4}{2} + \frac{7}{2} \): \[ \frac{4 + 7}{2} = \frac{11}{2} \]
-
Now put it all together: \[ 2y + \frac{11}{2} \]
Thus, the simplified expression is: \[ \boxed{2y + \frac{11}{2}} \]