To find the value of \(y\) in the system of equations:
- \(\frac{1}{3}x + \frac{1}{4}y = 1\)
- \(2x - 3y = -30\)
We can first eliminate the fractions in the first equation. Multiply the entire first equation by 12 (the least common multiple of 3 and 4):
\[ 12 \left( \frac{1}{3}x \right) + 12 \left( \frac{1}{4}y \right) = 12 \cdot 1 \]
This simplifies to:
\[ 4x + 3y = 12 \quad \text{(Equation 1)} \]
Now we have a new system of equations:
- \(4x + 3y = 12\)
- \(2x - 3y = -30\)
Next, we can add the two equations together to eliminate \(y\):
\[ (4x + 3y) + (2x - 3y) = 12 - 30 \]
This simplifies to:
\[ 6x = -18 \]
Now, solving for \(x\):
\[ x = -3 \]
Now that we have the value of \(x\), we can substitute it back into one of the equations to find \(y\). We'll use Equation 1:
\[ 4(-3) + 3y = 12 \]
This simplifies to:
\[ -12 + 3y = 12 \]
Adding 12 to both sides gives:
\[ 3y = 24 \]
Now, dividing by 3:
\[ y = 8 \]
Thus, the value of \(y\) is \(\boxed{8}\).
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