To solve the equation \( 3j - 5 = 8 + \frac{1}{2}j + \frac{1}{3}j \), we first need to combine the terms on the right side.
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Combine the \( j \) terms on the right side:
\[ \frac{1}{2}j + \frac{1}{3}j \]
To add these fractions, we need a common denominator, which is 6:
\[ \frac{1}{2}j = \frac{3}{6}j \quad \text{and} \quad \frac{1}{3}j = \frac{2}{6}j \]
Therefore:
\[ \frac{1}{2}j + \frac{1}{3}j = \frac{3}{6}j + \frac{2}{6}j = \frac{5}{6}j \]
Now, we substitute this back into the equation:
\[ 3j - 5 = 8 + \frac{5}{6}j \]
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Move the \( \frac{5}{6}j \) term to the left side by subtracting \( \frac{5}{6}j \) from both sides:
\[ 3j - \frac{5}{6}j - 5 = 8 \]
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To combine \( 3j \) and \( -\frac{5}{6}j \), convert \( 3j \) into sixths:
\[ 3j = \frac{18}{6}j \]
Thus:
\[ \frac{18}{6}j - \frac{5}{6}j = \frac{13}{6}j \]
Now the equation is:
\[ \frac{13}{6}j - 5 = 8 \]
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Add 5 to both sides to isolate the term with \( j \):
\[ \frac{13}{6}j = 8 + 5 = 13 \]
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Now, solve for \( j \) by multiplying both sides by \( \frac{6}{13} \):
\[ j = 13 \cdot \frac{6}{13} = 6 \]
Thus, the solution for \( j \) is:
\[ \boxed{6} \]
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