To solve the equation \( \frac{3}{7} + \frac{1}{5}x - \frac{1}{3} = \frac{4}{5}x \), we first need to get all the terms involving \( x \) on one side and the constant terms on the other side.
- Start by isolating the \( x \) terms. We can move \( \frac{1}{5}x \) to the right side of the equation:
\[ \frac{3}{7} - \frac{1}{3} = \frac{4}{5}x - \frac{1}{5}x \]
- Simplifying the right side gives us:
\[ \frac{4}{5}x - \frac{1}{5}x = \frac{3}{5}x \]
- Now we rewrite the equation:
\[ \frac{3}{7} - \frac{1}{3} = \frac{3}{5}x \]
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To simplify the left side, we need a common denominator for \( \frac{3}{7} \) and \( \frac{1}{3} \). The least common multiple of 7 and 3 is 21.
Rewrite \( \frac{3}{7} \) and \( \frac{1}{3} \) with the common denominator of 21:
\[ \frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21} \] \[ \frac{1}{3} = \frac{1 \times 7}{3 \times 7} = \frac{7}{21} \]
- Now, substitute these back into the equation:
\[ \frac{9}{21} - \frac{7}{21} = \frac{3}{5}x \]
- This simplifies to:
\[ \frac{2}{21} = \frac{3}{5}x \]
- To solve for \( x \), multiply both sides by the reciprocal of \( \frac{3}{5} \), which is \( \frac{5}{3} \):
\[ x = \frac{2}{21} \times \frac{5}{3} \]
- Multiply the fractions:
\[ x = \frac{2 \times 5}{21 \times 3} = \frac{10}{63} \]
Therefore, the solution to the equation is:
\[ \boxed{\frac{10}{63}} \]