To solve the linear equation
\[ \frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p, \]
let's first isolate \( p \).
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Start with the original equation: \[ \frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p. \]
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Next, subtract \(\frac{3}{5}p\) from both sides: \[ \frac{2}{5} + p - \frac{3}{5}p = \frac{4}{5}. \] This simplifies to: \[ \frac{2}{5} + \left(1 - \frac{3}{5}\right)p = \frac{4}{5} \] or \[ \frac{2}{5} + \frac{2}{5}p = \frac{4}{5}. \]
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Now, subtract \(\frac{2}{5}\) from both sides: \[ \frac{2}{5}p = \frac{4}{5} - \frac{2}{5}. \] Simplifying the right side gives: \[ \frac{2}{5}p = \frac{2}{5}. \]
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Finally, divide both sides by \(\frac{2}{5}\): \[ p = 1. \]
Therefore, the solution to the linear equation is
\[ \boxed{1}. \]