The problem is to solve the linear equation $\frac{2x}{5} = 3x + 1$ for $x$.

1. Problem Description

The problem is to solve the linear equation $\frac{2x}{5} = 3x + 1$ for $x$.

2. Solution Steps

First, we multiply both sides of the equation by 5 to eliminate the fraction:

$5 \cdot \frac{2x}{5} = 5 \cdot (3x + 1)$

$2x = 15x + 5$

Next, subtract $15x$ from both sides of the equation:

$2x - 15x = 15x + 5 - 15x$

$-13x = 5$

Finally, divide both sides of the equation by -13 to isolate $x$:

$\frac{-13x}{-13} = \frac{5}{-13}$

$x = -\frac{5}{13}$

3. Final Answer

$x = -\frac{5}{13}$