Solve the linear equation $\frac{2x}{5} = 3x + 1$ for $x$.

1. Problem Description

Solve the linear equation $\frac{2x}{5} = 3x + 1$ for $x$.

2. Solution Steps

First, 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 to isolate $x$ terms on one side.

$$$$

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

$$$$

$$$$

-13x = 5

$$$$

Finally, divide both sides by $-13$ to solve for $x$.

$$$$

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

$$$$

$$$$

x = -\frac{5}{13}

$$$$

3. Final Answer

The solution to the equation is $x = -\frac{5}{13}$.