Solve the equation $\frac{2x}{x-5} = \frac{2}{3}$ for $x$.

1. Problem Description

Solve the equation $\frac{2x}{x-5} = \frac{2}{3}$ for $x$.

2. Solution Steps

First, we cross-multiply to eliminate the fractions:

$2x \cdot 3 = 2 \cdot (x-5)$

Expand both sides of the equation:

$6x = 2x - 10$

Subtract $2x$ from both sides:

$6x - 2x = 2x - 10 - 2x$

$4x = -10$

Divide both sides by 4 to isolate $x$:

$\frac{4x}{4} = \frac{-10}{4}$

$x = -\frac{10}{4}$

Simplify the fraction:

$x = -\frac{5}{2}$

3. Final Answer

$x = -\frac{5}{2}$