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

1. Problem Description

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

2. Solution Steps

We are given the equation $\frac{2x}{x-5} = \frac{2}{3}$.

To solve for $x$, we can cross-multiply.

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

$6x = 2x - 10$

Now, we subtract $2x$ from both sides of the equation:

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

$4x = -10$

Now, we divide both sides by 4:

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

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

Simplify the fraction by dividing both the numerator and denominator by 2:

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

We should check if $x = -\frac{5}{2}$ is a valid solution. The original equation is $\frac{2x}{x-5} = \frac{2}{3}$. The denominator of the left side is $x-5$. If $x = -\frac{5}{2}$, then $x-5 = -\frac{5}{2} - 5 = -\frac{5}{2} - \frac{10}{2} = -\frac{15}{2}$, which is not zero. So $x = -\frac{5}{2}$ is a valid solution.

3. Final Answer

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