Solve the equation $\frac{2x}{x-3} = 4 + \frac{6}{x-3}$.

1. Problem Description

Solve the equation $\frac{2x}{x-3} = 4 + \frac{6}{x-3}$.

2. Solution Steps

First, we multiply both sides of the equation by $(x-3)$ to eliminate the fractions, keeping in mind that $x$ cannot be $3$ (since the denominator cannot be zero).

$(x-3) \cdot \frac{2x}{x-3} = (x-3) \cdot (4 + \frac{6}{x-3})$

$2x = 4(x-3) + 6$

Now, we simplify the equation:

$2x = 4x - 12 + 6$

$2x = 4x - 6$

Subtract $4x$ from both sides:

$2x - 4x = 4x - 6 - 4x$

$-2x = -6$

Divide both sides by $-2$:

$\frac{-2x}{-2} = \frac{-6}{-2}$

$x = 3$

However, we noted earlier that $x$ cannot be $3$ because the denominators in the original equation would be zero, making the fractions undefined. Therefore, there is no solution.

3. Final Answer

No solution.