We need to solve the following equation for $x$: $\frac{2}{3}(3x-5) - \frac{3}{5}(2x-3) = 3$.

1. Problem Description

We need to solve the following equation for $x$:

$\frac{2}{3}(3x-5) - \frac{3}{5}(2x-3) = 3$.

2. Solution Steps

First, distribute the fractions into the parentheses:

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

$2x - \frac{10}{3} - \frac{6}{5}x + \frac{9}{5} = 3$

Now, combine the terms with $x$:

$2x - \frac{6}{5}x = \frac{10}{5}x - \frac{6}{5}x = \frac{4}{5}x$

So the equation becomes:

$\frac{4}{5}x - \frac{10}{3} + \frac{9}{5} = 3$

Next, isolate the $x$ term. Add $\frac{10}{3}$ and subtract $\frac{9}{5}$ from both sides:

$\frac{4}{5}x = 3 + \frac{10}{3} - \frac{9}{5}$

Find a common denominator for the right side, which is

1

5. $3 = \frac{45}{15}$

$\frac{10}{3} = \frac{50}{15}$

$\frac{9}{5} = \frac{27}{15}$

So,

$\frac{4}{5}x = \frac{45}{15} + \frac{50}{15} - \frac{27}{15}$

$\frac{4}{5}x = \frac{45+50-27}{15} = \frac{68}{15}$

Now, multiply both sides by $\frac{5}{4}$ to solve for $x$:

$x = \frac{68}{15} \cdot \frac{5}{4}$

$x = \frac{68 \cdot 5}{15 \cdot 4}$

$x = \frac{340}{60}$

$x = \frac{34}{6}$

$x = \frac{17}{3}$

3. Final Answer

$x = \frac{17}{3}$