The problem asks us to solve the equation $\frac{5x}{2} - 7 = \frac{2x - 7}{6}$ for $x$.

1. Problem Description

The problem asks us to solve the equation $\frac{5x}{2} - 7 = \frac{2x - 7}{6}$ for $x$.

2. Solution Steps

First, we want to eliminate the fractions. Multiply both sides of the equation by 6:

$6 \left(\frac{5x}{2} - 7\right) = 6 \left(\frac{2x - 7}{6}\right)$

$6 \cdot \frac{5x}{2} - 6 \cdot 7 = 2x - 7$

$3 \cdot 5x - 42 = 2x - 7$

$15x - 42 = 2x - 7$

Now, we want to isolate $x$ on one side of the equation. Subtract $2x$ from both sides:

$15x - 2x - 42 = 2x - 2x - 7$

$13x - 42 = -7$

Add 42 to both sides:

$13x - 42 + 42 = -7 + 42$

$13x = 35$

Finally, divide both sides by 13:

$\frac{13x}{13} = \frac{35}{13}$

$x = \frac{35}{13}$

3. Final Answer

$x = \frac{35}{13}$