1. **State the problem:** Solve for $x$ in the equation $$\frac{2x - 3}{3} + \frac{3x + 4}{5} = 15.$$\n\n2. **Identify the formula and rules:** To solve this equation, we need to combine the fractions by finding a common denominator and then isolate $x$.\n\n3. **Find the common denominator:** The denominators are 3 and 5, so the least common denominator (LCD) is 15.\n\n4. **Rewrite each fraction with denominator 15:**\n$$\frac{2x - 3}{3} = \frac{(2x - 3) \times 5}{3 \times 5} = \frac{5(2x - 3)}{15}$$\n$$\frac{3x + 4}{5} = \frac{(3x + 4) \times 3}{5 \times 3} = \frac{3(3x + 4)}{15}$$\n\n5. **Rewrite the equation:**\n$$\frac{5(2x - 3)}{15} + \frac{3(3x + 4)}{15} = 15$$\n\n6. **Combine the fractions:**\n$$\frac{5(2x - 3) + 3(3x + 4)}{15} = 15$$\n\n7. **Multiply both sides by 15 to eliminate the denominator:**\n$$\cancel{15} \times \frac{5(2x - 3) + 3(3x + 4)}{\cancel{15}} = 15 \times 15$$\n$$5(2x - 3) + 3(3x + 4) = 225$$\n\n8. **Expand the terms:**\n$$5 \times 2x - 5 \times 3 + 3 \times 3x + 3 \times 4 = 225$$\n$$10x - 15 + 9x + 12 = 225$$\n\n9. **Combine like terms:**\n$$10x + 9x - 15 + 12 = 225$$\n$$19x - 3 = 225$$\n\n10. **Isolate $x$ by adding 3 to both sides:**\n$$19x - 3 + 3 = 225 + 3$$\n$$19x = 228$$\n\n11. **Divide both sides by 19:**\n$$\frac{19x}{\cancel{19}} = \frac{228}{\cancel{19}}$$\n$$x = 12$$\n\n**Final answer:** $x = 12$