1. **State the problem:** Solve the linear equation $$\frac{2x - 6}{4} = \frac{5x + 6}{3}$$. 2. **Formula and rules:** To solve equations with fractions, multiply both sides by the least common denominator (LCD) to eliminate the denominators. 3. **Find the LCD:** The denominators are 4 and 3, so the LCD is 12. 4. **Multiply both sides by 12:** $$12 \times \frac{2x - 6}{4} = 12 \times \frac{5x + 6}{3}$$ 5. **Simplify each side:** $$\cancel{12} \times \frac{2x - 6}{\cancel{4}} = \cancel{12} \times \frac{5x + 6}{\cancel{3}}$$ $$3(2x - 6) = 4(5x + 6)$$ 6. **Distribute:** $$6x - 18 = 20x + 24$$ 7. **Bring variables to one side and constants to the other:** $$6x - 20x = 24 + 18$$ 8. **Simplify:** $$-14x = 42$$ 9. **Divide both sides by -14:** $$\frac{-14x}{\cancel{-14}} = \frac{42}{\cancel{-14}}$$ $$x = -3$$ **Final answer:** $$x = -3$$