1. **State the problem:** Solve the equation $$\frac{9}{7}(1 + x) - \frac{6}{7}x + 1 = 1$$ for $x$. 2. **Write down the equation:** $$\frac{9}{7}(1 + x) - \frac{6}{7}x + 1 = 1$$ 3. **Distribute the $\frac{9}{7}$:** $$\frac{9}{7} \cdot 1 + \frac{9}{7} \cdot x - \frac{6}{7}x + 1 = 1$$ which simplifies to $$\frac{9}{7} + \frac{9}{7}x - \frac{6}{7}x + 1 = 1$$ 4. **Combine like terms for $x$:** $$\frac{9}{7}x - \frac{6}{7}x = \frac{3}{7}x$$ So the equation becomes $$\frac{9}{7} + \frac{3}{7}x + 1 = 1$$ 5. **Combine constants on the left:** $$\frac{9}{7} + 1 = \frac{9}{7} + \frac{7}{7} = \frac{16}{7}$$ So the equation is $$\frac{16}{7} + \frac{3}{7}x = 1$$ 6. **Subtract $\frac{16}{7}$ from both sides:** $$\frac{16}{7} + \frac{3}{7}x - \frac{16}{7} = 1 - \frac{16}{7}$$ which simplifies to $$\frac{3}{7}x = 1 - \frac{16}{7}$$ 7. **Rewrite 1 as $\frac{7}{7}$ and subtract:** $$1 - \frac{16}{7} = \frac{7}{7} - \frac{16}{7} = -\frac{9}{7}$$ So $$\frac{3}{7}x = -\frac{9}{7}$$ 8. **Solve for $x$ by dividing both sides by $\frac{3}{7}$:** $$x = \frac{-\frac{9}{7}}{\frac{3}{7}} = -\frac{9}{7} \times \frac{7}{3}$$ 9. **Cancel common factors:** $$x = -\frac{\cancel{9}}{\cancel{7}} \times \frac{\cancel{7}}{\cancel{3}} = -3$$ **Final answer:** $$x = -3$$