1. **State the problem:** Solve the equation $$x = - \frac{8}{9} y + \frac{5}{8}$$ for the variable $y$. 2. **Write the equation:** $$x = - \frac{8}{9} y + \frac{5}{8}$$ 3. **Isolate the term with $y$:** Subtract $\frac{5}{8}$ from both sides: $$x - \frac{5}{8} = - \frac{8}{9} y$$ 4. **Divide both sides by $- \frac{8}{9}$ to solve for $y$:** $$y = \frac{x - \frac{5}{8}}{- \frac{8}{9}}$$ 5. **Simplify the division by a fraction:** Dividing by $- \frac{8}{9}$ is the same as multiplying by its reciprocal $- \frac{9}{8}$: $$y = \left(x - \frac{5}{8}\right) \times \left(- \frac{9}{8}\right)$$ 6. **Distribute the multiplication:** $$y = - \frac{9}{8} x + \frac{9}{8} \times \frac{5}{8}$$ 7. **Multiply the fractions:** $$\frac{9}{8} \times \frac{5}{8} = \frac{45}{64}$$ 8. **Write the final simplified expression:** $$y = - \frac{9}{8} x + \frac{45}{64}$$ **Answer:** $$\boxed{y = - \frac{9}{8} x + \frac{45}{64}}$$