1. **State the problem:** Solve the equation $$\frac{7}{10}v + 1 = \frac{83}{15} - v$$ for $v$. 2. **Write down the equation:** $$\frac{7}{10}v + 1 = \frac{83}{15} - v$$ 3. **Goal:** Isolate $v$ on one side. 4. **Add $v$ to both sides:** $$\frac{7}{10}v + v + 1 = \frac{83}{15}$$ 5. **Combine like terms:** Note that $v = \frac{10}{10}v$, so $$\frac{7}{10}v + \frac{10}{10}v = \frac{17}{10}v$$ 6. **Rewrite equation:** $$\frac{17}{10}v + 1 = \frac{83}{15}$$ 7. **Subtract 1 from both sides:** $$\frac{17}{10}v = \frac{83}{15} - 1$$ 8. **Convert 1 to fraction with denominator 15:** $$1 = \frac{15}{15}$$ 9. **Calculate right side:** $$\frac{83}{15} - \frac{15}{15} = \frac{83 - 15}{15} = \frac{68}{15}$$ 10. **Equation now:** $$\frac{17}{10}v = \frac{68}{15}$$ 11. **Solve for $v$ by dividing both sides by $\frac{17}{10}$:** $$v = \frac{\frac{68}{15}}{\frac{17}{10}} = \frac{68}{15} \times \frac{10}{17}$$ 12. **Simplify multiplication:** $$v = \frac{68 \times 10}{15 \times 17}$$ 13. **Calculate numerator and denominator:** $$v = \frac{680}{255}$$ 14. **Simplify fraction by dividing numerator and denominator by 5:** $$v = \frac{\cancel{680}^{{136}}}{\cancel{255}^{{51}}}$$ 15. **Check for further simplification:** 136 and 51 share a common factor 17: $$v = \frac{\cancel{136}^{8}}{\cancel{51}^{3}}$$ 16. **Final simplified answer:** $$v = \frac{8}{3}$$ **Answer:** $v = \frac{8}{3}$