1. **State the problem:** We have 30 phone cases in total, divided into three colors: emerald, turquoise, and orange. 2. **Define variables:** Let $E$ be the number of emerald cases, $T$ the number of turquoise cases, and $O$ the number of orange cases. 3. **Write the equations from the problem:** - Total cases: $$E + T + O = 30$$ - Turquoise cases are 5 fewer than emerald: $$T = E - 5$$ - Orange cases are half as many as turquoise: $$O = \frac{T}{2}$$ 4. **Substitute $T$ and $O$ in the total equation:** $$E + (E - 5) + \frac{E - 5}{2} = 30$$ 5. **Multiply through by 2 to clear the fraction:** $$2E + 2(E - 5) + (E - 5) = 60$$ 6. **Simplify:** $$2E + 2E - 10 + E - 5 = 60$$ $$5E - 15 = 60$$ 7. **Add 15 to both sides:** $$5E - 15 + 15 = 60 + 15$$ $$5E = 75$$ 8. **Divide both sides by 5:** $$\cancel{5}E = \frac{75}{\cancel{5}}$$ $$E = 15$$ 9. **Find $T$ and $O$ using $E=15$:** $$T = E - 5 = 15 - 5 = 10$$ $$O = \frac{T}{2} = \frac{10}{2} = 5$$ 10. **Check the total:** $$15 + 10 + 5 = 30$$ which matches the total number of cases. **Final answer:** - Emerald: 15 - Turquoise: 10 - Orange: 5