1. **State the problem:** Paul and Kate bought a total of 6 CDs, paying 78 in total. Each CD Paul bought costs 12, and each CD Kate bought costs 15. We need to find how many CDs Kate bought. 2. **Define variables:** Let $x$ be the number of CDs Paul bought, and $y$ be the number of CDs Kate bought. 3. **Write the system of equations:** $$x + y = 6$$ $$12x + 15y = 78$$ 4. **Solve the first equation for $x$:** $$x = 6 - y$$ 5. **Substitute $x$ into the second equation:** $$12(6 - y) + 15y = 78$$ 6. **Distribute and simplify:** $$72 - 12y + 15y = 78$$ $$72 + 3y = 78$$ 7. **Isolate $y$:** $$3y = 78 - 72$$ $$3y = 6$$ 8. **Divide both sides by 3:** $$y = \cancel{\frac{3y}{3}} = \cancel{\frac{6}{3}} = 2$$ 9. **Interpret the result:** Kate bought 2 CDs. **Final answer:** Kate bought **Two** CDs.