1. **State the problem:** A farmer has two types of fertilizer packages: old and new. Old packages contain 60 pounds of long-term-growth supplement and 50 pounds of weed killer. New packages contain 65 pounds of long-term-growth supplement and 45 pounds of weed killer. The farmer needs 1550 pounds of long-term-growth supplement and 1200 pounds of weed killer. We need to find how many old and new packages to use. 2. **Set variables:** Let $x$ = number of old packages, $y$ = number of new packages. 3. **Write the system of equations based on supplements:** $$60x + 65y = 1550$$ $$50x + 45y = 1200$$ 4. **Solve the system:** Multiply the first equation by 9 and the second by 13 to align coefficients of $y$: $$9(60x + 65y) = 9(1550) \Rightarrow 540x + 585y = 13950$$ $$13(50x + 45y) = 13(1200) \Rightarrow 650x + 585y = 15600$$ 5. **Subtract the first from the second:** $$650x + 585y - (540x + 585y) = 15600 - 13950$$ $$650x - 540x + \cancel{585y} - \cancel{585y} = 1650$$ $$110x = 1650$$ 6. **Solve for $x$:** $$x = \frac{1650}{110} = 15$$ 7. **Substitute $x=15$ into the first equation:** $$60(15) + 65y = 1550$$ $$900 + 65y = 1550$$ $$65y = 1550 - 900 = 650$$ 8. **Solve for $y$:** $$y = \frac{650}{65} = 10$$ **Final answer:** The farmer should use 15 old packages and 10 new packages.