1. **State the problem:** We need to find the cost of 5 tulips and 6 roses given two purchases: 7 tulips and 9 roses cost 25.90, and 4 tulips and 8 roses cost 19.80. 2. **Define variables:** Let $t$ be the price of one tulip and $r$ be the price of one rose. 3. **Set up equations from the problem:** $$7t + 9r = 25.90$$ $$4t + 8r = 19.80$$ 4. **Solve the system of equations:** Multiply the second equation by 7 and the first by 4 to align tulip terms: $$4(7t + 9r) = 4(25.90) \Rightarrow 28t + 36r = 103.60$$ $$7(4t + 8r) = 7(19.80) \Rightarrow 28t + 56r = 138.60$$ 5. **Subtract the first new equation from the second:** $$\cancel{28t} + 56r - (\cancel{28t} + 36r) = 138.60 - 103.60$$ $$56r - 36r = 35.00$$ $$20r = 35.00$$ 6. **Solve for $r$:** $$r = \frac{35.00}{20} = 1.75$$ 7. **Substitute $r=1.75$ into one original equation to find $t$:** Using $4t + 8r = 19.80$: $$4t + 8(1.75) = 19.80$$ $$4t + 14 = 19.80$$ $$4t = 19.80 - 14 = 5.80$$ 8. **Solve for $t$:** $$t = \frac{5.80}{4} = 1.45$$ 9. **Calculate cost for 5 tulips and 6 roses:** $$5t + 6r = 5(1.45) + 6(1.75) = 7.25 + 10.50 = 17.75$$ **Final answer:** The cost for 5 tulips and 6 roses is $17.75$.