1. **State the problem:** A man buys some apples and oranges for 15. If he had bought 2 more apples and 3 fewer oranges, he would have spent 13. We need to find the price of one apple and one orange. 2. **Define variables:** Let the price of one apple be $a$ and the price of one orange be $b$. Let the original number of apples bought be $x$ and the number of oranges be $y$. 3. **Write equations from the problem:** - The cost for $x$ apples and $y$ oranges is 15: $$a x + b y = 15$$ - The cost for $(x+2)$ apples and $(y-3)$ oranges is 13: $$a(x+2) + b(y-3) = 13$$ 4. **Expand second equation:** $$a x + 2a + b y - 3b = 13$$ 5. **Subtract the first equation from the expanded second equation:** $$(a x + 2a + b y - 3b) - (a x + b y) = 13 - 15$$ $$2a - 3b = -2$$ 6. **Simplify:** $$2a - 3b = -2$$ 7. **Express one variable in terms of the other:** $$2a = 3b - 2$$ $$a = \frac{3b - 2}{2}$$ 8. **Substitute $a$ into the first equation:** $$a x + b y = 15$$ $$\left(\frac{3b - 2}{2}\right) x + b y = 15$$ 9. **Note:** We have two equations but four variables: $a,b,x,y$. We need more information or constraints to solve uniquely. Usually, number of fruits bought ($x,y$) is a positive integer. 10. **Assume $x,y$ are integers and prices are positive. Solve the system: We have:** - $$a x + b y = 15$$ - $$2a - 3b = -2$$ 11. **Try integer values for $x$ and $y$ to find integer $a$ and $b$: For example, solve for $a$ and $b$ given $x=3$ and $y=3$:** - Substituting $x=3,y=3$ in $a x + b y=15$: $$3a + 3b = 15$$ $$a + b = 5$$ - From earlier, $$2a - 3b = -2$$ 12. **Solve these two equations:** - $$a + b = 5$$ - $$2a - 3b = -2$$ Multiply the first by 3: $$3a + 3b = 15$$ Add this to the second: $$2a - 3b = -2$$ $$3a + 3b = 15$$ ---------------- $$5a = 13$$ $$a = \frac{13}{5} = 2.6$$ Substitute back for $b$: $$2.6 + b = 5 \,\Rightarrow\, b = 2.4$$ Prices are $a=2.6$ and $b=2.4$. 13. **Verification:** - Original cost: $3 \times 2.6 + 3 \times 2.4 = 7.8 + 7.2 = 15$ - New cost: $(3+2) \times 2.6 + (3-3) \times 2.4 = 5 \times 2.6 + 0 = 13$ This satisfies both conditions. **Final answer:** The price of an apple is $\boxed{2.6}$ and the price of an orange is $\boxed{2.4}$.