1. **Problem statement:** Given the ordered numbers $x, (x+1), 5, (2x+1), (y+2), 7, 8,$ and $(3x+2)$, the mean is 6.5 and the median is 7. Find $x$ and $y$. 2. **Mean formula:** The mean of $n$ numbers $a_1, a_2, \ldots, a_n$ is given by $$\text{Mean} = \frac{a_1 + a_2 + \cdots + a_n}{n}$$ 3. **Calculate the mean:** There are 8 numbers, so $$\frac{x + (x+1) + 5 + (2x+1) + (y+2) + 7 + 8 + (3x+2)}{8} = 6.5$$ 4. **Simplify numerator:** $$x + x + 1 + 5 + 2x + 1 + y + 2 + 7 + 8 + 3x + 2 = (x + x + 2x + 3x) + (1 + 5 + 1 + 2 + 7 + 8 + 2) + y = 9x + 26 + y$$ 5. **Set up equation:** $$\frac{9x + y + 26}{8} = 6.5$$ 6. **Multiply both sides by 8:** $$\cancel{8} \times \frac{9x + y + 26}{\cancel{8}} = 6.5 \times 8$$ $$9x + y + 26 = 52$$ 7. **Rearrange:** $$9x + y = 52 - 26$$ $$9x + y = 26 \quad \quad (1)$$ 8. **Median definition:** The median is the middle value when numbers are ordered. Since there are 8 numbers (even), the median is the average of the 4th and 5th numbers in the ordered list. 9. **Order the numbers:** We know median is 7, so the 4th and 5th numbers average to 7. 10. **Arrange numbers in increasing order:** The numbers depend on $x$ and $y$. We test possible orderings consistent with median 7. 11. **Assuming $x < 5$ (since 5 is fixed), the numbers are:** $$x, x+1, 5, 2x+1, y+2, 7, 8, 3x+2$$ 12. **Median is average of 4th and 5th numbers:** $$\frac{\text{4th} + \text{5th}}{2} = 7$$ 13. **Try 4th = $2x+1$, 5th = $y+2$:** $$\frac{2x + 1 + y + 2}{2} = 7$$ $$2x + y + 3 = 14$$ $$2x + y = 11 \quad \quad (2)$$ 14. **Solve system of equations (1) and (2):** $$9x + y = 26$$ $$2x + y = 11$$ 15. **Subtract second from first:** $$(9x + y) - (2x + y) = 26 - 11$$ $$7x = 15$$ $$x = \frac{15}{7}$$ 16. **Find $y$ using (2):** $$2 \times \frac{15}{7} + y = 11$$ $$\frac{30}{7} + y = 11$$ $$y = 11 - \frac{30}{7} = \frac{77}{7} - \frac{30}{7} = \frac{47}{7}$$ 17. **Final answers:** $$x = \frac{15}{7} \approx 2.14$$ $$y = \frac{47}{7} \approx 6.71$$