1. **Problem statement:** We are given a set of numbers: $x$, $(x+1)$, $5$, $(2x+1)$, $(y+2)$, $7$, $8$, and $(3x+2)$. The mean of these numbers is $6.5$, and the median is $7$. We need to find the values of $x$ and $y$. 2. **Formula for mean:** The mean of $n$ numbers $a_1, a_2, ..., a_n$ is given by $$\text{Mean} = \frac{a_1 + a_2 + \cdots + a_n}{n}$$ 3. **Step 1: Calculate the mean equation.** There are 8 numbers, so $$\frac{x + (x+1) + 5 + (2x+1) + (y+2) + 7 + 8 + (3x+2)}{8} = 6.5$$ 4. **Step 2: Simplify the numerator.** Combine like terms: $$x + (x+1) + 5 + (2x+1) + (y+2) + 7 + 8 + (3x+2) = (x + x + 2x + 3x) + (1 + 5 + 1 + 2 + 7 + 8 + 2) + y$$ $$= 9x + (1 + 5 + 1 + 2 + 7 + 8 + 2) + y = 9x + 26 + y$$ 5. **Step 3: Write the mean equation with simplified terms.** $$\frac{9x + y + 26}{8} = 6.5$$ 6. **Step 4: Multiply both sides by 8 to clear the denominator.** $$9x + y + 26 = 6.5 \times 8$$ $$9x + y + 26 = 52$$ 7. **Step 5: Isolate $y$.** $$y = 52 - 26 - 9x$$ $$y = 26 - 9x$$ 8. **Step 6: Use the median condition.** The median is the middle value when the numbers are ordered. Since there are 8 numbers, the median is the average of the 4th and 5th numbers after ordering. 9. **Step 7: Order the numbers.** The numbers are: $$x, x+1, 5, 2x+1, y+2, 7, 8, 3x+2$$ We want the 4th and 5th numbers to average to 7. 10. **Step 8: Find $x$ by testing order constraints.** Assuming $x$ is such that the order is: $$x < x+1 < 5 < 2x+1 < y+2 < 7 < 8 < 3x+2$$ The median is $$\frac{2x+1 + (y+2)}{2} = 7$$ 11. **Step 9: Write the median equation.** $$2x + 1 + y + 2 = 14$$ $$2x + y + 3 = 14$$ $$2x + y = 11$$ 12. **Step 10: Substitute $y$ from step 5 into the median equation.** $$2x + (26 - 9x) = 11$$ $$2x + 26 - 9x = 11$$ $$-7x + 26 = 11$$ 13. **Step 11: Solve for $x$.** $$-7x = 11 - 26$$ $$-7x = -15$$ $$x = \frac{-15}{-7} = \frac{15}{7}$$ 14. **Step 12: Find $y$ using $x = \frac{15}{7}$.** $$y = 26 - 9 \times \frac{15}{7} = 26 - \frac{135}{7} = \frac{182}{7} - \frac{135}{7} = \frac{47}{7}$$ 15. **Final answer:** $$x = \frac{15}{7} \approx 2.14, \quad y = \frac{47}{7} \approx 6.71$$