This problem requires a strong understanding of statistical measures, specifically the mean and variance. We will systematically use the given information to form a system of equations, solve for the unknown variables, and then calculate the required expression.

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1. Key Concepts and Formulas

For a set of $n$ observations $x_1, x_2, \ldots, x_n$:

• Mean ($\bar{x}$): The arithmetic average of all observations. It provides a linear relationship between the sum of observations and the mean. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

• Variance ($\sigma^2$): A measure of the spread of numbers from their mean. It is the average of the squared differences from the mean. This formula provides a quadratic relationship involving the unknown variables. $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$ An alternative form, $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$, is also valid and can be useful in some scenarios. For this problem, with a known integer mean, the first form is often more direct.

Problem Setup: We are given:

• 8 numbers: $x, y, 10, 12, 6, 12, 4, 8$.
• Number of observations, $n = 8$.
• Mean, $\bar{x} = 9$.
• Variance, $\sigma^2 = 9.25$.
• A condition: $x > y$.
• Our goal: Find the value of $3x - 2y$.

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2. Step-by-Step Solution

Step 1: Utilizing the Mean to Establish a Linear Relationship

• Why this step? We have two unknown variables, $x$ and $y$. To solve for them, we need at least two independent equations. The mean formula provides the simplest relationship, a linear equation involving $x$ and $y$.

First, let's sum the known observations: $10 + 12 + 6 + 12 + 4 + 8 = 52$ The total sum of all 8 observations, $\sum x_i$, is therefore: $\sum x_i = x + y + 52$ Now, substitute this sum, the given mean ($\bar{x}=9$), and the number of observations ($n=8$) into the mean formula: $\bar{x} = \frac{\sum x_i}{n}$ $9 = \frac{x+y+52}{8}$ Multiply both sides by 8 to clear the denominator: $9 \times 8 = x+y+52$ $72 = x+y+52$ Subtract 52 from both sides to find the sum of $x$ and $y$: $x+y = 72 - 52$ $\mathbf{x+y = 20 \quad \text{(Equation 1)}}$ This is our first equation relating $x$ and $y$.

Step 2: Employing the Variance to Form a Quadratic Relationship

• Why this step? We need a second independent equation to solve for $x$ and $y$. The variance formula provides this relationship. Since variance involves squared deviations from the mean, it will lead to a quadratic equation.

We will use the variance formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$. We know $\bar{x}=9$ and $\sigma^2 = 9.25$. Let's calculate the squared deviations $(x_i - \bar{x})^2$ for each of the 8 numbers:

Now, sum all the squared deviations: $\sum (x_i - \bar{x})^2 = (x-9)^2 + (y-9)^2 + 1 + 9 + 9 + 9 + 25 + 1$ $\sum (x_i - \bar{x})^2 = (x-9)^2 + (y-9)^2 + 54$ Substitute this sum, the given variance ($\sigma^2=9.25$), and $n=8$ into the variance formula: $9.25 = \frac{(x-9)^2 + (y-9)^2 + 54}{8}$ Multiply both sides by 8: $9.25 \times 8 = (x-9)^2 + (y-9)^2 + 54$ $74 = (x-9)^2 + (y-9)^2 + 54$ Subtract 54 from both sides: $74 - 54 = (x-9)^2 + (y-9)^2$ $20 = (x-9)^2 + (y-9)^2$ Expand the squared terms using the identity $(A-B)^2 = A^2 - 2AB + B^2$: $20 = (x^2 - 18x + 81) + (y^2 - 18y + 81)$ $20 = x^2 + y^2 - 18x - 18y + 162$ Factor out $-18$ from the linear terms: $20 = x^2 + y^2 - 18(x+y) + 162$ From Equation 1, we know $x+y = 20$. Substitute this value: $20 = x^2 + y^2 - 18(20) + 162$ $20 = x^2 + y^2 - 360 + 162$ $20 = x^2 + y^2 - 198$ Add 198 to both sides to isolate $x^2+y^2$: $x^2 + y^2 = 20 + 198$ $\mathbf{x^2 + y^2 = 218 \quad \text{(Equation 2)}}$

**Step 3: Solving the System of Equations for Individual Values of $x$ and $y$}

• Why this step? We now have a system of two equations with two variables:
  1. $x+y = 20$ (Linear equation)
  2. $x^2+y^2 = 218$ (Quadratic equation) We can solve this system to find the specific values of $x$ and $y$.

From Equation 1, we can express $y$ in terms of $x$: $y = 20-x$ Substitute this expression for $y$ into Equation 2: $x^2 + (20-x)^2 = 218$ Expand $(20-x)^2$: $x^2 + (400 - 40x + x^2) = 218$ Combine like terms to form a standard quadratic equation: $2x^2 - 40x + 400 = 218$ $2x^2 - 40x + 182 = 0$ Divide the entire equation by 2 to simplify: $x^2 - 20x + 91 = 0$ Now, solve this quadratic equation for $x$. We can factor it. We need two numbers that multiply to 91 and add up to -20. These numbers are -7 and -13. $(x-7)(x-13) = 0$ This gives two possible values for $x$:

• $x-7 = 0 \Rightarrow x=7$
• $x-13 = 0 \Rightarrow x=13$ For each value of $x$, find the corresponding value of $y$ using $y = 20-x$:
• If $x=7$, then $y = 20-7 = 13$.
• If $x=13$, then $y = 20-13 = 7$. So, we have two possible pairs for $(x,y)$: $(7, 13)$ and $(13, 7)$.

Step 4: Applying the Given Condition and Final Calculation

• Why this step? The problem provides a crucial condition, $x > y$. This condition allows us to uniquely determine which of the two possible $(x,y)$ pairs is the correct one.

Let's check our two pairs against the condition $x > y$:

• Case 1: $(x,y) = (7, 13)$ Here, $x=7$ and $y=13$. Since $7 \ngtr 13$, this pair does not satisfy the condition.
• Case 2: $(x,y) = (13, 7)$ Here, $x=13$ and $y=7$. Since $13 > 7$, this pair satisfies the condition. Therefore, the correct values are $x=13$ and $y=7$.

Finally, we calculate the required expression $3x - 2y$: $3x - 2y = 3(13) - 2(7)$ $3x - 2y = 39 - 14$ $3x - 2y = 25$

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3. Common Mistakes & Tips

• Arithmetic Errors: Double-check sums and multiplications, especially with negative numbers. A small mistake in summing known numbers or multiplying by $n$ can propagate through the entire solution.
• Algebraic Expansion: Be careful when expanding squared terms like $(x-9)^2$. Remember the middle term: $(A-B)^2 = A^2 - 2AB + B^2$.
• Sign Errors: Pay close attention to negative signs, particularly when combining terms or substituting values.
• Using the Right Variance Formula: While both variance formulas are equivalent, choosing $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ can sometimes simplify calculations if $\bar{x}$ is an integer, as the deviations are often simpler to work with.
• Applying Conditions: Always use all given conditions (like $x>y$) to narrow down possibilities and arrive at a unique solution.
• Verification: After finding $x$ and $y$, it's a good practice to quickly plug them back into the original mean and variance equations to ensure consistency.

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4. Summary

We began by utilizing the mean formula to establish a linear equation for $x$ and $y$. Next, we applied the variance formula, carefully calculating the sum of squared deviations from the mean, to derive a quadratic equation involving $x$ and $y$. This system of a linear and a quadratic equation was then solved using substitution, yielding two possible pairs for $(x,y)$. Finally, the given condition $x > y$ allowed us to select the unique correct pair of values for $x$ and $y$, which were then used to calculate the expression $3x - 2y$.

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The final answer is $\boxed{25}$.