To solve the problem step by step, we will follow the given conditions and calculate the values accordingly.
### Step 1: Set up the equations based on the mean
We are given the data set: 2, x, 7, 3, y, 9, 6. The mean of this data is 6.
The formula for the mean is:
\[
\text{Mean} = \frac{\text{Sum of data}}{\text{Number of data points}}
\]
Here, the number of data points is 7. Therefore, we can write:
\[
\frac{2 + x + 7 + 3 + y + 9 + 6}{7} = 6
\]
### Step 2: Simplify the equation
First, calculate the sum of the known numbers:
\[
2 + 7 + 3 + 9 + 6 = 27
\]
Now, substituting this into the equation gives:
\[
\frac{27 + x + y}{7} = 6
\]
### Step 3: Multiply both sides by 7
To eliminate the fraction, multiply both sides by 7:
\[
27 + x + y = 42
\]
### Step 4: Rearrange to find the first equation
Rearranging gives us:
\[
x + y = 42 - 27
\]
\[
x + y = 15 \quad \text{(Equation 1)}
\]
### Step 5: Set up the second condition
Now, when \( x \) is replaced by \( 3x + 1 \) and \( y \) is replaced by \( y + 3 \), the new mean is 8. The new data set is: 2, \( 3x + 1 \), 7, 3, \( y + 3 \), 9, 6.
Using the mean formula again:
\[
\frac{2 + (3x + 1) + 7 + 3 + (y + 3) + 9 + 6}{7} = 8
\]
### Step 6: Simplify the second equation
Calculating the sum of the known numbers gives:
\[
2 + 7 + 3 + 9 + 6 + 1 + 3 = 31
\]
So, substituting this into the equation gives:
\[
\frac{31 + 3x + y}{7} = 8
\]
### Step 7: Multiply both sides by 7
To eliminate the fraction, multiply both sides by 7:
\[
31 + 3x + y = 56
\]
### Step 8: Rearrange to find the second equation
Rearranging gives us:
\[
3x + y = 56 - 31
\]
\[
3x + y = 25 \quad \text{(Equation 2)}
\]
### Step 9: Solve the system of equations
Now we have two equations:
1. \( x + y = 15 \)
2. \( 3x + y = 25 \)
We can subtract Equation 1 from Equation 2:
\[
(3x + y) - (x + y) = 25 - 15
\]
This simplifies to:
\[
2x = 10
\]
### Step 10: Solve for \( x \)
Dividing both sides by 2 gives:
\[
x = 5
\]
### Final Answer
The value of \( x \) is \( 5 \).
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