High School

For the data set [tex]2, 9, x+6, 2x+3, 5, 10, 5[/tex], if the mean is 7, then the value of [tex]x[/tex] is:

(a) 9
(b) 6
(c) 5
(d) 3

Answer :

To find the value of [tex]\( x \)[/tex], given the data set [tex]\( 2, 9, x+6, 2x+3, 5, 10, 5 \)[/tex] and the mean of the data is 7, we can follow these steps:

1. List the Data Values: We have the numbers: [tex]\( 2, 9, x+6, 2x+3, 5, 10, 5 \)[/tex].

2. Calculate the Total Number of Data Points: There are 7 data points total.

3. Set Up the Mean Equation:
[tex]\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\][/tex]
We are given that the mean is 7, so:
[tex]\[
7 = \frac{2 + 9 + (x+6) + (2x+3) + 5 + 10 + 5}{7}
\][/tex]

4. Simplify the Sum of the Data Set:
- Combine like terms:
- Constants: [tex]\(2 + 9 + 6 + 3 + 5 + 10 + 5 = 40\)[/tex]
- Variables: [tex]\(x + 2x = 3x\)[/tex]
- So the sum is [tex]\(40 + 3x\)[/tex].

5. Set Up the Equation:
[tex]\[
7 = \frac{40 + 3x}{7}
\][/tex]

6. Solve for [tex]\( x \)[/tex]:
- Multiply both sides by 7 to eliminate the fraction:
[tex]\[
7 \times 7 = 40 + 3x
\][/tex]
[tex]\[
49 = 40 + 3x
\][/tex]

- Subtract 40 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
49 - 40 = 3x
\][/tex]
[tex]\[
9 = 3x
\][/tex]

- Divide both sides by 3:
[tex]\[
x = \frac{9}{3}
\][/tex]
[tex]\[
x = 3
\][/tex]

Therefore, the value of [tex]\( x \)[/tex] is 3. Thus, the correct answer is [tex]\( \boxed{3} \)[/tex].

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