Find the value of x if the arithmetic mean of 3 and 3x+5 is 8

To find the value of \( x \) given that the arithmetic mean of 3 and \( 3x + 5 \) is 8, we can follow these steps:

1. **Understanding the Arithmetic Mean:**
   The arithmetic mean of two numbers \( a \) and \( b \) is given by:
   \[
   \text{Mean} = \frac{a + b}{2}
   \]

2. **Setting Up the Equation:**
   Given the two numbers are 3 and \( 3x + 5 \), and their arithmetic mean is 8, we can set up the equation:
   \[
   \frac{3 + (3x + 5)}{2} = 8
   \]

3. **Simplifying the Equation:**
   Combine the terms in the numerator:
   \[
   \frac{3 + 3x + 5}{2} = 8
   \]
   \[
   \frac{3x + 8}{2} = 8
   \]

4. **Solving for \( x \):**
   Multiply both sides of the equation by 2 to eliminate the denominator:
   \[
   3x + 8 = 16
   \]
   Subtract 8 from both sides:
   \[
   3x = 8
   \]
   Divide both sides by 3:
   \[
   x = \frac{8}{3}
   \]

Therefore, the value of \( x \) is \( \frac{8}{3} \).

Would you like a detailed explanation or have any questions? Here are some related problems you might find interesting:

1. Find the arithmetic mean of 4 and \( 2x + 7 \) if the mean is 9.
2. Determine \( x \) if the arithmetic mean of 5 and \( 4x - 3 \) is 11.
3. Solve for \( x \) if the arithmetic mean of \( x \) and \( 2x + 10 \) is 15.
4. Find \( y \) if the arithmetic mean of 7 and \( 5y - 4 \) is 12.
5. Determine the value of \( x \) if the arithmetic mean of \( 6 \) and \( 2x + 9 \) is 10.
6. Solve for \( x \) if the arithmetic mean of \( 8 \) and \( 3x + 6 \) is 14.
7. Find \( x \) if the arithmetic mean of \( 10 \) and \( 4x + 2 \) is 13.
8. Determine \( x \) if the arithmetic mean of \( 2 \) and \( 3x + 1 \) is 7.

**Tip:** When solving equations involving arithmetic means, remember to clear the fraction by multiplying both sides of the equation by the denominator to simplify the solving process.

Learn how to find the value of x given the arithmetic mean of 3 and 3x + 5 is 8. This problem involves solving an arithmetic mean equation using basic algebraic principles. Suitable for students in Grades 7-9.

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