Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2

To find the value of \( x \) when the arithmetic mean of \( x + 2 \) and \( 4x + 5 \) is \( 3x + 2 \), follow these steps:

### Step 1: Write the formula for the arithmetic mean
The arithmetic mean of two numbers \( a \) and \( b \) is given by:
\[
\text{Arithmetic Mean} = \frac{a + b}{2}
\]
In this case, the two numbers are \( x + 2 \) and \( 4x + 5 \).

### Step 2: Set up the equation
We know the arithmetic mean of \( x + 2 \) and \( 4x + 5 \) is equal to \( 3x + 2 \). Therefore, we write:
\[
\frac{(x + 2) + (4x + 5)}{2} = 3x + 2
\]

### Step 3: Simplify the equation
First, simplify the expression inside the parentheses:
\[
\frac{x + 2 + 4x + 5}{2} = 3x + 2
\]
\[
\frac{5x + 7}{2} = 3x + 2
\]

### Step 4: Eliminate the fraction
Multiply both sides of the equation by 2 to eliminate the fraction:
\[
5x + 7 = 2(3x + 2)
\]

### Step 5: Expand and simplify
Distribute the 2 on the right side:
\[
5x + 7 = 6x + 4
\]

Next, move all terms involving \( x \) to one side and constants to the other side:
\[
5x - 6x = 4 - 7
\]
\[
-x = -3
\]

### Step 6: Solve for \( x \)
To solve for \( x \), divide both sides by \(-1\):
\[
x = 3
\]

### Final Answer:
The value of \( x \) is \( 3 \).

---

Would you like to explore this further or have any questions? Here are some related questions:

1. What are some other methods to solve equations involving arithmetic means?
2. How do you check the correctness of a solution in algebraic equations?
3. What are the applications of arithmetic mean in real-world problems?
4. How does the arithmetic mean compare to other types of means (e.g., geometric mean)?
5. Can an equation involving an arithmetic mean have more than one solution?

**Tip:** Always verify your final answer by substituting it back into the original equation to ensure it satisfies all parts of the problem.

Learn how to find the value of x when the arithmetic mean of x+2 and 4x+5 equals 3x+2. This problem involves arithmetic mean formula application and algebraic equation solving. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation