find the value of a when the arithmetic mean of a + 7 and a + 3 is 3a + 9

To find the value of \( a \), let's first set up the equation based on the given information.

The arithmetic mean of \( a + 7 \) and \( a + 3 \) is calculated as follows:

\[
\text{Arithmetic Mean} = \frac{(a + 7) + (a + 3)}{2}
\]

Simplifying this:

\[
\text{Arithmetic Mean} = \frac{2a + 10}{2} = a + 5
\]

We are given that this mean is equal to \( 3a + 9 \):

\[
a + 5 = 3a + 9
\]

Now, let's solve for \( a \):

1. Subtract \( a \) from both sides:

\[
5 = 2a + 9
\]

2. Subtract 9 from both sides:

\[
-4 = 2a
\]

3. Divide both sides by 2:

\[
a = -2
\]

Thus, the value of \( a \) is \( \boxed{-2} \).

Would you like further details, or do you have any other questions?

Here are some related questions for further practice:

1. What is the arithmetic mean of \( 2a + 4 \) and \( 3a + 8 \)?
2. Solve for \( a \) if the arithmetic mean of \( 4a + 2 \) and \( 6a + 10 \) equals \( 5a + 12 \).
3. What is the geometric mean of \( a + 2 \) and \( a + 6 \)?
4. If the arithmetic mean of \( a \) and \( b \) is 12, what is the sum of \( a \) and \( b \)?
5. Find \( a \) if the arithmetic mean of \( a + 5 \) and \( a + 10 \) equals \( 2a + 8 \).
6. Determine the value of \( a \) when the harmonic mean of \( a + 2 \) and \( a + 4 \) is \( \frac{2a + 6}{3} \).
7. Solve for \( b \) if the arithmetic mean of \( b + 3 \) and \( b + 7 \) is \( 5b + 2 \).
8. Calculate the arithmetic mean of \( a^2 + 1 \) and \( 2a^2 - 3 \).

**Tip:** When solving equations involving means, always start by simplifying the mean expression before setting it equal to the given value.

Learn how to find the value of 'a' in the arithmetic mean equation (a + 7 + a + 3) / 2 = 3a + 9 with a detailed step-by-step solution. This problem covers arithmetic mean concepts and equations. Suitable for Grades 7-9.

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